[FOM] Potential and actual infinities

[Marcin Mostowski]

Timothy Y. Chow observed:

I've never seen anyone define two separate axioms and declare one 
of them to be an "axiom of potential infinity" and the other an "axiom of 
actual infinity." 

No wonder, potential and actual infinity are views on mathematical truths, but on nature of mathematical reality.

Potential infinity appears in our computational experience. Always we have finite amount of memory and other given resources, but we can extend them – still by a finite amount.

Actual infinity appears when we try to comprehend our computational practice, when we try to find axioms correctly describing our computational experience. The problem is that we want to know possible results of extendinding of extending of our experience.

So potential infinity can be described by the following claims: 

the world is finite,

but it is greater than any given finite amount.

Of course this is inconsistent, when we try to comprehend the situation.


I investigated the case from logical point of view in: http://www.impan.pl/~kz/KR/talks/Truth_in_the_limit.pdf



Another relevant point is meaning of mathematical proofs. We search for mathematical truths, 
THE BEST ARE PROOFS, BUT THEY ARE NOT ONLY SOURCES OF MATHEMATICAL TRUTHS.

We accept PA consistency, imposibility of easy computing of discrete logarithms, and many other claims which cannot be decided on the basis of mathematical proofs.

Marcin Mostowski