Develop a framework of mathematics which removes the "divide by zero" problem, both in the general and discreet case, or possibly just one or the other at a time, but still works accurately for real world calculations.
If this is impossible, "prove" it is impossible.
If new axioms or theories of Mathematics are needed, develop them and "prove" they work.
If the above challenge is impossible, then what does that say about our framework of math and science, since our math involves equations which are undefined along their range or domain for some values, even when all component terms are defined or apparently defined individually?
Y = X / (X -1)
X and "X-1" are always defined, under present day math laws, yet Y is not.
if X = 1,
then Y = 1/0, which is Undefined.
Perhaps our concept of an "equalities, inequalities, and expressions," is flawed, since they necessarily give rise to "undefined" terms at times.
For example, if I simply write the term "5/0," the we can quickly argue "that's nonsense because division by zero is undefined."
But if I have the equality:
Y = X/(X-5)
If X = 5,
Then Y = 5/0, which is undefined.
The problem here is one math law requires that X is always defined, but the concept of an "equality" plus division forces the right hand side to produce a nonsense result.
I guess what I'm trying to say is this problem is bptj more complicated and more fundamental than it appears to be at first.
Perhaps the problem lies in the definition of Zero more so than the definition of division or equality?
This strikes me as THE biggest fundamental problem in both mathematics and physics today, and yet I've never seen a legitimate attempt to resolve it, or even an attempt to speculate on a line of reasoning that "might" resolve it.
Four apples minus five apples equals negative one apple.
What the HELL is a "negative apple"?
I can owe a person an apple, but that is not a negative number.
On a number line, a negative one exists, and yet in the real world, a "negative apple" or an "owed apple" does not exist.
Therefore we should correct the above statement as follows:
"Four apples minus five apples is undefined."
After all, if you have only FOUR apples, then the very act of subtracting FIVE apples is in fact logically impossible. Nobody can possibly take away FIVE apples from you if you've only got FOUR to start with.
but that is not sufficient. In the real world we COUNT each apple, one at a time.
If they ATTEMPT to remove five apples, and find only four apples, they stop at four apples, leaving not "zero Apples," but rather just "nothing," since Zero apples does not exist either.
You may say "an empty basket has Zero Apples."
Possibly, but I find that problematic as well, as by that reasoning it also contains "zero oranges" as well as "zero pears" and "zero Orangutans" and it even contains "zero zeros".
How can an EMPTY basket contain "Zero Oranges" and "Zero Zeros" at the same time?
Yet by our present definition of mathematics and physics, it apparently does just that, even though such is self contradictory at a conceptual level....
If the basket is EMPTY then it must contain "nothing" which is actually most closely expressed as the "empty set" in mathematics, and yet in the real world no true "empty sets" appear to exist, because in reality an "empty basket" is not really "empty" it contains air and space-time, if nothing else...
Nevertheless an "empty" basket contains zero of everything and zero zeros simultaneously, once again, even though that is self-contradictory on at least TWO conceptual levels...
Perhaps "Zero times Zero," a.k.a. "0*0," is the problem....
The standard claim that "Zero equals nothing" does not seem to hold true, for if "Zero equals nothing" then "Zero times Zero" or "0*0" should undoubtedly produce a blank space, rather than a symbolic solution.
0*0 = 0
For if I have zero zeros then I certainly cannot have one Zero at the same time.
Zero Zeros equals one Zero...***?!?
"Zero Zeros" should be an empty set, so then why does it equal "One Zero"?
If I have no Zeros then how can I have one Zero?
Therefore the expressions "0*0 = 0" and "zero times zero equals zero" are illogical.