Thanks for bumping this thread and I so have some things to add now:

1. The concept of 'infinity' is underspecified and indeterminant. Only a single unbounded, variable quantity should be used when computing limits (and it would appear natural to associate this with the time over which a process has occured).

If we define infinity and 0 to be:

infinity=n
0=1/n

and then allow for all irrational numbers to be defined by a single variable term n->infinity, to determine precision, then we no longer have indeterminism involved in statements like 0/0=1 of 0^0~=1-log(infinity)/infinity or infinity^2>infinity or infinity-infinity=0 etc.

And we can simply denote many irrational values as, for example e=(1+0)^infinity.

An additive 0, if necessary to describe for some reason (there are an infinite number of possible 0s we could add to an equation and so an addition of 0 isn't necessary or useful), we could construct is a cancellation of equal and opposing polarity terms such as 0-0 or x-x etc.

2. Irrational numbers have no specific precision (truly irrationals and likely even integers could be seen in the same context). A number line is dynamic and context dependent and consists of an ordering of elements at any particular moment and for example, the computation of 2+5=7 implicitly constructs the numberline:

0<1<2<5<7

(The numbers 0 and 1 I included because of their implicit necessity in both constructing 2 and 5 as (1+1) and (1+1+1+1+1) and the number 0 is also implicitly required in order that we can detect equality and inequality, because this is determined by whether or not x-y=0, in which case x=y, though we could remove 1 and define it as 0/0)

Numbers only have a meaning relative to other numbers and more fundamentally to the units by which numbers are constructed.

A more fundamental representation of "Real" numbers in this context could be seen as a range of possible solutions, which actually requires 2 dimensions to specify as either a minimum and maximum value or a central value with a tolerance, that are ordered numerically, with a fundamental representation as two rings, representing positive and negative quantities respectively and an implicit detached zero point for determining equality between these.

If we define +0, +1 and +infinity as starting points of a line segment (though we can define +1 to be either the geometric mean or the product of +0 and +infinity), then we have a positive number line ordered as:

+0<+1<+infinity

If we apply a reciprocal operation, this becomes the same as reversing the order of a set:

+1/+infinity<+1/+1=+1<+/+0

If we negate the original values, we again invert the ordering of this set and get:

-infinity<-1<-0

If we append these two, positive and negative line segments together, and place an element Z to indicate the additive identity for zero, we get:

-infinity<-1<-0<Z<+0<+1<+infinity

A potential problem arises though when an additive operation crosses from the positive line to the negative line as we have the potential inclusion of Z as an element and, for example +1-(+1) would not necessarily equal +0 or -0, but instead the additive zero, Z.

But at least the context in which a multiplicative 0 or infinite value occurs is deterministic by this system and as points are added and manipulated to this number line, a specific solution could be seen as arising from various reversals and shifts of segments of numbers along this axis and for some computation resulting in an approximation of pi, we could see the total context of a computation as, for example:

-(infinity^2)<-infinity<-1<-0<Z<+0<+1<+3<+3.1<pi<+3.2<+10<log(+infinity)<+infinity

And in this case pi is defined not as a single number but an ordered pairing of {3.1,3.2}, or alternately it might be a pairing of a current estimate and an iteration number etc.

3. A finite number is actually a number and not a variable. The statement "x is finite" is inaccurate, just as the statement that "all elements of 1,2,3,... are numbers" is inaccurate as '...' isn't a number and this implies that a recursive definition that an infinite sequence is a number.

And taken as a limit, there is no specific number that terminates the sequences 1,2,3,... and neither does a specific number terminate the sequence 1/1,1/2,1/3,.... The diferences in one case are infinite and in the other case the differences are infinitesimal.

4. An informationally compressive mapping from a set of possible solutions convey some quantity of information to an alternate mapping that doesn't convey identical information is not invertible and can't be considered an equality or provable identity, but instead a possible approximation or isolated solution from other elements and if a set is infinite, then any finite result would automatically not be an identity.