QUOTE (Cusa+Feb 26 2009, 09:13 PM)
Repeating 9's ever turn to One.
Mitch Raemsch
Then it's a good thing we are not talking about that isn't it.
0.99... is complete.
There is no repeating involved.
If you use any words ending in "ing" you are not talking about 0.99...
And there are no variables involved either.
Including the number of 9's involved, because there is no number of 9's involved.
Infinity is not a number.
Nothing is growing with 0.99... It is already complete or else it is not 0.99...
Mitch Raemsch
Then it's a good thing we are not talking about that isn't it.
0.99... is complete.
There is no repeating involved.
If you use any words ending in "ing" you are not talking about 0.99...
And there are no variables involved either.
Including the number of 9's involved, because there is no number of 9's involved.
Infinity is not a number.
Nothing is growing with 0.99... It is already complete or else it is not 0.99...
Hi all.
yes Bukh, It is seriously dependent on what you choose to measure. But I go back to the argument that "does energy DO anything differently just because we can perceive it".
Time is a construct....Would I still be here without time?...Of course I must. HOW would I be?...certainly in a very different form to how I perceive myself using a time construct.
Does space concern itself about time?...No. why should it? Time is only a local component of something measuring motion.When you stop measuring motion with time, the energy would still have to be as it was....
So nothing we do or perceive can affect what is already present when we are not using time. The result is set and the interference we think we have achieved was destined to take place anyway.
I you think of EVERY position in space HAS already been occupied (via another methodology which is not time based), But we are replaying the order in which they were occupied, you can see that we are in constant motion in a sequence that balances the universal seconds to keep space as we perceive it.
Cheers
Iseason
yes Bukh, It is seriously dependent on what you choose to measure. But I go back to the argument that "does energy DO anything differently just because we can perceive it".
Time is a construct....Would I still be here without time?...Of course I must. HOW would I be?...certainly in a very different form to how I perceive myself using a time construct.
Does space concern itself about time?...No. why should it? Time is only a local component of something measuring motion.When you stop measuring motion with time, the energy would still have to be as it was....
So nothing we do or perceive can affect what is already present when we are not using time. The result is set and the interference we think we have achieved was destined to take place anyway.
I you think of EVERY position in space HAS already been occupied (via another methodology which is not time based), But we are replaying the order in which they were occupied, you can see that we are in constant motion in a sequence that balances the universal seconds to keep space as we perceive it.
Cheers
Iseason
Iseason
- "does energy DO anything differently just because we can perceive it" ?
Once "object of sameness" is being used as perceiving tool - then time and energy and all other fundamentals inevitably follows. Once having said A one have to say B and C and --
Energy is defined as how we perceive changes, and we (I) have already decided to perceive physical universe via "how objects of sameness change their configurations" , and time is the order-line by which the changes takes place, and energy likewise is being defined out from such changes.
Now I am talking about how I - just I and no-one else see it -
Of course there is no true perceiving tool - it is highly subjective and connected to a mind.
Because observers (human) are very much the same construct - it is highly likely that there will be a common agreement on perceiving tool - and the logic consequences, respectively.
QUOTE: "When you stop measuring motion with time, the energy would still have to be as it was...."
No - the energy is being defined out from time and motion, so when one stop to measure motion with time - then energy would at the same "time" stop to be in existence. And now we are talking about energy in the same sense that we are talking about time and motion - as something that has a meaning to us.
What then will the Universe - the Everything - be about, if not being perceived ?
To this I would like just to say: " ? "
- "does energy DO anything differently just because we can perceive it" ?
Once "object of sameness" is being used as perceiving tool - then time and energy and all other fundamentals inevitably follows. Once having said A one have to say B and C and --
Energy is defined as how we perceive changes, and we (I) have already decided to perceive physical universe via "how objects of sameness change their configurations" , and time is the order-line by which the changes takes place, and energy likewise is being defined out from such changes.
Now I am talking about how I - just I and no-one else see it -
Of course there is no true perceiving tool - it is highly subjective and connected to a mind.
Because observers (human) are very much the same construct - it is highly likely that there will be a common agreement on perceiving tool - and the logic consequences, respectively.
QUOTE: "When you stop measuring motion with time, the energy would still have to be as it was...."
No - the energy is being defined out from time and motion, so when one stop to measure motion with time - then energy would at the same "time" stop to be in existence. And now we are talking about energy in the same sense that we are talking about time and motion - as something that has a meaning to us.
What then will the Universe - the Everything - be about, if not being perceived ?
To this I would like just to say: " ? "
QUOTE (buttershug+Feb 26 2009, 12:06 PM)
The 9's do not "go" at all.
And stop saying forever. That is a time based concept that has no place with understanding 0.99...
They STAY at an infinite amount of 9's.
Stop "adding" 9's after the last 9 that you see.
Start understanding that they are all represented as already being in place.
and 0.99.. has never represented anything phsysical.
So the nature of the universe has nothing to do with 0.99...
Can a repeating decimal be used as a multiplicand for purposes of a proof?
And stop saying forever. That is a time based concept that has no place with understanding 0.99...
They STAY at an infinite amount of 9's.
Stop "adding" 9's after the last 9 that you see.
Start understanding that they are all represented as already being in place.
and 0.99.. has never represented anything phsysical.
So the nature of the universe has nothing to do with 0.99...
Can a repeating decimal be used as a multiplicand for purposes of a proof?
INFINITE 9'S NEVER RAISE IT TO 1.
QUOTE (Cusa+Feb 27 2009, 11:01 PM)
INFINITE 9'S NEVER RAISE IT TO 1.
A single 9 will never "RAISE" to 1, much less "INFINITE 9'S." Were you attempting to say: .9r? And that wasn't the question.
A single 9 will never "RAISE" to 1, much less "INFINITE 9'S." Were you attempting to say: .9r? And that wasn't the question.
N is the set of natural numbers.
U is defined as the sum of elements 9/10^n for all n.
In symbols: U = SUM(9/10^n) for n=1 to inf
The lim of U as n approaches inf = S
Within the definition of limit is this requirement:
S exists if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have |S−U| < ε.
The value ε is never 0.
U never equals S, which equals 1.
The subject of the 4th statement is lim, not U.
S is a boundary value and is not contained in U.
If it was, the definition of limit would be pointless, and a simpler expression would suffice, U = S.
A geometric example.
The plot of y=|sqrt(x*x+1)| has u=x as an asymptote. As in the previous example, u is a boundary or limit, which y approaches but never touches.
When transforming a fraction x from base 9 to base 10,
(a/9)=.aaa... because there is a remainder.
(9/9)=1.0 because there is no remainder, i.e. not .999...
Therefore .9R does not represent a rational number.
If irrational numbers are defined as all possible infinite sequences of digits (base 10) to the right of a decimal point, then it can represent the largest irrational number less than 1.
To those who argue it's not a process, you're not considering the foundations of the 'language' of mathematics. The symbols represent statements consisting of objects and processes, as in lines 2 and 3. (if only good math fonts)
All concepts are definitions.
U is defined as the sum of elements 9/10^n for all n.
In symbols: U = SUM(9/10^n) for n=1 to inf
The lim of U as n approaches inf = S
Within the definition of limit is this requirement:
S exists if and only if for every real number ε > 0, there exists a natural number N such that for every n > N we have |S−U| < ε.
The value ε is never 0.
U never equals S, which equals 1.
The subject of the 4th statement is lim, not U.
S is a boundary value and is not contained in U.
If it was, the definition of limit would be pointless, and a simpler expression would suffice, U = S.
A geometric example.
The plot of y=|sqrt(x*x+1)| has u=x as an asymptote. As in the previous example, u is a boundary or limit, which y approaches but never touches.
When transforming a fraction x from base 9 to base 10,
(a/9)=.aaa... because there is a remainder.
(9/9)=1.0 because there is no remainder, i.e. not .999...
Therefore .9R does not represent a rational number.
If irrational numbers are defined as all possible infinite sequences of digits (base 10) to the right of a decimal point, then it can represent the largest irrational number less than 1.
To those who argue it's not a process, you're not considering the foundations of the 'language' of mathematics. The symbols represent statements consisting of objects and processes, as in lines 2 and 3. (if only good math fonts)
All concepts are definitions.
QUOTE (Derek1148+Feb 27 2009, 11:21 PM)
A single 9 will never "RAISE" to 1, much less "INFINITE 9'S." Were you attempting to say: .9r? And that wasn't the question.
No matter how many 9's it never is raised to 1.
If you agree then why are you arguing?
Mitch Raemsch
No matter how many 9's it never is raised to 1.
If you agree then why are you arguing?
Mitch Raemsch
QUOTE (phyti+Feb 28 2009, 02:49 AM)
When transforming a fraction x from base 9 to base 10,
(a/9)=.aaa... because there is a remainder.
(9/9)=1.0 because there is no remainder, i.e. not .999...
Therefore .9R does not represent a rational number.
If irrational numbers are defined as all possible infinite sequences of digits (base 10) to the right of a decimal point, then it can represent the largest irrational number less than 1.
To those who argue it's not a process, you're not considering the foundations of the 'language' of mathematics. The symbols represent statements consisting of objects and processes, as in lines 2 and 3. (if only good math fonts)
All concepts are definitions.
Therefore, would you say .3r cannot be employed as a multiplicand; for example; .3r x 3 = .9r? And any proof employing a repeating decimal as a multiplicand is flawed?
(a/9)=.aaa... because there is a remainder.
(9/9)=1.0 because there is no remainder, i.e. not .999...
Therefore .9R does not represent a rational number.
If irrational numbers are defined as all possible infinite sequences of digits (base 10) to the right of a decimal point, then it can represent the largest irrational number less than 1.
To those who argue it's not a process, you're not considering the foundations of the 'language' of mathematics. The symbols represent statements consisting of objects and processes, as in lines 2 and 3. (if only good math fonts)
All concepts are definitions.
Therefore, would you say .3r cannot be employed as a multiplicand; for example; .3r x 3 = .9r? And any proof employing a repeating decimal as a multiplicand is flawed?
QUOTE (phyti+Feb 28 2009, 02:49 AM)
If irrational numbers are defined as all possible infinite sequences of digits (base 10) to the right of a decimal point, then it can represent the largest irrational number less than 1.
That's a pretty big IF there... too bad that's not the definition of irrational numbers.
That's a pretty big IF there... too bad that's not the definition of irrational numbers.
QUOTE (Argyll+Feb 28 2009, 04:38 AM)
That's a pretty big IF there... too bad that's not the definition of irrational numbers.
Come on, that is just a minor technicality.
Come on, that is just a minor technicality.
We are still waiting for the realization to sink in that there is no largest rational number less than 1. We know that there is no such number, because to name it is to prove that it is not the largest rational number less than 1.
Proof: Call the largest rational number less than 1 to be A. So A < 1. B = 1 + A is also a rational number. C = B/2 is also a rational number. But 2A < B < 2 so A < C < 1. Therefore A cannot be the largest rational number less than 1. From the contradiction we learn that no number is the largest rational number less than 1.
So the fact that they aren't defining irrational numbers is just a tiny flaw in their vast incomprehension of mathematics.
Proof: Call the largest rational number less than 1 to be A. So A < 1. B = 1 + A is also a rational number. C = B/2 is also a rational number. But 2A < B < 2 so A < C < 1. Therefore A cannot be the largest rational number less than 1. From the contradiction we learn that no number is the largest rational number less than 1.
So the fact that they aren't defining irrational numbers is just a tiny flaw in their vast incomprehension of mathematics.
rpenner
"We are still waiting for the realization to sink in that there is no largest rational number less than 1. We know that there is no such number, because to name it is to prove that it is not the largest rational number less than 1."
Are you saying that .9.... is not equal to 1 ?
"We are still waiting for the realization to sink in that there is no largest rational number less than 1. We know that there is no such number, because to name it is to prove that it is not the largest rational number less than 1."
Are you saying that .9.... is not equal to 1 ?
I'm saying, among other things, that 0.9... can't be the largest number less than one.
Naturally there is no contradiction if we say 0.9... is equal to 1.
In a similar way, there is no actual problem with saying that 0.9... isn't even a number. But if you say that then you can't say word one about the real numbers or what 0.9... is.
You can't have your cake and eat it too.
Naturally there is no contradiction if we say 0.9... is equal to 1.
In a similar way, there is no actual problem with saying that 0.9... isn't even a number. But if you say that then you can't say word one about the real numbers or what 0.9... is.
You can't have your cake and eat it too.
QUOTE (rpenner+Feb 28 2009, 10:08 AM)
In a similar way, there is no actual problem with saying that 0.9... isn't even a number.
That is true and it shouldn't be treated as a number. But its value is 1.
That is true and it shouldn't be treated as a number. But its value is 1.
penner
"You can't have your cake and eat it too."
OK then we say that .9... is not a number but it is equal to the number 1. !
Perhaps this is a slightly different meaning to the concept of "equal to"
To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking.
"You can't have your cake and eat it too."
OK then we say that .9... is not a number but it is equal to the number 1. !
Perhaps this is a slightly different meaning to the concept of "equal to"
To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking.
QUOTE (rpenner+Feb 28 2009, 07:24 AM)
We are still waiting for the realization to sink in that there is no largest rational number less than 1. We know that there is no such number, because to name it is to prove that it is not the largest rational number less than 1.
Proof: Call the largest rational number less than 1 to be A. So A < 1. B = 1 + A is also a rational number. C = B/2 is also a rational number. But 2A < B < 2 so A < C < 1. Therefore A cannot be the largest rational number less than 1. From the contradiction we learn that no number is the largest rational number less than 1.
So the fact that they aren't defining irrational numbers is just a tiny flaw in their vast incomprehension of mathematics.
Notice that you actually proved that C could not be constructed because A was defined to be the largest rational number less than 1.
For example, if we have a set of numbers, {0,1/2,1}, then 1/2 is the largest rational number less than 1 in this set.
If you construct 3/4, this is also a rational number and would have to be in the set as well, but then 3/4 would be the largest rational number less than 1 instead of 1/2, so you simply proved that C instead of A was the largest rational number and that your definition of A was incorrect.
Notice that you could instead define C to be the largest natural number and then derive A=2C-1, (1/2=2*3/4-1) in which case the statement contains no paradox.
Obviously in a set of rational numbers that has at least one number less than 1, there is a largest ration number less than one, correct?
Being unable to determine (specifically) what that number is for all such sets does not mean this number does not exist.
Now I admit that I'm being somewhat hypocritical in that I've said that pi is actually a set of possible values, or that the representation 1,2,3,...,n actually describes a set of sets of natural numbers etc. and the same would be true for this "largest real number less than 1". Could it be equal to -5/6? Yes, but it truly depends on what set of real numbers are used because variables and unknowns aren't numbers. A good question is over the extent to which the term "rational number" in a statement is actually referring to a rational number.
If we had rational numbers n/m, then for a specific m, the largest number n/m<1 would be (m-1)/m and of course .999... satisifies this form.
Notice that a rational number line is actually identical to an integer number line with an additional scaling term for the common divisor.
If we place two rational numbers on a number line, p/q and n/m, they're effectively translated into integers on a number line using a common denominator q*m, so we place p*m and q*n as integers on a number line using units of 1/(q*m) thenn obviously the rational number immediately less than 1 or adjacent to 1 on this scale would be q*m-1. If we included rational numbers with relatively prime denominators to 10^j, then a rational number of the form ((10^-k)-1)/(10^-k) would not be locateable as an integer adjacent to 1 on this scale.
It's hard to think of a "number" that could appear to converge to 1 within any selected finite precision, but could not be proven to do so for all finite values, but as an example, if we took iterations of the logistic function x(0)=1/5 and x(t+1)=4*x(t)*(1-x(t)) and computed the maximum of the set of values for x(t)<=1/3, this would obviously converge toward 1/3 and it could probably be proven that for any selected rational number less than 1/3 there exists a value in the set larger than it, but I don't think this could be generalized into a proof that this is true for all rational number less than 1/3 because there are always rational numbers closer to 1/3 than we can select, though to be consistant, it would appear I'd have to agree with an argument saying that if we can't select or construct them, they aren't actually numbers but instead sets of possible values, or variables.
Proof: Call the largest rational number less than 1 to be A. So A < 1. B = 1 + A is also a rational number. C = B/2 is also a rational number. But 2A < B < 2 so A < C < 1. Therefore A cannot be the largest rational number less than 1. From the contradiction we learn that no number is the largest rational number less than 1.
So the fact that they aren't defining irrational numbers is just a tiny flaw in their vast incomprehension of mathematics.
Notice that you actually proved that C could not be constructed because A was defined to be the largest rational number less than 1.
For example, if we have a set of numbers, {0,1/2,1}, then 1/2 is the largest rational number less than 1 in this set.
If you construct 3/4, this is also a rational number and would have to be in the set as well, but then 3/4 would be the largest rational number less than 1 instead of 1/2, so you simply proved that C instead of A was the largest rational number and that your definition of A was incorrect.
Notice that you could instead define C to be the largest natural number and then derive A=2C-1, (1/2=2*3/4-1) in which case the statement contains no paradox.
Obviously in a set of rational numbers that has at least one number less than 1, there is a largest ration number less than one, correct?
Being unable to determine (specifically) what that number is for all such sets does not mean this number does not exist.
Now I admit that I'm being somewhat hypocritical in that I've said that pi is actually a set of possible values, or that the representation 1,2,3,...,n actually describes a set of sets of natural numbers etc. and the same would be true for this "largest real number less than 1". Could it be equal to -5/6? Yes, but it truly depends on what set of real numbers are used because variables and unknowns aren't numbers. A good question is over the extent to which the term "rational number" in a statement is actually referring to a rational number.
If we had rational numbers n/m, then for a specific m, the largest number n/m<1 would be (m-1)/m and of course .999... satisifies this form.
Notice that a rational number line is actually identical to an integer number line with an additional scaling term for the common divisor.
If we place two rational numbers on a number line, p/q and n/m, they're effectively translated into integers on a number line using a common denominator q*m, so we place p*m and q*n as integers on a number line using units of 1/(q*m) thenn obviously the rational number immediately less than 1 or adjacent to 1 on this scale would be q*m-1. If we included rational numbers with relatively prime denominators to 10^j, then a rational number of the form ((10^-k)-1)/(10^-k) would not be locateable as an integer adjacent to 1 on this scale.
It's hard to think of a "number" that could appear to converge to 1 within any selected finite precision, but could not be proven to do so for all finite values, but as an example, if we took iterations of the logistic function x(0)=1/5 and x(t+1)=4*x(t)*(1-x(t)) and computed the maximum of the set of values for x(t)<=1/3, this would obviously converge toward 1/3 and it could probably be proven that for any selected rational number less than 1/3 there exists a value in the set larger than it, but I don't think this could be generalized into a proof that this is true for all rational number less than 1/3 because there are always rational numbers closer to 1/3 than we can select, though to be consistant, it would appear I'd have to agree with an argument saying that if we can't select or construct them, they aren't actually numbers but instead sets of possible values, or variables.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
Notice that you actually proved that C could not be constructed because A was defined to be the largest rational number less than 1.
A is a rational number by assumption. Therefore it must obey the actual rules of rational numbers.
B is a legal construction fundamental to the rational numbers -- indeed all numbers.
C is a legal construction fundamental to the rational numbers.
Please demonstrate the error in construction or admit error.
B is a legal construction fundamental to the rational numbers -- indeed all numbers.
C is a legal construction fundamental to the rational numbers.
Please demonstrate the error in construction or admit error.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
For example, if we have a set of numbers, {0,1/2,1}, then 1/2 is the largest rational number less than 1 in this set.
If you construct 3/4, this is also a rational number and would have to be in the set as well, but then 3/4 would be the largest rational number less than 1 instead of 1/2, so you simply proved that C instead of A was the largest rational number and that your definition of A was incorrect.
If you construct 3/4, this is also a rational number and would have to be in the set as well, but then 3/4 would be the largest rational number less than 1 instead of 1/2, so you simply proved that C instead of A was the largest rational number and that your definition of A was incorrect.
And so on, and so on, and so on. Since every A leads to a C which is also a rational number, which is larger than A and less than 1, then there is no such A that doesn't lead to a C.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
Notice that you could instead define C to be the largest natural number and then derive A=2C-1, (1/2=2*3/4-1) in which case the statement contains no paradox.
It's not a paradox -- it's a contradiction. And your going backwards does nothing to address that whenever C is a rational number less than 1, D = (C+1)/2 is a rational number larger than C and less than 1.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
Obviously in a set of rational numbers that has at least one number less than 1, there is a largest ration number less than one, correct?
I'm not talking about sets you damn inconsistent finitist! I'm talking about the rational numbers which your axioms are not sufficiently strong to represent as a set. But mine are and always have been.. But in the class of rational numbers less than 1, there is no largest member. In the class of integers there is no largest member.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
Being unable to determine (specifically) what that number is for all such sets does not mean this number does not exist.
Correct, but I have shown using the definitions of rational number arithmetic that there is a larger number. So if you think different, you must not be using rational numbers either. Even the ancient Egyptians had rational numbers.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
Now I admit that I'm being somewhat hypocritical in that I've said that pi is actually a set of possible values, or that the representation 1,2,3,...,n actually describes a set of sets of natural numbers etc. and the same would be true for this "largest real number less than 1". Could it be equal to -5/6? Yes, but it truly depends on what set of real numbers are used because variables and unknowns aren't numbers. A good question is over the extent to which the term "rational number" in a statement is actually referring to a rational number.
Not a statement of math.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
If we had rational numbers n/m, then for a specific m, the largest number n/m<1 would be (m-1)/m and of course .999... satisifies this form.
For what natural number m? Not a statement of math.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
Notice that a rational number line is actually identical to an integer number line with an additional scaling term for the common divisor.
Not a statement of math.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
If we place two rational numbers on a number line, p/q and n/m, they're effectively translated into integers on a number line using a common denominator q*m, so we place p*m and q*n as integers on a number line using units of 1/(q*m) thenn obviously the rational number immediately less than 1 or adjacent to 1 on this scale would be q*m-1.
Misuse of word "translated" -- You are talking about the image after a transform, but fail to consider how this affects the other rational numbers which you have not considered.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
If we included rational numbers with relatively prime denominators to 10^j, then a rational number of the form ((10^-k)-1)/(10^-k) would not be locateable as an integer adjacent to 1 on this scale.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
It's hard to think of a "number" that could appear to converge
Because numbers don't move and don't converge.
QUOTE (SteveA2+Feb 28 2009, 01:41 PM)
to 1 within any selected finite precision, but could not be proven to do so for all finite values, but as an example, if we took iterations of the logistic function x(0)=1/5 and x(t+1)=4*x(t)*(1-x(t)) and computed the maximum of the set of values for x(t)<=1/3, this would obviously converge toward 1/3 and it could probably be proven that for any selected rational number less than 1/3 there exists a value in the set larger than it, but I don't think this could be generalized into a proof that this is true for all rational number less than 1/3 because there are always rational numbers closer to 1/3 than we can select, though to be consistant, it would appear I'd have to agree with an argument saying that if we can't select or construct them, they aren't actually numbers but instead sets of possible values, or variables.
That's why your position is finitist. That's fine, except you don't recognize that it puts limits on your mathematics that the rest of us don't have to subscribe to. You also haven't worked out the consequences of this yet and have rambled on for thousands of posts without saying anything both new and correct. And among your mistakes, you never start from your assumptions to build theorems and you never spell out your definitions at the beginning, and you misuse words which even finitists use in the standard mathematical sense.
That's why your position is finitist. That's fine, except you don't recognize that it puts limits on your mathematics that the rest of us don't have to subscribe to. You also haven't worked out the consequences of this yet and have rambled on for thousands of posts without saying anything both new and correct. And among your mistakes, you never start from your assumptions to build theorems and you never spell out your definitions at the beginning, and you misuse words which even finitists use in the standard mathematical sense.
QUOTE (rpenner+)
A is a rational number by assumption. Therefore it must obey the actual rules of rational numbers.
B is a legal construction fundamental to the rational numbers -- indeed all numbers.
C is a legal construction fundamental to the rational numbers.
Please demonstrate the error in construction or admit error.
If we have a list of natural numbers 1 or larger and A is the largest natural number and B is another number in this list, then you cannot construct C=A+B without changing the definitions so that C is larger than A.
Violation: You assume a finite list, either changing the definition of natural numbers or assuming an infinite list of all the natural numbers can be constructed in violation of finitist logic. The natural numbers have the property that there is no largest member. This is the Archimedean property of natural numbers which in Western mathematics dates back to at least the 4th century B.C. (Before birth of Archimedes.) http://en.wikipedia.org/wiki/Axiom_of_Archimedes
Similarly, if you instead create a paradox and say that A is the largest (positive) number of some list and then proceed to add B=A+1 to the list, then B is the largest number, if it's added to the set. It's not a paradox. It's a proof by contradiction. The assumption that there is such a largest element in the (infinite) list of natural numbers is the contradiction. The list (technically a class in your finitist view) cannot grow as it is already the complete class of all natural numbers.
Alternately, if you want, you can show that there exist two sets of numbers in which A is the largest less than 1 in one set and C is in the other (assuming B did not fullfill this if -1<A<0). You've switched without segue back from natural numbers to rational numbers. But you are not doing mathematics, you are futzing with the definition of "the class of all rational numbers less than one". That's not a statement in set theory -- it's a definition in mathematical logic.
So it was just the way you worded the problem that made it appear invalid. Non sequitur -- and therefore incorrect.
B is a legal construction fundamental to the rational numbers -- indeed all numbers.
C is a legal construction fundamental to the rational numbers.
Please demonstrate the error in construction or admit error.
If we have a list of natural numbers 1 or larger and A is the largest natural number and B is another number in this list, then you cannot construct C=A+B without changing the definitions so that C is larger than A.
Violation: You assume a finite list, either changing the definition of natural numbers or assuming an infinite list of all the natural numbers can be constructed in violation of finitist logic. The natural numbers have the property that there is no largest member. This is the Archimedean property of natural numbers which in Western mathematics dates back to at least the 4th century B.C. (Before birth of Archimedes.) http://en.wikipedia.org/wiki/Axiom_of_Archimedes
Similarly, if you instead create a paradox and say that A is the largest (positive) number of some list and then proceed to add B=A+1 to the list, then B is the largest number, if it's added to the set. It's not a paradox. It's a proof by contradiction. The assumption that there is such a largest element in the (infinite) list of natural numbers is the contradiction. The list (technically a class in your finitist view) cannot grow as it is already the complete class of all natural numbers.
Alternately, if you want, you can show that there exist two sets of numbers in which A is the largest less than 1 in one set and C is in the other (assuming B did not fullfill this if -1<A<0). You've switched without segue back from natural numbers to rational numbers. But you are not doing mathematics, you are futzing with the definition of "the class of all rational numbers less than one". That's not a statement in set theory -- it's a definition in mathematical logic.
So it was just the way you worded the problem that made it appear invalid. Non sequitur -- and therefore incorrect.
QUOTE (rpenner+)
And so on, and so on, and so on. Since every A leads to a C which is also a rational number, which is larger than A and less than 1, then there is no such A that doesn't lead to a C.
Then you'd need to declare C to be the largest natural number in the construction of this set than A. Point addressed above. It's not a set (in your axioms) and not "constructed" -- it's a logical sub-class of the class of natural numbers.
If the set is a dynamic construction, No set is a "dynamical construction" then we should refer to it as a time dependent construction No set is a time dependent construction. and if wasn't a time dependent construction, then it would not be a set of rational numbers but instead would contain irrational numbers. Non sequitur -- and therefore incorrect.
If you wanted to then create another sequence It's not a "sequence" it is proof by contradiction. via., for example, E=(D+2)/3 and have these exist within the set, then a deterministic application of the C or E processes in constructing a number would be necessary (we could simply need to determine how many operations of a(t+1)=(a(t)+1)/2 or b(t+1)=(b(t)+2)/3 have occured to derive a number over time.
If you do not use such deterministic constructions for your numbers, then they are not rational numbers as they're dependent upon an unspecified set of operations over time to construct.
Notice that rational numbers must terminate, hence you can't define a rational number to be derived from an infinite sequence of unspecified (orthogonal) operations. Increasing tendency towards word salad.
Then you'd need to declare C to be the largest natural number in the construction of this set than A. Point addressed above. It's not a set (in your axioms) and not "constructed" -- it's a logical sub-class of the class of natural numbers.
If the set is a dynamic construction, No set is a "dynamical construction" then we should refer to it as a time dependent construction No set is a time dependent construction. and if wasn't a time dependent construction, then it would not be a set of rational numbers but instead would contain irrational numbers. Non sequitur -- and therefore incorrect.
If you wanted to then create another sequence It's not a "sequence" it is proof by contradiction. via., for example, E=(D+2)/3 and have these exist within the set, then a deterministic application of the C or E processes in constructing a number would be necessary (we could simply need to determine how many operations of a(t+1)=(a(t)+1)/2 or b(t+1)=(b(t)+2)/3 have occured to derive a number over time.
If you do not use such deterministic constructions for your numbers, then they are not rational numbers as they're dependent upon an unspecified set of operations over time to construct.
Notice that rational numbers must terminate, hence you can't define a rational number to be derived from an infinite sequence of unspecified (orthogonal) operations. Increasing tendency towards word salad.
QUOTE (rpenner+)
It's not a paradox -- it's a contradiction. And your going backwards does nothing to address that whenever C is a rational number less than 1, D = (C+1)/2 is a rational number larger than C and less than 1.
We could assume D was the largest number less than 1
If we wanted to generalize this and we began with the set S(0)={0}, and began to construct S(t+1) as appending a(t+1) to S(t).
a(0)=0
a(t+1)=(a(t)+1)/2
Then we can construct S(1)={0,1/2} and S(2)={0,1/2,3/4} and determine that a(t) is the largest element in the set S(t).
Notice that if we removed the time dependency then we could no longer determine what elements were in S and S would contain irrational "numbers" because it would instead contain a set of possible sets of numbers.
A real mathematician would establish an order-preserving bijection between the class of natural numbers and a subclass of dyadic fractions such that f(n) = (2ⁿ−1)/2ⁿ which demonstrates that the Archimedean property of natural numbers is mapped to this subclass of dyadic fractions. In fact, to a real mathematician I have just done so without the word salad. But that's a different class than the subclass of all rational numbers less than one and is therefore not relevant unless you are making the new claim that the largest rational number less than one is also a dyadic fraction in this class.
We could assume D was the largest number less than 1
QUOTE
Not without changing the definitions, which is impermisable.
in this case and not A or C without contradiction, hence D would now be the largest natural number less than 1. Shifting goal posts.If we wanted to generalize this and we began with the set S(0)={0}, and began to construct S(t+1) as appending a(t+1) to S(t).
a(0)=0
a(t+1)=(a(t)+1)/2
Then we can construct S(1)={0,1/2} and S(2)={0,1/2,3/4} and determine that a(t) is the largest element in the set S(t).
Notice that if we removed the time dependency then we could no longer determine what elements were in S and S would contain irrational "numbers" because it would instead contain a set of possible sets of numbers.
A real mathematician would establish an order-preserving bijection between the class of natural numbers and a subclass of dyadic fractions such that f(n) = (2ⁿ−1)/2ⁿ which demonstrates that the Archimedean property of natural numbers is mapped to this subclass of dyadic fractions. In fact, to a real mathematician I have just done so without the word salad. But that's a different class than the subclass of all rational numbers less than one and is therefore not relevant unless you are making the new claim that the largest rational number less than one is also a dyadic fraction in this class.
QUOTE (->
| QUOTE |
| Not without changing the definitions, which is impermisable. |
in this case and not A or C without contradiction, hence D would now be the largest natural number less than 1. Shifting goal posts.
If we wanted to generalize this and we began with the set S(0)={0}, and began to construct S(t+1) as appending a(t+1) to S(t).
a(0)=0
a(t+1)=(a(t)+1)/2
Then we can construct S(1)={0,1/2} and S(2)={0,1/2,3/4} and determine that a(t) is the largest element in the set S(t).
Notice that if we removed the time dependency then we could no longer determine what elements were in S and S would contain irrational "numbers" because it would instead contain a set of possible sets of numbers.
A real mathematician would establish an order-preserving bijection between the class of natural numbers and a subclass of dyadic fractions such that f(n) = (2ⁿ−1)/2ⁿ which demonstrates that the Archimedean property of natural numbers is mapped to this subclass of dyadic fractions. In fact, to a real mathematician I have just done so without the word salad. But that's a different class than the subclass of all rational numbers less than one and is therefore not relevant unless you are making the new claim that the largest rational number less than one is also a dyadic fraction in this class.
I'm not talking about sets you damn inconsistent finitist! I'm talking about the rational numbers which your axioms are not sufficiently strong to represent as a set
Many of your comments are actually related to the properties of sets of rational numbers and not numbers. If you really want to describe a rational number, then select a rational number. You have selected the topic already: The rational number which is the largest rational number smaller than one. I point out that this selection criteria does not and cannot identify a number of the form p/q where p is an integer and q is a non-zero natural number. But you responded not with mathematical definitions and logic but with non sequiturs and word salad unconnected to definitions. Thus I find that once again you are out of bounds in the finitist mathematics you wish to talk about.
Would you agree that a rational number possess specific numberic properties, such as being representable by a pair of integers n/m with no common divisors shared between n and m (m!=0)? Correct?
If you know of even a single rational number that does not satisfy this, please point it out so that we can avoid confusion as to what properties you feel rational numbers should possess.
We should also be able to, for example, multiply a rational number by 4 and determine if the result is an integer or not, correct? Those are the definitions I am using. But this is not a conversation. You need to start with definitions and proceed to conclusions. You don't like the conclusions and proceed to change the definitions.
If we wanted to generalize this and we began with the set S(0)={0}, and began to construct S(t+1) as appending a(t+1) to S(t).
a(0)=0
a(t+1)=(a(t)+1)/2
Then we can construct S(1)={0,1/2} and S(2)={0,1/2,3/4} and determine that a(t) is the largest element in the set S(t).
Notice that if we removed the time dependency then we could no longer determine what elements were in S and S would contain irrational "numbers" because it would instead contain a set of possible sets of numbers.
A real mathematician would establish an order-preserving bijection between the class of natural numbers and a subclass of dyadic fractions such that f(n) = (2ⁿ−1)/2ⁿ which demonstrates that the Archimedean property of natural numbers is mapped to this subclass of dyadic fractions. In fact, to a real mathematician I have just done so without the word salad. But that's a different class than the subclass of all rational numbers less than one and is therefore not relevant unless you are making the new claim that the largest rational number less than one is also a dyadic fraction in this class.
I'm not talking about sets you damn inconsistent finitist! I'm talking about the rational numbers which your axioms are not sufficiently strong to represent as a set
Many of your comments are actually related to the properties of sets of rational numbers and not numbers. If you really want to describe a rational number, then select a rational number. You have selected the topic already: The rational number which is the largest rational number smaller than one. I point out that this selection criteria does not and cannot identify a number of the form p/q where p is an integer and q is a non-zero natural number. But you responded not with mathematical definitions and logic but with non sequiturs and word salad unconnected to definitions. Thus I find that once again you are out of bounds in the finitist mathematics you wish to talk about.
Would you agree that a rational number possess specific numberic properties, such as being representable by a pair of integers n/m with no common divisors shared between n and m (m!=0)? Correct?
If you know of even a single rational number that does not satisfy this, please point it out so that we can avoid confusion as to what properties you feel rational numbers should possess.
We should also be able to, for example, multiply a rational number by 4 and determine if the result is an integer or not, correct? Those are the definitions I am using. But this is not a conversation. You need to start with definitions and proceed to conclusions. You don't like the conclusions and proceed to change the definitions.
QUOTE (rpenner+)
But mine are and always have been.. But in the class of rational numbers less than 1, there is no largest member. In the class of integers there is no largest member.
I can't see how referring to the definition of a set as a class changes anything, though feel free to give me link to or definition of greater specifics as to what this class defines. A class stands outside the limits of set theory. Since you reject Cantor's axiom of infinity, you cannot accept that the natural numbers or any class equinumerous with the natural numbers forms a set. But one can speak of the class of natural numbers and the class of natural numbers so long as you never assume that any subclass of these classes is a set without proving that statement from whatever axioms of set theory you have actual adopted. Since you give no clues in that direction, I work strictly with classes.
In that regard a class is a conceptual class of all objects for which the defining logical property of a class holds true. Thus "all rational numbers less than one" is a subclass of "all rational numbers" and a number is always in that class if it is true that it is a rational number and that it is less than one. It is just a wrapper for logical predicates.
If it determines whether or not a number is rational, then the reference to "rational numbers less than 1" implies a set. Nope, because you reject Cantor's axiom of infinity which says that sets of infinite size exist. And both the class of rational numbers and the class of rational numbers less than one are too large for your finitist set theory.
Notice that even if we only have a single rational number, it can still be considered to be the largest number in its set. But that is a statement true of ordered finite sets in every version of set theory. I prefer to work with what was given: the rational numbers less than one. To do so, I am not allowed to use my set theory because you reject it, and I am not allowed to use your set theory because you specifically rejected axioms which would allow the class to be a set.
If there is a set that does not have a largest element, then it would have to be an empty set, unless you have some alternative to offer. In my set theory, objects which are not totally ordered may have multiple largest elements which cannot be compared to each other. In my set theory, infinite sets -- even when triality applies -- may not have a largest element. Thus in my set theory the infinite set of all natural numbers exists and has no largest element.
Otherwise, if we have at least one rational number in a set, there exists a largest (ignoring duplicates unless we want to allow more than 1 largest number). But because of your rejection of the axiom of infinity, you forbid yourself from ever mentioning the set of all rational numbers less than 1, because such a set would prove the axiom of infinity to be true.
I can't see how referring to the definition of a set as a class changes anything, though feel free to give me link to or definition of greater specifics as to what this class defines. A class stands outside the limits of set theory. Since you reject Cantor's axiom of infinity, you cannot accept that the natural numbers or any class equinumerous with the natural numbers forms a set. But one can speak of the class of natural numbers and the class of natural numbers so long as you never assume that any subclass of these classes is a set without proving that statement from whatever axioms of set theory you have actual adopted. Since you give no clues in that direction, I work strictly with classes.
In that regard a class is a conceptual class of all objects for which the defining logical property of a class holds true. Thus "all rational numbers less than one" is a subclass of "all rational numbers" and a number is always in that class if it is true that it is a rational number and that it is less than one. It is just a wrapper for logical predicates.
If it determines whether or not a number is rational, then the reference to "rational numbers less than 1" implies a set. Nope, because you reject Cantor's axiom of infinity which says that sets of infinite size exist. And both the class of rational numbers and the class of rational numbers less than one are too large for your finitist set theory.
Notice that even if we only have a single rational number, it can still be considered to be the largest number in its set. But that is a statement true of ordered finite sets in every version of set theory. I prefer to work with what was given: the rational numbers less than one. To do so, I am not allowed to use my set theory because you reject it, and I am not allowed to use your set theory because you specifically rejected axioms which would allow the class to be a set.
If there is a set that does not have a largest element, then it would have to be an empty set, unless you have some alternative to offer. In my set theory, objects which are not totally ordered may have multiple largest elements which cannot be compared to each other. In my set theory, infinite sets -- even when triality applies -- may not have a largest element. Thus in my set theory the infinite set of all natural numbers exists and has no largest element.
Otherwise, if we have at least one rational number in a set, there exists a largest (ignoring duplicates unless we want to allow more than 1 largest number). But because of your rejection of the axiom of infinity, you forbid yourself from ever mentioning the set of all rational numbers less than 1, because such a set would prove the axiom of infinity to be true.
QUOTE (rpenner+)
Correct, but I have shown using the definitions of rational number arithmetic that there is a larger number. So if you think different, you must not be using rational numbers either. Even the ancient Egyptians had rational numbers.
You showed that there can exist a rational number C that is larger than A, but in that case C is the largest number, but in both cases there was a largest rational number less than 1, which agrees with my claim that there always exists a largest number in a set that has at least one number. No -- I demonstrated that in one step I can always construct a number less than 1 which is greater that any number you give me. Thus it would only take one step to prove C is not the largest number less than one either.
If we were to take your claim seriously, that there are no such largest numbers then there should not have existed either A or C as largest elements, so it appears my claim was consistantly valid throughout the computations. A proof of contradiction always assumes the reverse of what is to be proven. Thus I assumed A was the rational number in question, proved that led to contradiction and nonsense, and used that to reject the assumption.
You showed that there can exist a rational number C that is larger than A, but in that case C is the largest number, but in both cases there was a largest rational number less than 1, which agrees with my claim that there always exists a largest number in a set that has at least one number. No -- I demonstrated that in one step I can always construct a number less than 1 which is greater that any number you give me. Thus it would only take one step to prove C is not the largest number less than one either.
If we were to take your claim seriously, that there are no such largest numbers then there should not have existed either A or C as largest elements, so it appears my claim was consistantly valid throughout the computations. A proof of contradiction always assumes the reverse of what is to be proven. Thus I assumed A was the rational number in question, proved that led to contradiction and nonsense, and used that to reject the assumption.
QUOTE (rpenner+)
... Because numbers don't move and don't converge.
True, I agree with that.
True, I agree with that.
QUOTE (rpenner+)
That's why your position is finitist. That's fine, except you don't recognize that it puts limits on your mathematics that the rest of us don't have to subscribe to. You also haven't worked out the consequences of this yet and have rambled on for thousands of posts without saying anything both new and correct. And among your mistakes, you never start from your assumptions to build theorems and you never spell out your definitions at the beginning, and you misuse words which even finitists use in the standard mathematical sense.
Actually I think I've pointed some rather novel insights, including the fact that there don't exist logical generalizations for counting - counting is an inherent physical ability that logic can't generalize upon. Mathematics should be built upon a preexisting ability to count and give explicit physical and temporal considerations to this as well. Of course we can still generalize and extrapolate, but (capital I) Infinity would have to be the finest grained process over time and that's already true for existing mathematics but it's not generally recognized.
Acknowledging a limit is not the same as creating a limit and I don't believe ignoring it or blurring over it assists in overcoming it or in some cases taking advantage of it.
I've typically found that by recognizing what limits exist allows one to at least partly overcome them and ironically, oftentimes there are even better alternatives that would not have been noticed had the limit not been recognized and the limit turns into a step leading to something greater (though some limits can also be frustratingly stubborn and potentially impossible to overcome - one unknown thing is just as unknown as something else and one impossibility is just as impossible as a collection of them. From that standpoint, they have an identical significance).
Well, I disagree and will red-pen your posts as I feel necessary. Since I started as moderator, I have read your posts and I have found you are the student of none and are the teacher of none and do not engage in productive discussions on mathematical topics.
Actually I think I've pointed some rather novel insights, including the fact that there don't exist logical generalizations for counting - counting is an inherent physical ability that logic can't generalize upon. Mathematics should be built upon a preexisting ability to count and give explicit physical and temporal considerations to this as well. Of course we can still generalize and extrapolate, but (capital I) Infinity would have to be the finest grained process over time and that's already true for existing mathematics but it's not generally recognized.
Acknowledging a limit is not the same as creating a limit and I don't believe ignoring it or blurring over it assists in overcoming it or in some cases taking advantage of it.
I've typically found that by recognizing what limits exist allows one to at least partly overcome them and ironically, oftentimes there are even better alternatives that would not have been noticed had the limit not been recognized and the limit turns into a step leading to something greater (though some limits can also be frustratingly stubborn and potentially impossible to overcome - one unknown thing is just as unknown as something else and one impossibility is just as impossible as a collection of them. From that standpoint, they have an identical significance).
Well, I disagree and will red-pen your posts as I feel necessary. Since I started as moderator, I have read your posts and I have found you are the student of none and are the teacher of none and do not engage in productive discussions on mathematical topics.
QUOTE (Cusa+Feb 28 2009, 09:47 PM)
If the odds of something are One out of Infinity then it could never happen more than once.
Mitch Raemsch
I had a dream recently along those lines. One thing that been bothering is how we can count things when every thing is unique. Obviously we set up vague statistical classes for objects and then count things with similar features, but it still appears there need to physically exist objects that are indistinguishable except for representing specific numbers.
Something to consider is that just as the influence of a collection of unknowns can be reduced to the influence of a single unknown, you can also reverse this and describe a single unknown as being an arbitrary quantity of otherwise indistinguishable unknowns.
For example, "nothing" could be similar to an unknown in that it's not anything specific and two nothings are just as influencial as one, etc.
So similar to 0=0+0, we could say that 1*0=2*0=3*0=... and begin to count nothing and have it remain non-influencial. In a sense we'd be counting identical (non) objects.
There's a reciprocal view in that if these were actually infinitesimals instead of 0s, then we could, over a specific Infinite count reach 1 and so a reciprocal view could be taken of 1 unknown and over time, cutting it into finer and finer "unknowns", much like an expansion of space beyond detection.
There would still exist some mechanism for constructing these boundaries between unknowns, but at least there could exist something that is truly indistinguishable and identical to count - an unknown - because there's nothing specific to distinguish between them and make them individually distinguishable, so they could exist as unperceived quantities (which has a quite interesting correlation to photon detections).
Notice also that we could establish a trichotomy between possibilities, unknowns and impossibilities.
Things that were possible would not be entirely unknown, so they would be unknowns that became possible and impossibilities would similarly be unknowns that became impossible, so we could similarly take an expanding space of unknowns and witness these additions in terms of those things that have become possible in a space, versus those things that have become impossible in a space and this would be similar to a space filled with binary values true or false with regard to whether or not an object described by that location in space is possible.
If such an object is possible, then as you said, it should only exist once, if it's truly unique, so we could then have a function of time determine which possibilities were realized at a specific moment of time as well as whether or not it's already occured in the past, so you could see the possibilities as always beginning as future possibilities and then becoming present realizations and moving into past histories and there are some interesting possibilities on how these could be interpreted as reading a binary string but collecting elements of it into larger units with greater distinctions between their traits.
It's very natural to refer to possibilities as a plural - there are many possibilities, but that there's not much sense in referring to multiple impossible things or "impossibilities" as one thing that is impossible is just as "real" as anything else impossible or multiple impossible things, but by defining a set of impossible properties we could actually work with a space of impossibilities that would have many complimentary properties to the space of things that are possible or real or historical etc. In the middle or on the boundaries, depending on the view, we should have a space lying between these described by an unknown that becomes countable quantities of identical units of unknowns over time. So the process converting a single unknown into quantities would be the single source of time (for a space) and it should be effectively counting. Every number should be unique, and if it's not independently unique, then it would instead be unique within the context of the current space, and be unseperable from space, so the total quantity of space itself would be doing the counting (which, in that case would allow some measure of "will" in the determination of what locations in this space would be possible, impossible, present or past - though there's a conflict in there and likely simplification because if the time determined the count then the values within space would not specifically be future/possible, past or present, but instead would just remain possible or impossible with the specific time or current count defining which subset of the possible were specifically present or past. So I'd need to refine the details better and determine a specific construction for space also, though this should resemble ideas in physics regarding the abstract speed of light in that deterministic spaces can only be built upon things that are already determined and so we'd likely begin with an origin (might as well use the Unknown for that) and use quantities of identical unknowns to determine the distances between points in space - an n dimensional space can be defined by determining n distances to other objects within the space and this space need not be uniformly fillable nor have perfectly orthogonal dimesions, we just need some measure of independence in the quantities).
Mitch Raemsch
I had a dream recently along those lines. One thing that been bothering is how we can count things when every thing is unique. Obviously we set up vague statistical classes for objects and then count things with similar features, but it still appears there need to physically exist objects that are indistinguishable except for representing specific numbers.
Something to consider is that just as the influence of a collection of unknowns can be reduced to the influence of a single unknown, you can also reverse this and describe a single unknown as being an arbitrary quantity of otherwise indistinguishable unknowns.
For example, "nothing" could be similar to an unknown in that it's not anything specific and two nothings are just as influencial as one, etc.
So similar to 0=0+0, we could say that 1*0=2*0=3*0=... and begin to count nothing and have it remain non-influencial. In a sense we'd be counting identical (non) objects.
There's a reciprocal view in that if these were actually infinitesimals instead of 0s, then we could, over a specific Infinite count reach 1 and so a reciprocal view could be taken of 1 unknown and over time, cutting it into finer and finer "unknowns", much like an expansion of space beyond detection.
There would still exist some mechanism for constructing these boundaries between unknowns, but at least there could exist something that is truly indistinguishable and identical to count - an unknown - because there's nothing specific to distinguish between them and make them individually distinguishable, so they could exist as unperceived quantities (which has a quite interesting correlation to photon detections).
Notice also that we could establish a trichotomy between possibilities, unknowns and impossibilities.
Things that were possible would not be entirely unknown, so they would be unknowns that became possible and impossibilities would similarly be unknowns that became impossible, so we could similarly take an expanding space of unknowns and witness these additions in terms of those things that have become possible in a space, versus those things that have become impossible in a space and this would be similar to a space filled with binary values true or false with regard to whether or not an object described by that location in space is possible.
If such an object is possible, then as you said, it should only exist once, if it's truly unique, so we could then have a function of time determine which possibilities were realized at a specific moment of time as well as whether or not it's already occured in the past, so you could see the possibilities as always beginning as future possibilities and then becoming present realizations and moving into past histories and there are some interesting possibilities on how these could be interpreted as reading a binary string but collecting elements of it into larger units with greater distinctions between their traits.
It's very natural to refer to possibilities as a plural - there are many possibilities, but that there's not much sense in referring to multiple impossible things or "impossibilities" as one thing that is impossible is just as "real" as anything else impossible or multiple impossible things, but by defining a set of impossible properties we could actually work with a space of impossibilities that would have many complimentary properties to the space of things that are possible or real or historical etc. In the middle or on the boundaries, depending on the view, we should have a space lying between these described by an unknown that becomes countable quantities of identical units of unknowns over time. So the process converting a single unknown into quantities would be the single source of time (for a space) and it should be effectively counting. Every number should be unique, and if it's not independently unique, then it would instead be unique within the context of the current space, and be unseperable from space, so the total quantity of space itself would be doing the counting (which, in that case would allow some measure of "will" in the determination of what locations in this space would be possible, impossible, present or past - though there's a conflict in there and likely simplification because if the time determined the count then the values within space would not specifically be future/possible, past or present, but instead would just remain possible or impossible with the specific time or current count defining which subset of the possible were specifically present or past. So I'd need to refine the details better and determine a specific construction for space also, though this should resemble ideas in physics regarding the abstract speed of light in that deterministic spaces can only be built upon things that are already determined and so we'd likely begin with an origin (might as well use the Unknown for that) and use quantities of identical unknowns to determine the distances between points in space - an n dimensional space can be defined by determining n distances to other objects within the space and this space need not be uniformly fillable nor have perfectly orthogonal dimesions, we just need some measure of independence in the quantities).
I have another question steve.
How did Avogadro the chemist count his first mole.
How did he count the atoms he weighed?
It looks like that is forever beyond our power.
Mitch Raemsch
How did Avogadro the chemist count his first mole.
How did he count the atoms he weighed?
It looks like that is forever beyond our power.
Mitch Raemsch
QUOTE (SteveA+)
rpenner states ...
Screw it. I'm gone. I already told you not to edit my posts.
Live with your beliefs and impose them on the forum. I've already shown your views to be inconsistant and instead of learning and adapting to complexity, you curl up into an armadillo.
Cya guys
Screw it. I'm gone. I already told you not to edit my posts.
Live with your beliefs and impose them on the forum. I've already shown your views to be inconsistant and instead of learning and adapting to complexity, you curl up into an armadillo.
Cya guys
QUOTE (SteveA2+Mar 1 2009, 12:03 AM)
Screw it. I'm gone.
Not to live in the past, but I've heard that before.
QUOTE (SteveA2+Mar 1 2009, 12:03 AM)
I already told you not to edit my posts.
This is the first time I have edited one of your posts. Prior to this, I only had the power to delete them and move them. Your request was considered and did not meet the current requirements of this forum, as I see it.
QUOTE (SteveA2+Mar 1 2009, 12:03 AM)
Live with your beliefs and impose them on the forum.
Thank you, I certainly shall. I have long been a supporter of you living by your own beliefs. If you believe that you are incapable of being corrected then you might have the makings of a bank president or Hollywood celebrity. In mathematical exposition, however, there are regrettably always editors.
QUOTE (SteveA2+Mar 1 2009, 12:03 AM)
I've already shown your views to be inconsistant and instead of learning and adapting to complexity, you curl up into an armadillo.
My views are mutually inconsistent with your views. But then your use of terms of art in fields of mathematics which you do not subscribe to were also inconsistent with your views. The armadillo analogy seems to be unsupported.
QUOTE (SteveA2+Mar 1 2009, 12:03 AM)
Cya guys
TTFN.
I wouldn't be suprised if I am going to be a continued target because I believe science comes from God.
Mitch Raemsch
Mitch Raemsch
That has absolutely nothing to do with anything being discussed here, but you knew that already...
Regardless of where science comes from, your posts are often single line unsupported claims of extraordinary nature that -- and this is really important -- do not connect with the thread.
We would all prefer that you invest more thought in the posting process. This isn't Twitter, you know. But even so, we have established a preserve for your "observations" -- perhaps if you spent time reading them you would see that they don't form a conventional expository thread and could work one improving your writing style.
Some notes from others:
But the good news is that this situation is one you can change.
We would all prefer that you invest more thought in the posting process. This isn't Twitter, you know. But even so, we have established a preserve for your "observations" -- perhaps if you spent time reading them you would see that they don't form a conventional expository thread and could work one improving your writing style.
Some notes from others:
QUOTE
Would you please do something about this ***, Cusa?
Why is he immune from the post deletions many of us have had when responding to his blathering?
More quasi-religious non-science.
Reported for the thread's sheer ridiculousness, both in the title and the opening post. Mitch Raemsch isn't exactly prone to rational debate but the absurdity of the statement he's making in general - with no solid reasoning behind it - is just mind-boggling.
Bogus thread. Request deletion.
Cusa is continually off-topic and disruptive of any thread he spams.
I refuse to follow up on any thread he posts to.
Please, have him banned. He is of no use to anyone but himself.
So in light of that, your posts are a primary source of concern to me. I'm not deleting them fast enough by these opinions. They are not valued.Why is he immune from the post deletions many of us have had when responding to his blathering?
More quasi-religious non-science.
Reported for the thread's sheer ridiculousness, both in the title and the opening post. Mitch Raemsch isn't exactly prone to rational debate but the absurdity of the statement he's making in general - with no solid reasoning behind it - is just mind-boggling.
Bogus thread. Request deletion.
Cusa is continually off-topic and disruptive of any thread he spams.
I refuse to follow up on any thread he posts to.
Please, have him banned. He is of no use to anyone but himself.
But the good news is that this situation is one you can change.
QUOTE (rpenner+Mar 1 2009, 01:25 PM)
Not to live in the past, but I've heard that before.
This is the first time I have edited one of your posts. Prior to this, I only had the power to delete them and move them. Your request was considered and did not meet the current requirements of this forum, as I see it.
Thank you, I certainly shall. I have long been a supporter of you living by your own beliefs. If you believe that you are incapable of being corrected then you might have the makings of a bank president or Hollywood celebrity. In mathematical exposition, however, there are regrettably always editors.
My views are mutually inconsistent with your views. But then your use of terms of art in fields of mathematics which you do not subscribe to were also inconsistent with your views. The armadillo analogy seems to be unsupported.
TTFN.
Hi rpenner
This is the first I've heard of posts being edited. I would be concerned if it were for being incorrect. ....It is certainly true that the forum suffered from abusive postings and if you are keeping that out , then I support you whole heartedly. I can't see what Stevea2 would be putting into posts that needs editing as he doesn't generally abuse other posters.
Anyway. If your keeping abusive spam at bay...good on you .
Cheers
Iseason
This is the first time I have edited one of your posts. Prior to this, I only had the power to delete them and move them. Your request was considered and did not meet the current requirements of this forum, as I see it.
Thank you, I certainly shall. I have long been a supporter of you living by your own beliefs. If you believe that you are incapable of being corrected then you might have the makings of a bank president or Hollywood celebrity. In mathematical exposition, however, there are regrettably always editors.
My views are mutually inconsistent with your views. But then your use of terms of art in fields of mathematics which you do not subscribe to were also inconsistent with your views. The armadillo analogy seems to be unsupported.
TTFN.
Hi rpenner
This is the first I've heard of posts being edited. I would be concerned if it were for being incorrect. ....It is certainly true that the forum suffered from abusive postings and if you are keeping that out , then I support you whole heartedly. I can't see what Stevea2 would be putting into posts that needs editing as he doesn't generally abuse other posters.
Anyway. If your keeping abusive spam at bay...good on you .
Cheers
Iseason
QUOTE (iseason+Mar 1 2009, 03:18 AM)
Hi rpenner
This is the first I've heard of posts being edited. I would be concerned if it were for being incorrect. ....It is certainly true that the forum suffered from abusive postings and if you are keeping that out , then I support you whole heartedly. I can't see what Stevea2 would be putting into posts that needs editing as he doesn't generally abuse other posters.
Anyway. If your keeping abusive spam at bay...good on you .
Cheers
Iseason
Idiocy is the ultimate offense against some people. When people refuse to learn and to be rational it is abusive to me.
This is the first I've heard of posts being edited. I would be concerned if it were for being incorrect. ....It is certainly true that the forum suffered from abusive postings and if you are keeping that out , then I support you whole heartedly. I can't see what Stevea2 would be putting into posts that needs editing as he doesn't generally abuse other posters.
Anyway. If your keeping abusive spam at bay...good on you .
Cheers
Iseason
Idiocy is the ultimate offense against some people. When people refuse to learn and to be rational it is abusive to me.
DuzmA
"When people refuse to learn and to be rational it is abusive to me."
Learning - what is right what is wrong - what is the "truth" - rationality - ???
All these concepts are highly subjective and it is more a question about how many other similar individuals, that said person wants to share his/her ideas with.
To me it cannot be abusive if other people take their right to decide what they believe in - and if You decides it to be abusive - I see it as something that You take on your shoulder.
Sometimes history demonstrates how the minority thinkers sooner or later take over the "truth".
"When people refuse to learn and to be rational it is abusive to me."
Learning - what is right what is wrong - what is the "truth" - rationality - ???
All these concepts are highly subjective and it is more a question about how many other similar individuals, that said person wants to share his/her ideas with.
To me it cannot be abusive if other people take their right to decide what they believe in - and if You decides it to be abusive - I see it as something that You take on your shoulder.
Sometimes history demonstrates how the minority thinkers sooner or later take over the "truth".
QUOTE (bukh+Mar 1 2009, 01:01 PM)
DuzmA
"When people refuse to learn and to be rational it is abusive to me."
Learning - what is right what is wrong - what is the "truth" - rationality - ???
All these concepts are highly subjective and it is more a question about how many other similar individuals, that said person wants to share his/her ideas with.
To me it cannot be abusive if other people take their right to decide what they believe in - and if You decides it to be abusive - I see it as something that You take on your shoulder.
Sometimes history demonstrates how the minority thinkers sooner or later take over the "truth".
Well I don't find it abuse to challenge people's beliefs with evidence yet I am constantly cited as abusive in my day to day life for doing just that. The idea that everyone's beliefs should be respected no matter how bogus they are is a foolish idea and an idea that will I will not honor. If you believe something that is clearly opposed to the evidence, prepare to have its absurdity thrown at you. As for everything being subjective: I hear this from time to time, it is more or less a flavor of relativism. Minority thinkers don't take over the truth by spamming beliefs that are based on NO evidence, much less those that are in contradiction to the evidence. Words like truth in the sense you are using it serve only to make the discussion murky, the evidence does not lie. Since people's upbringings and experiences make interpretation subjective to a degree you want to just accept everything that everyone says? Are you a sociologist?
"When people refuse to learn and to be rational it is abusive to me."
Learning - what is right what is wrong - what is the "truth" - rationality - ???
All these concepts are highly subjective and it is more a question about how many other similar individuals, that said person wants to share his/her ideas with.
To me it cannot be abusive if other people take their right to decide what they believe in - and if You decides it to be abusive - I see it as something that You take on your shoulder.
Sometimes history demonstrates how the minority thinkers sooner or later take over the "truth".
Well I don't find it abuse to challenge people's beliefs with evidence yet I am constantly cited as abusive in my day to day life for doing just that. The idea that everyone's beliefs should be respected no matter how bogus they are is a foolish idea and an idea that will I will not honor. If you believe something that is clearly opposed to the evidence, prepare to have its absurdity thrown at you. As for everything being subjective: I hear this from time to time, it is more or less a flavor of relativism. Minority thinkers don't take over the truth by spamming beliefs that are based on NO evidence, much less those that are in contradiction to the evidence. Words like truth in the sense you are using it serve only to make the discussion murky, the evidence does not lie. Since people's upbringings and experiences make interpretation subjective to a degree you want to just accept everything that everyone says? Are you a sociologist?
DuzmA
"Are you a sociologist?"
Is that meant positively ?
"Are you a sociologist?"
Is that meant positively ?
QUOTE (bukh+Mar 1 2009, 01:56 PM)
DuzmA
"Are you a sociologist?"
Is that meant positively ?
Its not really meant positively or negatively, your sentiments just reflect those of several sociologists that I know. I realize that there are many non-sociologists that feel similar, I just thought I'd ask.
"Are you a sociologist?"
Is that meant positively ?
Its not really meant positively or negatively, your sentiments just reflect those of several sociologists that I know. I realize that there are many non-sociologists that feel similar, I just thought I'd ask.
QUOTE (bukh+Feb 28 2009, 11:38 AM)
penner
"You can't have your cake and eat it too."
OK then we say that .9... is not a number but it is equal to the number 1. !
Perhaps this is a slightly different meaning to the concept of "equal to"
To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking.
You can say that two different fruits are equal in nutrition.
Have you heard the word "equivalent" used in mathematics?
To be blunt have you considered that "equal" doesn't mean exactly what most people think it means (in mathematical terms)?
and furthermore that the issue has already been dealt with?
"You can't have your cake and eat it too."
OK then we say that .9... is not a number but it is equal to the number 1. !
Perhaps this is a slightly different meaning to the concept of "equal to"
To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking.
You can say that two different fruits are equal in nutrition.
Have you heard the word "equivalent" used in mathematics?
To be blunt have you considered that "equal" doesn't mean exactly what most people think it means (in mathematical terms)?
and furthermore that the issue has already been dealt with?
buttershug
"To be blunt have you considered that "equal" doesn't mean exactly what most people think it means (in mathematical terms)?
and furthermore that the issue has already been dealt with?"
Yeah - already in my answer I noted that I wrongly said "equal" - when I meant "identical"
"To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking."
Thanks -
BTW - do you agree that they are not identical ?
[COLOR=green]
"To be blunt have you considered that "equal" doesn't mean exactly what most people think it means (in mathematical terms)?
and furthermore that the issue has already been dealt with?"
Yeah - already in my answer I noted that I wrongly said "equal" - when I meant "identical"
"To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking."
Thanks -
BTW - do you agree that they are not identical ?
[COLOR=green]
QUOTE (bukh+Mar 1 2009, 03:55 PM)
BTW - do you agree that they are not identical ?
[COLOR=green]
Yes, do you agree that NO ONE has claimed they are?
0.99...=1 is what is said.
I'm not sure how to make an equivalent sign in post. It's three parallel lines.
I hope Rpenner or AN tell me if it is appropriate to say that 0.99... is not equilvalent to 1.
I'm not aware of a mathematical definition of "identical".
[COLOR=green]
Yes, do you agree that NO ONE has claimed they are?
0.99...=1 is what is said.
I'm not sure how to make an equivalent sign in post. It's three parallel lines.
I hope Rpenner or AN tell me if it is appropriate to say that 0.99... is not equilvalent to 1.
I'm not aware of a mathematical definition of "identical".
QUOTE (buttershug+Mar 1 2009, 05:09 PM)
Yes, do you agree that NO ONE has claimed they are?
0.99...=1 is what is said.
I'm not sure how to make an equivalent sign in post. It's three parallel lines.
I hope Rpenner or AN tell me if it is appropriate to say that 0.99... is not equilvalent to 1.
I'm not aware of a mathematical definition of "identical".
I believe that equivalence is the same as equality when neither side has any variables.
The difference is when one or both sides have variables... equivalence implies that both sides are equal regardless of the value of the variables:
2a is equivalent to 4a/2 (true for any value of a)
2a is equal to a^2 (only true for specific values of a)
But that 'definition' (am I being slightly optimistic with that word???) may be secondary school, over-simplistic guff.
0.99...=1 is what is said.
I'm not sure how to make an equivalent sign in post. It's three parallel lines.
I hope Rpenner or AN tell me if it is appropriate to say that 0.99... is not equilvalent to 1.
I'm not aware of a mathematical definition of "identical".
I believe that equivalence is the same as equality when neither side has any variables.
The difference is when one or both sides have variables... equivalence implies that both sides are equal regardless of the value of the variables:
2a is equivalent to 4a/2 (true for any value of a)
2a is equal to a^2 (only true for specific values of a)
But that 'definition' (am I being slightly optimistic with that word???) may be secondary school, over-simplistic guff.
buttershug
"Yes, do you agree that NO ONE has claimed they are?"
How should I know ! I do not know NO ONE -
"Yes, do you agree that NO ONE has claimed they are?"
How should I know ! I do not know NO ONE -
QUOTE (bukh+Mar 1 2009, 03:55 PM)
Yeah - already in my answer I noted that I wrongly said "equal" - when I meant "identical"
"To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking."
Thanks -
BTW - do you agree that they are not identical ?
[COLOR=green]
If someone gives you two halves of an apple: did they give you an apple or something that is equivalent to an apple?
"To me it is the same as trying to compare apples and grapes - and I still favor the concept that they are not equal - but I think that we have definitions in place - and the rest is a question of words. I am happy with the above - it is in line with my thinking."
Thanks -
BTW - do you agree that they are not identical ?
[COLOR=green]
If someone gives you two halves of an apple: did they give you an apple or something that is equivalent to an apple?
Whenever you have two quantities joined by the equals sign, the left and right are equal and interchangeable for all mathematical purposes. A weaker statement of equivalence is isomorphism, which is saying that the left and right sides have the same internal structure. So a letter U is isomorphic to a letter C because you can show they have the same structure in various mathematical ways.
By the rules of arithmetic 1 + 1 defines a number. By convention 2 defines a number. 1 + 1 = 2 is a statement that these both refer to the same number. Thus they are equal, which is to say they are equivalent and can be used interchangeably for every mathematical purpose. Naturally, it would be inconvenient to build computer systems such that users had the freedom to type The smallest prime number, the positive square root of 4, the natural number that follows 1, the maximum of 1 + 2x − x˛, 0 + 2, etc. But all of these are names for the same number. [Edit -- not 3 + 2x − x˛ -- my mind wandered.]
In precisely the same way 0.1 + 0.01 + 0.001 + 0.0001 + ... is an infinite series which can be evaluated using methods familiar to the ancient Greeks. It sums to 1/9. If a_n is forced to lie between 0 and 9, then a_1 × 0.1 + a_2 × 0.01 + a_3 × 0.001 + a_4 × 0.0001 + ... must also define a well-defined number between 0 and 9×(1/9) = 1. The only time it is equal to 1 is when all the a_n are 9's. Since this is the formal definition of 0.999... then it is entirely correct to say 0.999... = 1.
http://us.metamath.org/mpegif/0.999....html
This is not "real" mathematics in the way that professionals approach new ideas. But when they publish papers, they assume that they have done enough work so that someone could prove their statements to this level of detail.
1 ⊢ (k ∈ ℕ → k ∈ ℕ₀) If a number is a positive integer, then it is a natural number.
2 ⊢ 9 ∈ ℝ Nine is a real number.
3 ⊢ 9 ∈ ℂ Therefore, the number nine is a complex number.
4 ⊢ ((9 ∈ ℂ ⋀ (10↑k) ∈ ℂ ⋀ (10↑k) ≠ 0) → (9 / (10↑k)) = (9 · (1 / (10↑k)))) If nine is a complex number, and 10^k is a complex number and 10^k is a non-zero complex number then dividing 9 by 10^k is the same as multiplying 9 and the reciprocal to 10^k.
5 ⊢ (((10↑k) ∈ ℂ ⋀ (10↑k) ≠ 0) → (9 / (10↑k)) = (9 · (1 / (10↑k)))) But we can simplify that statement because nine is a complex number.
6 ⊢ 10 ∈ ℝ Ten is a real number.
7 ⊢ 10 ∈ ℂ Therefore the number ten is a complex number.
8 ⊢ ((10 ∈ ℂ ⋀ k ∈ ℕ₀) → (10↑k) ∈ ℂ) If ten is a complex number and k is a natural number, then 10^k is a complex number.
9 ⊢ (k ∈ ℕ₀ → (10↑k) ∈ ℂ) But we can simplify that statement because 10 is a complex number.
10 ⊢ 0 < 10 Ten is positive.
11 ⊢ 10 ≠ 0 Since ten is a real, positive number, we know it is not zero.
12 ⊢ ((10 ∈ ℂ ⋀ k ∈ ℕ₀ ⋀ 10 ≠ 0) → (10↑k) ≠ 0) If 10 is a non-zero, complex number and k is a natural number and then 10^k is not zero.
13 ⊢ (k ∈ ℕ₀ → (10↑k) ≠ 0) But we can simplify that statement because ten is a non-zero, complex number.
14 ⊢ (k ∈ ℕ₀ → (9 / (10↑k)) = (9 · (1 / (10↑k)))) From what has been seen above, it follows that if k is a natural number, then dividing 9 by 10^k is the same as multiplying 9 and the reciprocal to 10^k.
15 ⊢ ((10 ∈ ℂ ⋀ k ∈ ℕ₀ ⋀ 10 ≠ 0) → ((1 / 10)↑k) = (1 / (10↑k))) If ten is a non-zero complex number, and k is a natural number, then the kth power of the reciprocal of ten is the same as the reciprocal of the kth power of ten.
16 7, 11, 15 mp3an13 910 . . . . . 6 ⊢ (k ∈ ℕ₀ → ((1 / 10)↑k) = (1 / (10↑k))) But we can simplify that statement because ten is a non-zero, complex number.
17 ⊢ (k ∈ ℕ₀ → (9 · ((1 / 10)↑k)) = (9 · (1 / (10↑k)))) And we can multiply on the left by nine.
18 ⊢ (k ∈ ℕ₀ → (9 / (10↑k)) = (9 · ((1 / 10)↑k))) And we can now reformat the equation more.
19 ⊢ (k ∈ ℕ → (9 / (10↑k)) = (9 · ((1 / 10)↑k))) And now that we've proven the general case, all of this follows from the more specific case when k is a positive integer.
20 ⊢ Σ k ∈ ℕ (9 / (10↑k)) = Σ k ∈ ℕ (9 · ((1 / 10)↑k)) So the sum over all positive integers is the same if we write the summand in either of these forms.
21 ⊢ (1 / 10) ∈ ℝ Since ten is a non-zero, real number, its reciprocal is a real number.
22 ⊢ (1 / 10) ∈ ℂ Which is to say it is a complex number.
23 ⊢ 0 ∈ ℝ And zero is a real number.
24 ⊢ 0 < (1 / 10) And since ten is greater than zero, its reciprocal is greater than zero.
25 ⊢ 0 ≤ (1 / 10) And greater than zero means one tenth is also greater-to-or-equal-to zero.
26 ⊢ (0 ≤ (1 / 10) → (abs ‘(1 / 10)) = (1 / 10)) From the definition of absolute value, if one tenth is greater-to-or-equal-to zero then we know it is equal to its absolute value.
27 ⊢ (abs ‘(1 / 10)) = (1 / 10) So this is the case.
28 ⊢ 0 < 9 Nine is a positive number.
29 ⊢ 1 ∈ ℝ One is a real number.
30 ⊢ ((9 ∈ ℝ ⋀ 1 ∈ ℝ) → (0 < 9 ↔ 1 < (9 + 1))) If nine is a real number and one is a real number, then saying nine is positive is the same thing as saying nine plus one is greater than one.
31 ⊢ (0 < 9 ↔ 1 < (9 + 1)) So we say it.
32 ⊢ 1 < (9 + 1) Having said it, we prove nine plus one is greater than 1.
33 ⊢ 10 = (9 + 1) But ten is defined as nine plus one.
34 ⊢ 1 < 10 So we have proved ten is greater than one.
35 ⊢ ((10 ∈ ℝ ⋀ 0 < 10) → (1 < 10 ↔ (1 / 10) < 1)) Now if ten is a positive real number, then saying ten is greater than one is the same as saying the reciprocal of ten is less than one.
36 ⊢ (1 < 10 ↔ (1 / 10) < 1) So we say it.
37 ⊢ (1 / 10) < 1 And so we have proved that one tenth is less than one.
38 ⊢ (abs ‘(1 / 10)) < 1 And, something that will immediately be useful, we have proved that one tenth has an absolute value less than 1.
39 ⊢ ((9 ∈ ℂ ⋀ (1 / 10) ∈ ℂ ⋀ (abs ‘(1 / 10)) < 1) → Σ k ∈ ℕ (9 · ((1 / 10)↑k)) = ((9 · (1 / 10)) / (1 − (1 / 10)))) A general theorem has proven than if nine is a complex number, and one tenth is a complex number with a absolute value less than one, then the sum over all positive integers of nine times the kth power of one tenth is equal to nine times one tenth divided by the quantity one minus one tenth.
40 ⊢ Σ k ∈ ℕ (9 · ((1 / 10)↑k)) = ((9 · (1 / 10)) / (1 − (1 / 10))) But by this point we have proved all the antecedents, and we say: The sum over all positive integers of nine times the kth power of one tenth is equal to nine times one tenth divided by the quantity one minus one tenth.
Algebra to reformat answer:
41 ⊢ (9 / 10) = (9 · (1 / 10)) Nine divided by ten is the same as nine times one tenth.
42 ⊢ (10 · (9 / 10)) = 9 Ten times nine divided by ten is nine. We will need these results later.
43 ⊢ 1 ∈ ℂ One is a complex number.
44 ⊢ (10 · (1 − (1 / 10))) = ((10 · 1) − (10 · (1 / 10))) Ten distributes over subtraction.
45 ⊢ (10 · 1) = 10 Ten times one is ten.
46 ⊢ (10 · (1 / 10)) = 1 Ten times one tenth is one.
47 ⊢ ((10 · 1) − (10 · (1 / 10))) = (10 − 1) Ten times one minus ten times one tenth is ten minus one.
48 ⊢ (1 + 9) = (9 + 1) One plus nine is the same as nine plus one.
49 ⊢ (1 + 9) = 10 So one plus nine is ten.
50 ⊢ (10 − 1) = 9 So ten minus one is nine.
51 ⊢ 9 = (10 · (1 − (1 / 10))) So nine is equal to ten times the quantity one minus one tenth.
52 ⊢ (10 · (9 / 10)) = (10 · (1 − (1 / 10))) So ten times nine over ten is equal to ten times the quantity one minus one tenth.
53 ⊢ (9 / 10) ∈ ℝ Nine over ten is a real number.
54 ⊢ (9 / 10) ∈ ℂ Therefore nine over ten is a complex number.
55 ⊢ (1 − (1 / 10)) ∈ ℂ And one minus one tenth is complex number.
56 ⊢ ((10 · (9 / 10)) = (10 · (1 − (1 / 10))) ↔ (9 / 10) = (1 − (1 / 10))) Since ten is non-zero, we can cancel it from both sides of the equals sign.
57 ⊢ (9 / 10) = (1 − (1 / 10)) So we know nine over ten is the same as one minus one tenth.
58 ⊢ ((9 / 10) / (9 / 10)) = ((9 · (1 / 10)) / (1 − (1 / 10))) So we can see two different ways of writing nine over ten. Dividing one by the other gives the same result no matter how we write it.
59 ⊢ 0 < (9 / 10) But nine over ten is greater than zero.
60 ⊢ (9 / 10) ≠ 0 Therefore it is non-zero.
61 ⊢ ((9 / 10) / (9 / 10)) = 1 So when you divide it by itself the result is one.
62 ⊢ ((9 · (1 / 10)) / (1 − (1 / 10))) = 1 So nine times one tenth divided by the quantity one minus one tenth is equal to one.
63 ⊢ Σ k ∈ ℕ (9 / (10↑k)) = 1 Therefore the sum over all positive integers of nine divided by the kth power of ten is equal to 1.
By the rules of arithmetic 1 + 1 defines a number. By convention 2 defines a number. 1 + 1 = 2 is a statement that these both refer to the same number. Thus they are equal, which is to say they are equivalent and can be used interchangeably for every mathematical purpose. Naturally, it would be inconvenient to build computer systems such that users had the freedom to type The smallest prime number, the positive square root of 4, the natural number that follows 1, the maximum of 1 + 2x − x˛, 0 + 2, etc. But all of these are names for the same number. [Edit -- not 3 + 2x − x˛ -- my mind wandered.]
In precisely the same way 0.1 + 0.01 + 0.001 + 0.0001 + ... is an infinite series which can be evaluated using methods familiar to the ancient Greeks. It sums to 1/9. If a_n is forced to lie between 0 and 9, then a_1 × 0.1 + a_2 × 0.01 + a_3 × 0.001 + a_4 × 0.0001 + ... must also define a well-defined number between 0 and 9×(1/9) = 1. The only time it is equal to 1 is when all the a_n are 9's. Since this is the formal definition of 0.999... then it is entirely correct to say 0.999... = 1.
http://us.metamath.org/mpegif/0.999....html
This is not "real" mathematics in the way that professionals approach new ideas. But when they publish papers, they assume that they have done enough work so that someone could prove their statements to this level of detail.
1 ⊢ (k ∈ ℕ → k ∈ ℕ₀) If a number is a positive integer, then it is a natural number.
2 ⊢ 9 ∈ ℝ Nine is a real number.
3 ⊢ 9 ∈ ℂ Therefore, the number nine is a complex number.
4 ⊢ ((9 ∈ ℂ ⋀ (10↑k) ∈ ℂ ⋀ (10↑k) ≠ 0) → (9 / (10↑k)) = (9 · (1 / (10↑k)))) If nine is a complex number, and 10^k is a complex number and 10^k is a non-zero complex number then dividing 9 by 10^k is the same as multiplying 9 and the reciprocal to 10^k.
5 ⊢ (((10↑k) ∈ ℂ ⋀ (10↑k) ≠ 0) → (9 / (10↑k)) = (9 · (1 / (10↑k)))) But we can simplify that statement because nine is a complex number.
6 ⊢ 10 ∈ ℝ Ten is a real number.
7 ⊢ 10 ∈ ℂ Therefore the number ten is a complex number.
8 ⊢ ((10 ∈ ℂ ⋀ k ∈ ℕ₀) → (10↑k) ∈ ℂ) If ten is a complex number and k is a natural number, then 10^k is a complex number.
9 ⊢ (k ∈ ℕ₀ → (10↑k) ∈ ℂ) But we can simplify that statement because 10 is a complex number.
10 ⊢ 0 < 10 Ten is positive.
11 ⊢ 10 ≠ 0 Since ten is a real, positive number, we know it is not zero.
12 ⊢ ((10 ∈ ℂ ⋀ k ∈ ℕ₀ ⋀ 10 ≠ 0) → (10↑k) ≠ 0) If 10 is a non-zero, complex number and k is a natural number and then 10^k is not zero.
13 ⊢ (k ∈ ℕ₀ → (10↑k) ≠ 0) But we can simplify that statement because ten is a non-zero, complex number.
14 ⊢ (k ∈ ℕ₀ → (9 / (10↑k)) = (9 · (1 / (10↑k)))) From what has been seen above, it follows that if k is a natural number, then dividing 9 by 10^k is the same as multiplying 9 and the reciprocal to 10^k.
15 ⊢ ((10 ∈ ℂ ⋀ k ∈ ℕ₀ ⋀ 10 ≠ 0) → ((1 / 10)↑k) = (1 / (10↑k))) If ten is a non-zero complex number, and k is a natural number, then the kth power of the reciprocal of ten is the same as the reciprocal of the kth power of ten.
16 7, 11, 15 mp3an13 910 . . . . . 6 ⊢ (k ∈ ℕ₀ → ((1 / 10)↑k) = (1 / (10↑k))) But we can simplify that statement because ten is a non-zero, complex number.
17 ⊢ (k ∈ ℕ₀ → (9 · ((1 / 10)↑k)) = (9 · (1 / (10↑k)))) And we can multiply on the left by nine.
18 ⊢ (k ∈ ℕ₀ → (9 / (10↑k)) = (9 · ((1 / 10)↑k))) And we can now reformat the equation more.
19 ⊢ (k ∈ ℕ → (9 / (10↑k)) = (9 · ((1 / 10)↑k))) And now that we've proven the general case, all of this follows from the more specific case when k is a positive integer.
20 ⊢ Σ k ∈ ℕ (9 / (10↑k)) = Σ k ∈ ℕ (9 · ((1 / 10)↑k)) So the sum over all positive integers is the same if we write the summand in either of these forms.
21 ⊢ (1 / 10) ∈ ℝ Since ten is a non-zero, real number, its reciprocal is a real number.
22 ⊢ (1 / 10) ∈ ℂ Which is to say it is a complex number.
23 ⊢ 0 ∈ ℝ And zero is a real number.
24 ⊢ 0 < (1 / 10) And since ten is greater than zero, its reciprocal is greater than zero.
25 ⊢ 0 ≤ (1 / 10) And greater than zero means one tenth is also greater-to-or-equal-to zero.
26 ⊢ (0 ≤ (1 / 10) → (abs ‘(1 / 10)) = (1 / 10)) From the definition of absolute value, if one tenth is greater-to-or-equal-to zero then we know it is equal to its absolute value.
27 ⊢ (abs ‘(1 / 10)) = (1 / 10) So this is the case.
28 ⊢ 0 < 9 Nine is a positive number.
29 ⊢ 1 ∈ ℝ One is a real number.
30 ⊢ ((9 ∈ ℝ ⋀ 1 ∈ ℝ) → (0 < 9 ↔ 1 < (9 + 1))) If nine is a real number and one is a real number, then saying nine is positive is the same thing as saying nine plus one is greater than one.
31 ⊢ (0 < 9 ↔ 1 < (9 + 1)) So we say it.
32 ⊢ 1 < (9 + 1) Having said it, we prove nine plus one is greater than 1.
33 ⊢ 10 = (9 + 1) But ten is defined as nine plus one.
34 ⊢ 1 < 10 So we have proved ten is greater than one.
35 ⊢ ((10 ∈ ℝ ⋀ 0 < 10) → (1 < 10 ↔ (1 / 10) < 1)) Now if ten is a positive real number, then saying ten is greater than one is the same as saying the reciprocal of ten is less than one.
36 ⊢ (1 < 10 ↔ (1 / 10) < 1) So we say it.
37 ⊢ (1 / 10) < 1 And so we have proved that one tenth is less than one.
38 ⊢ (abs ‘(1 / 10)) < 1 And, something that will immediately be useful, we have proved that one tenth has an absolute value less than 1.
39 ⊢ ((9 ∈ ℂ ⋀ (1 / 10) ∈ ℂ ⋀ (abs ‘(1 / 10)) < 1) → Σ k ∈ ℕ (9 · ((1 / 10)↑k)) = ((9 · (1 / 10)) / (1 − (1 / 10)))) A general theorem has proven than if nine is a complex number, and one tenth is a complex number with a absolute value less than one, then the sum over all positive integers of nine times the kth power of one tenth is equal to nine times one tenth divided by the quantity one minus one tenth.
40 ⊢ Σ k ∈ ℕ (9 · ((1 / 10)↑k)) = ((9 · (1 / 10)) / (1 − (1 / 10))) But by this point we have proved all the antecedents, and we say: The sum over all positive integers of nine times the kth power of one tenth is equal to nine times one tenth divided by the quantity one minus one tenth.
Algebra to reformat answer:
41 ⊢ (9 / 10) = (9 · (1 / 10)) Nine divided by ten is the same as nine times one tenth.
42 ⊢ (10 · (9 / 10)) = 9 Ten times nine divided by ten is nine. We will need these results later.
43 ⊢ 1 ∈ ℂ One is a complex number.
44 ⊢ (10 · (1 − (1 / 10))) = ((10 · 1) − (10 · (1 / 10))) Ten distributes over subtraction.
45 ⊢ (10 · 1) = 10 Ten times one is ten.
46 ⊢ (10 · (1 / 10)) = 1 Ten times one tenth is one.
47 ⊢ ((10 · 1) − (10 · (1 / 10))) = (10 − 1) Ten times one minus ten times one tenth is ten minus one.
48 ⊢ (1 + 9) = (9 + 1) One plus nine is the same as nine plus one.
49 ⊢ (1 + 9) = 10 So one plus nine is ten.
50 ⊢ (10 − 1) = 9 So ten minus one is nine.
51 ⊢ 9 = (10 · (1 − (1 / 10))) So nine is equal to ten times the quantity one minus one tenth.
52 ⊢ (10 · (9 / 10)) = (10 · (1 − (1 / 10))) So ten times nine over ten is equal to ten times the quantity one minus one tenth.
53 ⊢ (9 / 10) ∈ ℝ Nine over ten is a real number.
54 ⊢ (9 / 10) ∈ ℂ Therefore nine over ten is a complex number.
55 ⊢ (1 − (1 / 10)) ∈ ℂ And one minus one tenth is complex number.
56 ⊢ ((10 · (9 / 10)) = (10 · (1 − (1 / 10))) ↔ (9 / 10) = (1 − (1 / 10))) Since ten is non-zero, we can cancel it from both sides of the equals sign.
57 ⊢ (9 / 10) = (1 − (1 / 10)) So we know nine over ten is the same as one minus one tenth.
58 ⊢ ((9 / 10) / (9 / 10)) = ((9 · (1 / 10)) / (1 − (1 / 10))) So we can see two different ways of writing nine over ten. Dividing one by the other gives the same result no matter how we write it.
59 ⊢ 0 < (9 / 10) But nine over ten is greater than zero.
60 ⊢ (9 / 10) ≠ 0 Therefore it is non-zero.
61 ⊢ ((9 / 10) / (9 / 10)) = 1 So when you divide it by itself the result is one.
62 ⊢ ((9 · (1 / 10)) / (1 − (1 / 10))) = 1 So nine times one tenth divided by the quantity one minus one tenth is equal to one.
63 ⊢ Σ k ∈ ℕ (9 / (10↑k)) = 1 Therefore the sum over all positive integers of nine divided by the kth power of ten is equal to 1.
rpenner
Quote: "Whenever you have two quantities joined by the equals sign, the left and right are equal and interchangeable for all mathematical purposes."
I am under the impression that we are discussion perhaps a little in west and east.
My take is a "physical" approach, where I reach the conclusion that .9... is not identical to 1, and this is according to my own and very simple way of thinking physical.
From a strict mathematical point of view I have no idea, so for all "mathematical purposes" it is for others to discuss.
I think it is fair to assume that numbers can be seen both as virtual numbers - bearing no relationship to physical - and they can be seen as representatives for physical world, respectively.
Mathematically it is meaningless to use the word "identical" - physically it is equally meaningless - and therefore I think that equivalency is a better term (yes - I know that I jump a little around) - so my final version is that 0.9... is not equal to 1 when numbers are being used as representatives for physical world.
Quote: "Whenever you have two quantities joined by the equals sign, the left and right are equal and interchangeable for all mathematical purposes."
I am under the impression that we are discussion perhaps a little in west and east.
My take is a "physical" approach, where I reach the conclusion that .9... is not identical to 1, and this is according to my own and very simple way of thinking physical.
From a strict mathematical point of view I have no idea, so for all "mathematical purposes" it is for others to discuss.
I think it is fair to assume that numbers can be seen both as virtual numbers - bearing no relationship to physical - and they can be seen as representatives for physical world, respectively.
Mathematically it is meaningless to use the word "identical" - physically it is equally meaningless - and therefore I think that equivalency is a better term (yes - I know that I jump a little around) - so my final version is that 0.9... is not equal to 1 when numbers are being used as representatives for physical world.
QUOTE (bukh+Mar 1 2009, 09:39 PM)
My take is a "physical" approach, where I reach the conclusion that .9... is not identical to 1, and this is according to my own and very simple way of thinking physical.
Yes -- 0.999... looks different than 1.
But 0.9... looks different that 0.999... and they also represent the same thing.
Because there is no physicality at all to a literally infinite string of 9's.
But they are just different names for the same thing.
The Number One -- the number of heads on you -- the number of moles on my leg -- the number of bottle caps on this bottle I hold -- is not a physical thing of itself. It's a name for a concept. An infinite string of 9's is not physical at all -- it's a concept, but it's a concept we use when we say 0.999... = 1.
Yes -- 0.999... looks different than 1.
But 0.9... looks different that 0.999... and they also represent the same thing.
Because there is no physicality at all to a literally infinite string of 9's.
But they are just different names for the same thing.
The Number One -- the number of heads on you -- the number of moles on my leg -- the number of bottle caps on this bottle I hold -- is not a physical thing of itself. It's a name for a concept. An infinite string of 9's is not physical at all -- it's a concept, but it's a concept we use when we say 0.999... = 1.
rpenner
"An infinite string of 9's is not physical at all -- it's a concept, but it's a concept we use when we say 0.999... = 1."
OK - then let me put it this way: Can numbers - can math be used to describe physical world accurately ?
"An infinite string of 9's is not physical at all -- it's a concept, but it's a concept we use when we say 0.999... = 1."
OK - then let me put it this way: Can numbers - can math be used to describe physical world accurately ?
Numbers "describe" nothing -- mathmatical models describe how the universe behaves at least well enough to let us predict it, and numbers are key to testing how well these models describe reality.
http://relativity.livingreviews.org/Articles/lrr-2006-3/
http://relativity.livingreviews.org/Articles/lrr-2005-5/
http://relativity.livingreviews.org/Articles/lrr-2003-5/
And this has been true at least since Newton compared the gravity of falling objects on Earth's surface to the movement of the Moon.
http://relativity.livingreviews.org/Articles/lrr-2006-3/
http://relativity.livingreviews.org/Articles/lrr-2005-5/
http://relativity.livingreviews.org/Articles/lrr-2003-5/
And this has been true at least since Newton compared the gravity of falling objects on Earth's surface to the movement of the Moon.
QUOTE (bukh+Mar 1 2009, 10:41 PM)
rpenner
"An infinite string of 9's is not physical at all -- it's a concept, but it's a concept we use when we say 0.999... = 1."
OK - then let me put it this way: Can numbers - can math be used to describe physical world accurately ?
Suppose they didn't.
0.99... would still equal 1.
If you limit yourself to the physical then 0.99... is outside of what you discuss.
If you don't want to discuss something that is not based on physicality then don't.
I think you and StevenA should come up with your own symbology.
maybe 0.99***
"An infinite string of 9's is not physical at all -- it's a concept, but it's a concept we use when we say 0.999... = 1."
OK - then let me put it this way: Can numbers - can math be used to describe physical world accurately ?
Suppose they didn't.
0.99... would still equal 1.
If you limit yourself to the physical then 0.99... is outside of what you discuss.
If you don't want to discuss something that is not based on physicality then don't.
I think you and StevenA should come up with your own symbology.
maybe 0.99***
rpenner
Mathematics is the only arena where anything can be "identical". That Relativity is true 'causes', us to look at each position or particle value . Even allowing for the unlikely "sameness" to be in two places at once, the position is unique in space and time and disqualifies a repeat of the same event. (since the rest of the universe has moved on).
This means that a medium like mathematics which IS an exacting science is limited by the fact that it cannot predict past a certain likelihood....this is why often the conclusion that things are infinite gives a result where no result would be found otherwise.
Cheers
Iseason
Mathematics is the only arena where anything can be "identical". That Relativity is true 'causes', us to look at each position or particle value . Even allowing for the unlikely "sameness" to be in two places at once, the position is unique in space and time and disqualifies a repeat of the same event. (since the rest of the universe has moved on).
This means that a medium like mathematics which IS an exacting science is limited by the fact that it cannot predict past a certain likelihood....this is why often the conclusion that things are infinite gives a result where no result would be found otherwise.
Cheers
Iseason
Iseason
"Mathematics is the only arena where anything can be "identical"."
Thanks - That is actually also what I tried to think - but I have not been clear enough in my way of expressing - and thinking - but this is exactly what it is about.
"Mathematics is the only arena where anything can be "identical"."
Thanks - That is actually also what I tried to think - but I have not been clear enough in my way of expressing - and thinking - but this is exactly what it is about.
QUOTE (bukh+Mar 2 2009, 07:18 AM)
Iseason
"Mathematics is the only arena where anything can be "identical"."
Thanks - That is actually also what I tried to think - but I have not been clear enough in my way of expressing - and thinking - but this is exactly what it is about.
And 0.99... is only in the arena of math.
And if you don't want to deal with the non-physical then you can't deal with 0.99...
And that's your loss.
"Mathematics is the only arena where anything can be "identical"."
Thanks - That is actually also what I tried to think - but I have not been clear enough in my way of expressing - and thinking - but this is exactly what it is about.
And 0.99... is only in the arena of math.
And if you don't want to deal with the non-physical then you can't deal with 0.99...
And that's your loss.
QUOTE (Argyll+Feb 28 2009, 12:38 AM)
That's a pretty big IF there... too bad that's not the definition of irrational numbers.
Even if the definition used is not an 'official' definition, show me a decimal expansion between 0 and 1 that is not included.
Even if the definition used is not an 'official' definition, show me a decimal expansion between 0 and 1 that is not included.
QUOTE (phyti+Mar 2 2009, 02:54 PM)
Even if the definition used is not an 'official' definition, show me a decimal expansion between 0 and 1 that is not included.
Your 'definition' includes all numbers, irrational and rational.
So, your deduction should become:
"0.999... can represent the largest irrational or rational number less than or equal to one."
Which would be correct; one is the largest irrational or rational number less than or equal to one.
___
Alternatively, if you change your 'definition' to:
"all possible infinite non-repeating sequences of digits (base 10) to the right of a decimal point"
then you would be much closer to the definition of an irrational, but obviously would no longer be talking about 0.9r
Your 'definition' includes all numbers, irrational and rational.
So, your deduction should become:
"0.999... can represent the largest irrational or rational number less than or equal to one."
Which would be correct; one is the largest irrational or rational number less than or equal to one.
___
Alternatively, if you change your 'definition' to:
"all possible infinite non-repeating sequences of digits (base 10) to the right of a decimal point"
then you would be much closer to the definition of an irrational, but obviously would no longer be talking about 0.9r
buttershug
"And 0.99... is only in the arena of math.
And if you don't want to deal with the non-physical then you can't deal with 0.99...
And that's your loss."
Loss or loss - I am just convincing myself that infinity is not part of physical world and infinity cannot be used in a meaningful manner when dealing with physical.
"And 0.99... is only in the arena of math.
And if you don't want to deal with the non-physical then you can't deal with 0.99...
And that's your loss."
Loss or loss - I am just convincing myself that infinity is not part of physical world and infinity cannot be used in a meaningful manner when dealing with physical.
If you spent more time understanding simple math and less time "convincing" yourself that a limited grasp of math isn't a hindrance to to your understanding of the physical world, this conversation would be unnecessary.
Granouille
"If you spent more time understanding simple math and less time "convincing" yourself that a limited grasp of math isn't a hindrance to to your understanding of the physical world, this conversation would be unnecessary."
Are you indicating that infinite is a concept solely connected with math - and that the concept cannot be applied in physics ?
"If you spent more time understanding simple math and less time "convincing" yourself that a limited grasp of math isn't a hindrance to to your understanding of the physical world, this conversation would be unnecessary."
Are you indicating that infinite is a concept solely connected with math - and that the concept cannot be applied in physics ?
No. You will infer what you wish, but I stated my point as simply as I could, with you in mind. 
QUOTE (Granouille+Mar 3 2009, 07:09 AM)
No. You will infer what you wish, but I stated my point as simply as I could, with you in mind.
My turn....Are you inferring that two points CAN be identical (remembering we are referring to relativity..
Cheers
Iseason
My turn....Are you inferring that two points CAN be identical (remembering we are referring to relativity..
Cheers
Iseason
Wouldn't that be 'implying'?
And I wasn't talking about relativity. Where did you get that bit?
In case you've forgotten, here is the original post:
Renormalization is not only practical but also necessary in order to predict anything. And accuracy will always be limited by the lack of sufficient calculating power -
So Yes - I agree with you that "this is why often the conclusion that things are infinite gives a result where no result would be found otherwise"
he bukh, iseason
Exactly. That is what you have symbolic computing for. To deal with infinities easily without dealing with them computationally.
That is also why you have projective geometry for- to deal with incidences which are exact instead of comparing distances which are practically non-computable.
Mathematically it is simple , but how does Universe perform limit taking in practice, physically? Since Universe does compute infinite limits somehow.
Like 1 is "understood" as the equivalent to 0.9...
Whether or not they are equal is irrelevant to the fact that all the 9s are already there.
Personally, I've seen compelling arguments both ways, so I'm not 100% sure either way, but I'm leaning towards the "equals" side.
And I wasn't talking about relativity. Where did you get that bit?
In case you've forgotten, here is the original post:
QUOTE (iseason+May 12 2008, 03:23 AM)
I had a discussion in a forum with a couple of maths guys here , but felt the answer was still somewhat unresolved. I've seen posts on "what is the biggest number?".
I still think infinity (in actuality) is impossible. Here's why.
If infinity existed (in any form) then the outer borders and the inner borders don't exist. Therefore there would be no friction in the middle to create any variance whatsoever. The fact that we can measure at all proves that infinity is false.
thoughts?
Cheers
Iseason
So.
I still think infinity (in actuality) is impossible. Here's why.
If infinity existed (in any form) then the outer borders and the inner borders don't exist. Therefore there would be no friction in the middle to create any variance whatsoever. The fact that we can measure at all proves that infinity is false.
thoughts?
Cheers
Iseason
So.
Granouille
Question: "Are you indicating that infinite is a concept solely connected with math - and that the concept cannot be applied in physics"
Answer: "No. You will infer what you wish, but I stated my point as simply as I could, with you in mind."
This gives me an impression of "trick-answering" -
So let me see if I ask the question like this:
"Is infinite a concept that can be used in the definition - description - communicating of physical world ? "
Question: "Are you indicating that infinite is a concept solely connected with math - and that the concept cannot be applied in physics"
Answer: "No. You will infer what you wish, but I stated my point as simply as I could, with you in mind."
This gives me an impression of "trick-answering" -
So let me see if I ask the question like this:
"Is infinite a concept that can be used in the definition - description - communicating of physical world ? "
Bravo! You did ask the question like that.
What the hell are you talking about? What is "definition-description-communicating"?
What the hell are you talking about? What is "definition-description-communicating"?
QUOTE (bukh+Mar 2 2009, 05:44 PM)
Granouille
"If you spent more time understanding simple math and less time "convincing" yourself that a limited grasp of math isn't a hindrance to to your understanding of the physical world, this conversation would be unnecessary."
Are you indicating that infinite is a concept solely connected with math - and that the concept cannot be applied in physics ?
The infinitely big is a concept we cannot physically deal with. But the infinitely small shows up as a natural extension to the concept of infinity and applies to a number line or to Einstein's space-time continuum. For instance if the odds for something is infinitely small it can never repeat itself.
Mitch Raemsch
"If you spent more time understanding simple math and less time "convincing" yourself that a limited grasp of math isn't a hindrance to to your understanding of the physical world, this conversation would be unnecessary."
Are you indicating that infinite is a concept solely connected with math - and that the concept cannot be applied in physics ?
The infinitely big is a concept we cannot physically deal with. But the infinitely small shows up as a natural extension to the concept of infinity and applies to a number line or to Einstein's space-time continuum. For instance if the odds for something is infinitely small it can never repeat itself.
Mitch Raemsch
You seem to do alright with an infinitely big mouth, given your infinitely small mind. Spammer.
Granouille
"What the hell are you talking about? What is "definition-description-communicating"?
Well - I am afraid that I am wasting your time - spent your genius on something better - your presence in this thread is at the best pathetic.
It would seem pretty obvious that you have undertaken the role of being the clown.
"What the hell are you talking about? What is "definition-description-communicating"?
Well - I am afraid that I am wasting your time - spent your genius on something better - your presence in this thread is at the best pathetic.
It would seem pretty obvious that you have undertaken the role of being the clown.
Something seems obvious to you? 
Incredible on the face of it, considering your inability to do arithmetic.
And you still got it wrong.
Incredible on the face of it, considering your inability to do arithmetic.And you still got it wrong.
QUOTE (Granouille+Mar 2 2009, 11:17 PM)
You seem to do alright with an infinitely big mouth, given your infinitely small mind.
A number line is an example of something composed of the infinitely small. There simply is no infinitely big.
A number line is an example of something composed of the infinitely small. There simply is no infinitely big.
QUOTE (Cusa+Mar 2 2009, 10:58 PM)
The infinitely big is a concept we cannot physically deal with. But the infinitely small shows up as a natural extension to the concept of infinity and applies to a number line or to Einstein's space-time continuum. For instance if the odds for something is infinitely small it can never repeat itself.
Mitch Raemsch
If the odds of something are infinitly small then it can not happen.
And if it can happen once it can happen again. There is no such thing as can only happen once when talking about a probability of something.
Mitch Raemsch
If the odds of something are infinitly small then it can not happen.
And if it can happen once it can happen again. There is no such thing as can only happen once when talking about a probability of something.
QUOTE (buttershug+Mar 3 2009, 03:55 AM)
If the odds of something are infinitly small then it can not happen.
And if it can happen once it can happen again. There is no such thing as can only happen once when talking about a probability of something.
You can pick points on a number line forever and they will never repeat and will never exhaust.
Take a knob. Position it. Reposition it. You can never get it back exactly at either of those infinitesimal or infinitely small places.
Mitch Raemsch
And if it can happen once it can happen again. There is no such thing as can only happen once when talking about a probability of something.
You can pick points on a number line forever and they will never repeat and will never exhaust.
Take a knob. Position it. Reposition it. You can never get it back exactly at either of those infinitesimal or infinitely small places.
Mitch Raemsch
QUOTE (Cusa+Mar 3 2009, 05:04 PM)
You can pick points on a number line forever and they will never repeat and will never exhaust.
Take a knob. Position it. Reposition it. You can never get it back exactly at either of those infinitesimal or infinitely small places.
Mitch Raemsch
Hi Mitch.
your usage of infinite here are exactly my point. You have NO evidence that that statement is true. If you meant "unmeasurable", then I could agree, But I'm quite sure the general usage being applied is "never ending".......Which you cannot prove...Only assume.
Cheers
Iseason
Take a knob. Position it. Reposition it. You can never get it back exactly at either of those infinitesimal or infinitely small places.
Mitch Raemsch
Hi Mitch.
your usage of infinite here are exactly my point. You have NO evidence that that statement is true. If you meant "unmeasurable", then I could agree, But I'm quite sure the general usage being applied is "never ending".......Which you cannot prove...Only assume.
Cheers
Iseason
Hi all
just to clarify my last post. I am very much of the opinion that you can never return to a like point. Many of my posts are clearly stating that a position can only be used once. This is quite different from the "chances" of finding an exact position again. So although I agree that the chances of finding the position again are NIL, this is different from leaving an infinite window of opportunity to do so.
Cheers
Iseason
just to clarify my last post. I am very much of the opinion that you can never return to a like point. Many of my posts are clearly stating that a position can only be used once. This is quite different from the "chances" of finding an exact position again. So although I agree that the chances of finding the position again are NIL, this is different from leaving an infinite window of opportunity to do so.
Cheers
Iseason
Granouille
"Something seems obvious to you? Incredible on the face of it, considering your inability to do arithmetic."
It does not take a master in arithmetic to see that you behave like a clown.
"Something seems obvious to you? Incredible on the face of it, considering your inability to do arithmetic."
It does not take a master in arithmetic to see that you behave like a clown.
QUOTE (Cusa+Mar 3 2009, 04:04 AM)
You can pick points on a number line forever and they will never repeat and will never exhaust.
Take a knob. Position it. Reposition it. You can never get it back exactly at either of those infinitesimal or infinitely small places.
Mitch Raemsch
Then you are not picking them randomly.
Take a knob. Position it. Reposition it. You can never get it back exactly at either of those infinitesimal or infinitely small places.
Mitch Raemsch
Then you are not picking them randomly.
iseason
QUOTE: "This means that a medium like mathematics which IS an exacting science is limited by the fact that it cannot predict past a certain likelihood....this is why often the conclusion that things are infinite gives a result where no result would be found otherwise."
You say that math is limited by the fact that "it cannot predict past a certain likelihood --"
I would like to say that math as a system probably would be able to calculate the probability to last decimal - provided that there was sufficient amount of calculating power at hand - but there is not - for very obvious reasons there is not - because that would imply that the poor person faced with such a calculating task would need the entire universal calculating power - smallest unit and highest pace - or close to if we provide the person with say a period of a life-span to do it.
Renormalization is not only practical but also necessary in order to predict anything. And accuracy will always be limited by the lack of sufficient calculating power -
So Yes - I agree with you that "this is why often the conclusion that things are infinite gives a result where no result would be found otherwise"
QUOTE: "This means that a medium like mathematics which IS an exacting science is limited by the fact that it cannot predict past a certain likelihood....this is why often the conclusion that things are infinite gives a result where no result would be found otherwise."
You say that math is limited by the fact that "it cannot predict past a certain likelihood --"
I would like to say that math as a system probably would be able to calculate the probability to last decimal - provided that there was sufficient amount of calculating power at hand - but there is not - for very obvious reasons there is not - because that would imply that the poor person faced with such a calculating task would need the entire universal calculating power - smallest unit and highest pace - or close to if we provide the person with say a period of a life-span to do it.
Renormalization is not only practical but also necessary in order to predict anything. And accuracy will always be limited by the lack of sufficient calculating power -
So Yes - I agree with you that "this is why often the conclusion that things are infinite gives a result where no result would be found otherwise"
QUOTE (bukh+Mar 3 2009, 01:12 PM)
Renormalization is not only practical but also necessary in order to predict anything. And accuracy will always be limited by the lack of sufficient calculating power -
So Yes - I agree with you that "this is why often the conclusion that things are infinite gives a result where no result would be found otherwise"
he bukh, iseason
Exactly. That is what you have symbolic computing for. To deal with infinities easily without dealing with them computationally.
That is also why you have projective geometry for- to deal with incidences which are exact instead of comparing distances which are practically non-computable.
Mathematically it is simple , but how does Universe perform limit taking in practice, physically? Since Universe does compute infinite limits somehow.
Concerning infinity:
Because there is no actual experience to form a basis for a concept, it is pure imagination. Examining the current constructs, we find them to be extrapolations of finite ideas, i.e. finite elements manipulated with finite operations, but requiring an 'infinite amount of time' (whatever that is).
Concerning infinite decimal expansions involving 1 digit:
1/3 .3333 .00003R
2/3 .6666 .00006R
3/3 .9999 .00009R
The 3 columns show the correspondence of fractional addition to decimal addition.
In practice the decimal equivalent is truncated for reasons such as time constraints, device memory, precision specs, etc. The decimal is therefore less than equivalent by the value in column 3. Despite this fact the decimal mode still maintains a consistent correspondence with the fractions.
If the last element in column 3 was assumed to be equivalent to .0001, what adjustments could be made to the remaining elements in column 3 to maintain consistency within the decimals?
Concerning limits:
The limit for an infinite decimal expansion as discussed, is by definition, a boundary value that the expansion converges on but never reaches.
If we assume at some arbitrary step k, the expansion d = b the limit, then at step k+1, d > b, because each increment is >0. This contradicts the definition (at both steps), therefore the assumption is incorrect.
Reality check:
How many people do you know who divide 9 by 9, and get .999R?
Because there is no actual experience to form a basis for a concept, it is pure imagination. Examining the current constructs, we find them to be extrapolations of finite ideas, i.e. finite elements manipulated with finite operations, but requiring an 'infinite amount of time' (whatever that is).
Concerning infinite decimal expansions involving 1 digit:
1/3 .3333 .00003R
2/3 .6666 .00006R
3/3 .9999 .00009R
The 3 columns show the correspondence of fractional addition to decimal addition.
In practice the decimal equivalent is truncated for reasons such as time constraints, device memory, precision specs, etc. The decimal is therefore less than equivalent by the value in column 3. Despite this fact the decimal mode still maintains a consistent correspondence with the fractions.
If the last element in column 3 was assumed to be equivalent to .0001, what adjustments could be made to the remaining elements in column 3 to maintain consistency within the decimals?
Concerning limits:
The limit for an infinite decimal expansion as discussed, is by definition, a boundary value that the expansion converges on but never reaches.
If we assume at some arbitrary step k, the expansion d = b the limit, then at step k+1, d > b, because each increment is >0. This contradicts the definition (at both steps), therefore the assumption is incorrect.
Reality check:
How many people do you know who divide 9 by 9, and get .999R?
Hi Ivars - iseason
Ivars: - yes - "mathematically is is simple to handle the concept of infinity - and how does universe perform limit taking in practice, physically? Since Universe does compute infinite limits somehow - "
I have difficulties in seeing how Universe does compute infinite limits somehow or anyhow.
Infinite limit is the equivalent of nil information,
I favor the idea that physical per definition cannot originate from nil information -
Instead, I stick to this idea (obsession ) that origin condition in terms of "PHYSICAL" is "object of sameness" - there must be a volume in order to carry information, information about how universe develops - out from a template-like origin. A holographic principle.
And there must be something that can account for "ENERGY"
Energy must be be available infinitely - and energy must be available in exact amounts and tuned to fit circumstances.
Object of sameness can solve the question about infinite source of energy delivered in finely tuned amounts exactly adjusted to need - because energy and information is being founded in same principles.
Energy is fundamentally founded on the principle of best fit - how smallest objects of sameness arrange and re-arrange themselves in space to achieve least free void or to achieve most evenly distribution of free void.
The principles outlined by Close suggest a very simple cubic object as smallest - and such an object carry no information - and cannot serve as a template for a holographic construct and cannot serve as an infinite source of energy via best fit - because best fit will be achieved immediately.
String theories are matrix descriptions and limited in the same way (if I understand rightly) - and matrix is fundamentally defined out from a simple quadrant principle - so every time a dimension or a scaling down is being performed - it is loss of information and eventually ends up in a simple information-depleted smallest object.
I like to repeat what I have said previously - "it takes a 3D object of sameness to create a 0D - 1D and 2D". Non-dimensionless points - lines and areas are mind-concepts created by a mind that owes its "existence" to the availability of 3D objects.
Ivars: - yes - "mathematically is is simple to handle the concept of infinity - and how does universe perform limit taking in practice, physically? Since Universe does compute infinite limits somehow - "
I have difficulties in seeing how Universe does compute infinite limits somehow or anyhow.
Infinite limit is the equivalent of nil information,
I favor the idea that physical per definition cannot originate from nil information -
Instead, I stick to this idea (obsession
) that origin condition in terms of "PHYSICAL" is "object of sameness" - there must be a volume in order to carry information, information about how universe develops - out from a template-like origin. A holographic principle.And there must be something that can account for "ENERGY"
Energy must be be available infinitely - and energy must be available in exact amounts and tuned to fit circumstances.
Object of sameness can solve the question about infinite source of energy delivered in finely tuned amounts exactly adjusted to need - because energy and information is being founded in same principles.
Energy is fundamentally founded on the principle of best fit - how smallest objects of sameness arrange and re-arrange themselves in space to achieve least free void or to achieve most evenly distribution of free void.
The principles outlined by Close suggest a very simple cubic object as smallest - and such an object carry no information - and cannot serve as a template for a holographic construct and cannot serve as an infinite source of energy via best fit - because best fit will be achieved immediately.
String theories are matrix descriptions and limited in the same way (if I understand rightly) - and matrix is fundamentally defined out from a simple quadrant principle - so every time a dimension or a scaling down is being performed - it is loss of information and eventually ends up in a simple information-depleted smallest object.
I like to repeat what I have said previously - "it takes a 3D object of sameness to create a 0D - 1D and 2D". Non-dimensionless points - lines and areas are mind-concepts created by a mind that owes its "existence" to the availability of 3D objects.
QUOTE (phyti+Mar 3 2009, 04:37 PM)
In practice the decimal equivalent is truncated for reasons such as time constraints, device memory, precision specs, etc.
But it doesn't have to be truncated.
You can say 0.99...
Where the "..." represent the infinite amount of 9's that follow the two shown.
0.99... does not represent .99 with 9's being added to it. The other 9's don't need to be shown to be understood that they are all there. (all infinity of them).
They are understood to be there without having to be shown.
But it doesn't have to be truncated.
You can say 0.99...
Where the "..." represent the infinite amount of 9's that follow the two shown.
0.99... does not represent .99 with 9's being added to it. The other 9's don't need to be shown to be understood that they are all there. (all infinity of them).
They are understood to be there without having to be shown.
QUOTE (buttershug+Mar 3 2009, 05:25 PM)
They are understood to be there without having to be shown.
Or, NOT understood, as the case seems to be with many posters on this forum...
Or, NOT understood, as the case seems to be with many posters on this forum...
QUOTE (buttershug+Mar 3 2009, 11:50 AM)
Then you are not picking them randomly.
picking them at all is random.
picking them at all is random.
Argyll
"Or, NOT understood, as the case seems to be with many posters on this forum..."
Like 1 is "understood" as the equivalent to 0.9...
"Or, NOT understood, as the case seems to be with many posters on this forum..."
Like 1 is "understood" as the equivalent to 0.9...
QUOTE (bukh+Mar 3 2009, 06:13 PM)
Like 1 is "understood" as the equivalent to 0.9...
Whether or not they are equal is irrelevant to the fact that all the 9s are already there.
Personally, I've seen compelling arguments both ways, so I'm not 100% sure either way, but I'm leaning towards the "equals" side.
QUOTE (Argyll+Mar 3 2009, 06:35 PM)
Whether or not they are equal is irrelevant to the fact that all the 9s are already there.
Personally, I've seen compelling arguments both ways, so I'm not 100% sure either way, but I'm leaning towards the "equals" side.
If it's not equal to 1 then it is not 0.99...
Because if they are not equal then what is the difference?
The only difference can be at the end of the 9's.
But by very definition there is no end.
They are already there by definition.
Personally, I've seen compelling arguments both ways, so I'm not 100% sure either way, but I'm leaning towards the "equals" side.
If it's not equal to 1 then it is not 0.99...
Because if they are not equal then what is the difference?
The only difference can be at the end of the 9's.
But by very definition there is no end.
They are already there by definition.
QUOTE (phyti+Mar 3 2009, 05:37 PM)
If we assume at some arbitrary step k, the expansion d = b the limit
Why would anyone do that? Sounds a lot like the kind of argument StevenA would make, and is pretty daft.
phyti, you seem to have a good grasp of some physics concepts, how come you are struggling so much with this? Even the style of your arguments on this subject have slipped from fairly reasoned and sensible to ridiculousness like this.
Why would anyone do that? Sounds a lot like the kind of argument StevenA would make, and is pretty daft.
phyti, you seem to have a good grasp of some physics concepts, how come you are struggling so much with this? Even the style of your arguments on this subject have slipped from fairly reasoned and sensible to ridiculousness like this.
QUOTE (buttershug+Mar 3 2009, 07:00 PM)
If it's not equal to 1 then it is not 0.99...
Because if they are not equal then what is the difference?
.9 repeating is short by the infinitely small nonzero.
Because if they are not equal then what is the difference?
.9 repeating is short by the infinitely small nonzero.
QUOTE (Cusa+Mar 3 2009, 08:04 PM)
infinitely small nonzero.
No such Real number.
No such Real number.
QUOTE (bm1957+Mar 3 2009, 07:05 PM)
No such Real number.
It is well defined. It is a concept; like infinity is only a concept.
Mitch Raemsch
It is well defined. It is a concept; like infinity is only a concept.
Mitch Raemsch
QUOTE (Cusa+Mar 3 2009, 07:04 PM)
.9 repeating is short by the infinitely small nonzero.
Good thing we are not talking about that then.
I was talking about 0.99... which has no infinitely small nonzero.
Good thing we are not talking about that then.
I was talking about 0.99... which has no infinitely small nonzero.
QUOTE (buttershug+Mar 3 2009, 07:25 PM)
I was talking about 0.99... which has no infinitely small nonzero.
We are talking about what is short of One by the infinitely small nonzero.
Mitch Raemsch
We are talking about what is short of One by the infinitely small nonzero.
Mitch Raemsch
QUOTE (bukh+Mar 4 2009, 06:15 AM)
Hi Ivars - iseason
Ivars: - yes - "mathematically is is simple to handle the concept of infinity - and how does universe perform limit taking in practice, physically? Since Universe does compute infinite limits somehow - "
I have difficulties in seeing how Universe does compute infinite limits somehow or anyhow.
Infinite limit is the equivalent of nil information,
I favor the idea that physical per definition cannot originate from nil information -
Instead, I stick to this idea (obsession ) that origin condition in terms of "PHYSICAL" is "object of sameness" - there must be a volume in order to carry information, information about how universe develops - out from a template-like origin. A holographic principle.
And there must be something that can account for "ENERGY"
Energy must be be available infinitely - and energy must be available in exact amounts and tuned to fit circumstances.
Object of sameness can solve the question about infinite source of energy delivered in finely tuned amounts exactly adjusted to need - because energy and information is being founded in same principles.
Energy is fundamentally founded on the principle of best fit - how smallest objects of sameness arrange and re-arrange themselves in space to achieve least free void or to achieve most evenly distribution of free void.
The principles outlined by Close suggest a very simple cubic object as smallest - and such an object carry no information - and cannot serve as a template for a holographic construct and cannot serve as an infinite source of energy via best fit - because best fit will be achieved immediately.
String theories are matrix descriptions and limited in the same way (if I understand rightly) - and matrix is fundamentally defined out from a simple quadrant principle - so every time a dimension or a scaling down is being performed - it is loss of information and eventually ends up in a simple information-depleted smallest object.
I like to repeat what I have said previously - "it takes a 3D object of sameness to create a 0D - 1D and 2D". Non-dimensionless points - lines and areas are mind-concepts created by a mind that owes its "existence" to the availability of 3D objects.
Hi Bukh,Ivars et all.....
Heaps of good debate.
Bukh.
Perhaps we are discussing a universe which never had a "smallest object/division". I know I have been using the term at length in this forum, but maybe we are discussing 'shades of variation' of the largest object. from where we are set , this would of course seem to be an effect of one quanta at a time. However there is no real way for us to know this for sure. It could well be that every variation is equal but EVENLY distributed . In effect this would equal a scale universe which only ever contained the same amount of variation (mass/energy), but varied where it could be found (motion of mass/changes in energy).
i am again drawn to the 9 verses 10 argument as to methodology and am better convinced by this than that there is a "smallest energy". I prefer energy to be universally consistent in order to satisfy cause and effect. If nothing "changed", then there was no first cause.....no God particle..
Variation on exact scale universally does not conflict with our ability to measure based on variables because universally the measure is always equal, but locally/relatively, the information is read at different levels..
gravity is a case in point..our gravity is a collaboration of local information over universal, but must comply universally by default.
Cheers
Iseason
Ivars: - yes - "mathematically is is simple to handle the concept of infinity - and how does universe perform limit taking in practice, physically? Since Universe does compute infinite limits somehow - "
I have difficulties in seeing how Universe does compute infinite limits somehow or anyhow.
Infinite limit is the equivalent of nil information,
I favor the idea that physical per definition cannot originate from nil information -
Instead, I stick to this idea (obsession
) that origin condition in terms of "PHYSICAL" is "object of sameness" - there must be a volume in order to carry information, information about how universe develops - out from a template-like origin. A holographic principle.And there must be something that can account for "ENERGY"
Energy must be be available infinitely - and energy must be available in exact amounts and tuned to fit circumstances.
Object of sameness can solve the question about infinite source of energy delivered in finely tuned amounts exactly adjusted to need - because energy and information is being founded in same principles.
Energy is fundamentally founded on the principle of best fit - how smallest objects of sameness arrange and re-arrange themselves in space to achieve least free void or to achieve most evenly distribution of free void.
The principles outlined by Close suggest a very simple cubic object as smallest - and such an object carry no information - and cannot serve as a template for a holographic construct and cannot serve as an infinite source of energy via best fit - because best fit will be achieved immediately.
String theories are matrix descriptions and limited in the same way (if I understand rightly) - and matrix is fundamentally defined out from a simple quadrant principle - so every time a dimension or a scaling down is being performed - it is loss of information and eventually ends up in a simple information-depleted smallest object.
I like to repeat what I have said previously - "it takes a 3D object of sameness to create a 0D - 1D and 2D". Non-dimensionless points - lines and areas are mind-concepts created by a mind that owes its "existence" to the availability of 3D objects.
Hi Bukh,Ivars et all.....
Heaps of good debate.
Bukh.
Perhaps we are discussing a universe which never had a "smallest object/division". I know I have been using the term at length in this forum, but maybe we are discussing 'shades of variation' of the largest object. from where we are set , this would of course seem to be an effect of one quanta at a time. However there is no real way for us to know this for sure. It could well be that every variation is equal but EVENLY distributed . In effect this would equal a scale universe which only ever contained the same amount of variation (mass/energy), but varied where it could be found (motion of mass/changes in energy).
i am again drawn to the 9 verses 10 argument as to methodology and am better convinced by this than that there is a "smallest energy". I prefer energy to be universally consistent in order to satisfy cause and effect. If nothing "changed", then there was no first cause.....no God particle..
Variation on exact scale universally does not conflict with our ability to measure based on variables because universally the measure is always equal, but locally/relatively, the information is read at different levels..
gravity is a case in point..our gravity is a collaboration of local information over universal, but must comply universally by default.
Cheers
Iseason
It is in fact possiblle to randomly select from an infinite list.
The key is to recognize that any such selection will be biased towards small numbers, but a biased selection is still a random one none the less.
Consider the following experiment.
An atom with a half life of one minute, that undergoes an Alpha decay is contained within a chamber.
The walls of the chamber are capable of detecting the emission of the Alpha particle.
The chamber is hardwired to a number generator that ouputs one random number between 0 and 9 every second, storing the string of numbers produced.
When the chamber detects that the atom has decayed, it outputs the string as a number.
The machine has a 50% chance of producing a 60 digit number, a 25% chance of producing a 120 digit number, a 12.5% chance of producing a 180 digit number and so on and so forth, so, the machine is biased towards producing short numbers.
However.
If the machine produces a number after 1 half life (ie, a 60 digit number) then it's just as likely to produce 000000000000000000000000000000000000000000000000000000000000 as it is to produce 999999999999999999999999999999999999999999999999999999999999 as it is to produce 528452050871912763517683940057261527311983672637281983214783
The length of the number is randomized, the number is randomized, and the machine has the potential to operate for any period of time.
And the correct answer to "What are the chances of randomly selecting the same number twice from an infinite pool" is "Kronecker delta".
The key is to recognize that any such selection will be biased towards small numbers, but a biased selection is still a random one none the less.
Consider the following experiment.
An atom with a half life of one minute, that undergoes an Alpha decay is contained within a chamber.
The walls of the chamber are capable of detecting the emission of the Alpha particle.
The chamber is hardwired to a number generator that ouputs one random number between 0 and 9 every second, storing the string of numbers produced.
When the chamber detects that the atom has decayed, it outputs the string as a number.
The machine has a 50% chance of producing a 60 digit number, a 25% chance of producing a 120 digit number, a 12.5% chance of producing a 180 digit number and so on and so forth, so, the machine is biased towards producing short numbers.
However.
If the machine produces a number after 1 half life (ie, a 60 digit number) then it's just as likely to produce 000000000000000000000000000000000000000000000000000000000000 as it is to produce 999999999999999999999999999999999999999999999999999999999999 as it is to produce 528452050871912763517683940057261527311983672637281983214783
The length of the number is randomized, the number is randomized, and the machine has the potential to operate for any period of time.
And the correct answer to "What are the chances of randomly selecting the same number twice from an infinite pool" is "Kronecker delta".
Trippy
QUOTE:
- "It is in fact possible to randomly select from an infinite list."
- "The length of the number is randomized, the number is randomized, and the machine has the potential to operate for any period of time."
Yes the machine has the POTENTIAL to operate for any period of time
but the number is not being taken from an infinite source - the source is being limited by the limitation put by time -
mathematical concepts are pure mind concepts - anything is possible as long as it is in accordance with set axioms - infinity is no problem.
unlike situations where a physical condition is being involved - then suddenly there is a co-player - a co-player that require that the defined set frame can be realized. A potential is not good enough to qualify for infinity as long as it is not being fully realized. There is no guarantee that Alpha decay can continue infinitely.
QUOTE:
- "It is in fact possible to randomly select from an infinite list."
- "The length of the number is randomized, the number is randomized, and the machine has the potential to operate for any period of time."
Yes the machine has the POTENTIAL to operate for any period of time
but the number is not being taken from an infinite source - the source is being limited by the limitation put by time -
mathematical concepts are pure mind concepts - anything is possible as long as it is in accordance with set axioms - infinity is no problem.
unlike situations where a physical condition is being involved - then suddenly there is a co-player - a co-player that require that the defined set frame can be realized. A potential is not good enough to qualify for infinity as long as it is not being fully realized. There is no guarantee that Alpha decay can continue infinitely.
QUOTE (bm1957+Mar 3 2009, 03:03 PM)
Why would anyone do that? Sounds a lot like the kind of argument StevenA would make, and is pretty daft.
phyti, you seem to have a good grasp of some physics concepts, how come you are struggling so much with this? Even the style of your arguments on this subject have slipped from fairly reasoned and sensible to ridiculousness like this.
The reasoning enforces the definition of limit. The definition does not state the function equals the limit, for if it did, it would not be true.
Fractions like 1/3, 2/7, etc. produce repeating patterns, because the denominators and 10 are relatively prime. The fraction 9/9 as an integer, does not produce the repeating pattern .9R, therefore 1 does not produce it either. This is in total agreement with the limit statement that .9R does not equal 1. You can round it up if you wish.
When calculations are done, the theoretical sequences are never used, only appoximations, and then the values are rounded for the desired precision.
All measurements and all computers have limited precision.
If I'm making a wheel for an ox cart, pi=3 is sufficient.
It seems StevenA has left the building.
phyti, you seem to have a good grasp of some physics concepts, how come you are struggling so much with this? Even the style of your arguments on this subject have slipped from fairly reasoned and sensible to ridiculousness like this.
The reasoning enforces the definition of limit. The definition does not state the function equals the limit, for if it did, it would not be true.
Fractions like 1/3, 2/7, etc. produce repeating patterns, because the denominators and 10 are relatively prime. The fraction 9/9 as an integer, does not produce the repeating pattern .9R, therefore 1 does not produce it either. This is in total agreement with the limit statement that .9R does not equal 1. You can round it up if you wish.
When calculations are done, the theoretical sequences are never used, only appoximations, and then the values are rounded for the desired precision.
All measurements and all computers have limited precision.
If I'm making a wheel for an ox cart, pi=3 is sufficient.
It seems StevenA has left the building.
I would like to look at the lower limit. It is the concept of the infinitey small. It is the next thing to zero without actually being.
You can have an infinite amount of the infinitely small and you would have a finite.
Mitch Raemsch
You can have an infinite amount of the infinitely small and you would have a finite.
Mitch Raemsch
QUOTE (Cusa+Mar 5 2009, 12:53 AM)
I would like to look at the lower limit. It is the concept of the infinitey small. It is the next thing to zero without actually being.
You can have an infinite amount of the infinitely small and you would have a finite.
Mitch Raemsch
Genius.
You can have an infinite amount of the infinitely small and you would have a finite.
Mitch Raemsch
Genius.
QUOTE (bukh+Mar 5 2009, 10:44 AM)
Trippy
QUOTE:
- "It is in fact possible to randomly select from an infinite list."
- "The length of the number is randomized, the number is randomized, and the machine has the potential to operate for any period of time."
Yes the machine has the POTENTIAL to operate for any period of time
but the number is not being taken from an infinite source - the source is being limited by the limitation put by time -
mathematical concepts are pure mind concepts - anything is possible as long as it is in accordance with set axioms - infinity is no problem.
unlike situations where a physical condition is being involved - then suddenly there is a co-player - a co-player that require that the defined set frame can be realized. A potential is not good enough to qualify for infinity as long as it is not being fully realized. There is no guarantee that Alpha decay can continue infinitely.
It only needs to have the potential to do so.
The number is being chosen from an infinite source that's biased towards numbers.
With the above setup, I can tell you exactly what the probability of selecting any number between 0 and megistron (and beyond) is.
With the above setup, it does not matter how large your number is, I can assign a probability to it.
QUOTE:
- "It is in fact possible to randomly select from an infinite list."
- "The length of the number is randomized, the number is randomized, and the machine has the potential to operate for any period of time."
Yes the machine has the POTENTIAL to operate for any period of time
but the number is not being taken from an infinite source - the source is being limited by the limitation put by time -
mathematical concepts are pure mind concepts - anything is possible as long as it is in accordance with set axioms - infinity is no problem.
unlike situations where a physical condition is being involved - then suddenly there is a co-player - a co-player that require that the defined set frame can be realized. A potential is not good enough to qualify for infinity as long as it is not being fully realized. There is no guarantee that Alpha decay can continue infinitely.
It only needs to have the potential to do so.
The number is being chosen from an infinite source that's biased towards numbers.
With the above setup, I can tell you exactly what the probability of selecting any number between 0 and megistron (and beyond) is.
With the above setup, it does not matter how large your number is, I can assign a probability to it.
Trippy
"It only needs to have the potential to do so.
The number is being chosen from an infinite source that's biased towards numbers.
With the above setup, I can tell you exactly what the probability of selecting any number between 0 and megistron (and beyond) is.
With the above setup, it does not matter how large your number is, I can assign a probability to it."
Well - I have a different opinion.
You can tell the exact probability once the operation for said probability has been executed - and Kronecker delta is being based upon quantized conditions - discreteness - and as such disqualifying infinite.
To me it is valid that infinity and continuous cannot be implemented into a physical world - and cannot be implemented in thought experiments that is being based upon physical conditions.
"It only needs to have the potential to do so.
The number is being chosen from an infinite source that's biased towards numbers.
With the above setup, I can tell you exactly what the probability of selecting any number between 0 and megistron (and beyond) is.
With the above setup, it does not matter how large your number is, I can assign a probability to it."
Well - I have a different opinion.
You can tell the exact probability once the operation for said probability has been executed - and Kronecker delta is being based upon quantized conditions - discreteness - and as such disqualifying infinite.
To me it is valid that infinity and continuous cannot be implemented into a physical world - and cannot be implemented in thought experiments that is being based upon physical conditions.
Hi all.
This is exactly what this thread is about. Can mathematics be applied in a physical world where the concepts like "now" are fixed points that are supposed to appear in a universe where no fixed point would be possible(if infinite).
If the universe was indeed infinite, then how could you be relative to anything when the "probability" of any random point in space was equally likely. This seems at total odds with the flow of time and space. More inclining is a closed system that allows each point an order, even if they don't appear that way in review.
I'm not opposed to the universe as being laid down according to a "probability scale", in fact I'm most in favor of it. It suggests order, since each point has the same chance of being there. That also suggests that energy is evenly distributed, but not as in time and space(for that is a secondary measure). however it needs to be a closed system to work. Infinity as far as the physical universe goes is a "physical impossibility".
Cheers
Iseason
This is exactly what this thread is about. Can mathematics be applied in a physical world where the concepts like "now" are fixed points that are supposed to appear in a universe where no fixed point would be possible(if infinite).
If the universe was indeed infinite, then how could you be relative to anything when the "probability" of any random point in space was equally likely. This seems at total odds with the flow of time and space. More inclining is a closed system that allows each point an order, even if they don't appear that way in review.
I'm not opposed to the universe as being laid down according to a "probability scale", in fact I'm most in favor of it. It suggests order, since each point has the same chance of being there. That also suggests that energy is evenly distributed, but not as in time and space(for that is a secondary measure). however it needs to be a closed system to work. Infinity as far as the physical universe goes is a "physical impossibility".
Cheers
Iseason
Iseason
"I'm not opposed to the universe as being laid down according to a "probability scale", in fact I'm most in favor of it. It suggests order, since each point has the same chance of being there."
I like to put it a bit differently.
When you say that each point has the same chance of being there - I get the impression that this is related to the way that we see the space-time STRUCTURE, as an ideal ordered 3D arrangement - a lattice - and probability is not in play.
But the WAY that this structure express itself has to do with probabilities - in that all physical expressions is being founded in the configuration of the "points" of this lattice - in the ever shifting configurations.
And now probability can be used in forecast calculations.
It is the exact configuration of the foregoing flash expression that determines the probability for the next coming.
And number of points is finite - and therefore it is possible (potentially but not in reality) to calculate exact forecasts.
Provided of curse that sufficient amount of calculating power is at hand and provided that the rules for calculation are known with sufficient precision -
Of course such pre-conditions for exact calculation are not available - and therefore we have to accept that everything is with a certain inaccuracy and will always be.
And YES - infinity is not and cannot be part of physical world - nice to be in agreement on that - as we have always been
"I'm not opposed to the universe as being laid down according to a "probability scale", in fact I'm most in favor of it. It suggests order, since each point has the same chance of being there."
I like to put it a bit differently.
When you say that each point has the same chance of being there - I get the impression that this is related to the way that we see the space-time STRUCTURE, as an ideal ordered 3D arrangement - a lattice - and probability is not in play.
But the WAY that this structure express itself has to do with probabilities - in that all physical expressions is being founded in the configuration of the "points" of this lattice - in the ever shifting configurations.
And now probability can be used in forecast calculations.
It is the exact configuration of the foregoing flash expression that determines the probability for the next coming.
And number of points is finite - and therefore it is possible (potentially but not in reality) to calculate exact forecasts.
Provided of curse that sufficient amount of calculating power is at hand and provided that the rules for calculation are known with sufficient precision -
Of course such pre-conditions for exact calculation are not available - and therefore we have to accept that everything is with a certain inaccuracy and will always be.
And YES - infinity is not and cannot be part of physical world - nice to be in agreement on that - as we have always been
QUOTE (bukh+Mar 5 2009, 08:49 PM)
Trippy
"It only needs to have the potential to do so.
The number is being chosen from an infinite source that's biased towards numbers.
With the above setup, I can tell you exactly what the probability of selecting any number between 0 and megistron (and beyond) is.
With the above setup, it does not matter how large your number is, I can assign a probability to it."
Well - I have a different opinion.
You can tell the exact probability once the operation for said probability has been executed - and Kronecker delta is being based upon quantized conditions - discreteness - and as such disqualifying infinite.
To me it is valid that infinity and continuous cannot be implemented into a physical world - and cannot be implemented in thought experiments that is being based upon physical conditions.
Bzzzzt.
Wrong answer.
The probability represents a simple sum to infinity.
The calculations are relatively simple.
"It only needs to have the potential to do so.
The number is being chosen from an infinite source that's biased towards numbers.
With the above setup, I can tell you exactly what the probability of selecting any number between 0 and megistron (and beyond) is.
With the above setup, it does not matter how large your number is, I can assign a probability to it."
Well - I have a different opinion.
You can tell the exact probability once the operation for said probability has been executed - and Kronecker delta is being based upon quantized conditions - discreteness - and as such disqualifying infinite.
To me it is valid that infinity and continuous cannot be implemented into a physical world - and cannot be implemented in thought experiments that is being based upon physical conditions.
Bzzzzt.
Wrong answer.
The probability represents a simple sum to infinity.
The calculations are relatively simple.
QUOTE (phyti+Mar 5 2009, 01:29 AM)
The fraction 9/9 as an integer, does not produce the repeating pattern .9R, therefore 1 does not produce it either.
Yes, it can.
I take it you know long division (i.e. haven't forgotten it since high school!).
Let's calculate 9|9.00000000000...
First, how many 9s in 9? One answer is 1 r0. Write the 1 in the units and carry the 0, repeat ad infinitum and you get the answer 1.000...
Another equally valid answer is 0 r9. So we write 0 in the units and carry the 9. How many 9s in 90? 9 r9. We write 9 in the tenths and carry the 9. Repeat ad infinitum and you get the equally valid answer 0.999...
But by definition, 0.9r is equal to the limit of the sum of the infinite sequence. It is not equal to the infinite sequence itself.
But by definition, 0.9r is equal to the limit of the sum of the infinite sequence. It is not equal to the infinite sequence itself.
When calculations are done, the theoretical sequences are never used, only appoximations, and then the values are rounded for the desired precision.
If that was true you'd never get those 'eureka' moments when you find a pi^2/pi and are able to cancel to infinitely precise values to arrive at a neater answer.
It certainly does. With all the stamina he showed when arguing, I am surprised he chose that option.
Yes, it can.
I take it you know long division (i.e. haven't forgotten it since high school!).
Let's calculate 9|9.00000000000...
First, how many 9s in 9? One answer is 1 r0. Write the 1 in the units and carry the 0, repeat ad infinitum and you get the answer 1.000...
Another equally valid answer is 0 r9. So we write 0 in the units and carry the 9. How many 9s in 90? 9 r9. We write 9 in the tenths and carry the 9. Repeat ad infinitum and you get the equally valid answer 0.999...
QUOTE
This is in total agreement with the limit statement that .9R does not equal 1.
But by definition, 0.9r is equal to the limit of the sum of the infinite sequence. It is not equal to the infinite sequence itself.
QUOTE (->
| QUOTE |
| This is in total agreement with the limit statement that .9R does not equal 1. |
But by definition, 0.9r is equal to the limit of the sum of the infinite sequence. It is not equal to the infinite sequence itself.
When calculations are done, the theoretical sequences are never used, only appoximations, and then the values are rounded for the desired precision.
If that was true you'd never get those 'eureka' moments when you find a pi^2/pi and are able to cancel to infinitely precise values to arrive at a neater answer.
QUOTE
It seems StevenA has left the building.
It certainly does. With all the stamina he showed when arguing, I am surprised he chose that option.
QUOTE (phyti+Mar 5 2009, 12:29 AM)
The fraction 9/9 as an integer, does not produce the repeating pattern .9R, therefore 1 does not produce it either.
Is there a fraction that will produce .9r?
Is there a fraction that will produce .9r?
Trippy
"Bzzzzt.
Wrong answer.
The probability represents a simple sum to infinity.
The calculations are relatively simple."
What is meant by "a simple sum to infinity" - In my optics infinity is not and cannot be represented by a sum - infinity is a mind concept - infinity cannot be defined - and definitely not being calculated - infinity cannot be used in calculations - except if one wants to get rid of smallest inaccuracies.
You suggest to compare apples with oranges and then get a well defined "probability"
"Bzzzzt.
Wrong answer.
The probability represents a simple sum to infinity.
The calculations are relatively simple."
What is meant by "a simple sum to infinity" - In my optics infinity is not and cannot be represented by a sum - infinity is a mind concept - infinity cannot be defined - and definitely not being calculated - infinity cannot be used in calculations - except if one wants to get rid of smallest inaccuracies.
You suggest to compare apples with oranges and then get a well defined "probability"
QUOTE (bukh+Mar 5 2009, 11:12 PM)
Trippy
"Bzzzzt.
Wrong answer.
The probability represents a simple sum to infinity.
The calculations are relatively simple."
What is meant by "a simple sum to infinity" - In my optics infinity is not and cannot be represented by a sum - infinity is a mind concept - infinity cannot be defined - and definitely not being calculated - infinity cannot be used in calculations - except if one wants to get rid of smallest inaccuracies.
You suggest to compare apples with oranges and then get a well defined "probability"
Sum to infinity
Geometric series
If I look at my machine after n half lifes, then (for example) the chances of it having produced a number that will be read as a "1" is:
P(x=1)+P(x=01)+P(x=001)+P(X=0001)+P(X=00001)...
Which is simple the sum to infinity of:
(1/(2^n))*(1/(10^n))
For the above case it works out at something like 2.5%
"Bzzzzt.
Wrong answer.
The probability represents a simple sum to infinity.
The calculations are relatively simple."
What is meant by "a simple sum to infinity" - In my optics infinity is not and cannot be represented by a sum - infinity is a mind concept - infinity cannot be defined - and definitely not being calculated - infinity cannot be used in calculations - except if one wants to get rid of smallest inaccuracies.
You suggest to compare apples with oranges and then get a well defined "probability"
Sum to infinity
Geometric series
If I look at my machine after n half lifes, then (for example) the chances of it having produced a number that will be read as a "1" is:
P(x=1)+P(x=01)+P(x=001)+P(X=0001)+P(X=00001)...
Which is simple the sum to infinity of:
(1/(2^n))*(1/(10^n))
For the above case it works out at something like 2.5%
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