@Mahmud I'd suggest something that contains this concern about inaccessibility and paradoxes -- the current formulation of "what happens when numbers get really big?" definitely doesn't seem an ideal way to introduce people to the problem

I am sorry I do not follow. Do you mean the 'abstract' or where I talked about "number leading to self-refernce"? Please clarify.

@Mahmud I think the recent edit is definitely helpful. I guess I am just having a hard time understanding the theoretical concern -- why you would expect some radical change in behavior with large numbers?

@JosephWeissman Take an infinite-Turing tape. There is no limitation to the generation of a number starting with 1 and followed by 0 on that tape. Now imagine God or a Being with non-terminating life press 1 on that tape and press 0. As the number of 0s increase indefinitely it exceeds the number of particles and exceeds single perturbation count. Can there be a theoretical number so large so as to be self-referential? Reference welcome.

It may just be that I don't share the intuition that anything is special about (very) large numbers, I would really like to hear a bit more from you on this point -- you might also like this paper of JDH's which talks about infinite time Turing Machines, which could decide membership in any computably enumerable set

arxiv.org/PS_cache/math/pdf/0212/0212047v1.pdf

Thank you for the link. The concept of infinite time Turing Machine led to this blog article which deals heavily with my contention and may be suggested to tackle in a new answer.

Alternately, what I was inquiring was about these paradoxes of set theory such as Burali-Forti