First, some context. Ever since I was a high schooler, I have been fascinated with large numbers. As I have grown in mathematical maturity, I have become both disappointed and fascinated to see that the process of naming larger and larger numbers requires a sort of philosophical tradeoff.

For example, an ultrafinitist might reject $10^{10^{100}}$ as being a valid representation of a number. A finitist might reject numbers defined using ordinal collapsing functions with large cardinals. A person who believes only in the constructible universe might reject the same ordinal collapsing functions if cardinals incompatible with $V=L$ are used. In general, the more you are willing to accept philosophically, the "larger" a number you can name concisely.

The interesting thing about these tradeoffs is that some of these philosophical stances have concrete impacts on the "real" world. For example, there is a number $n$ for which new axioms in addition to those of ZFC would be needed to prove the value of $BB(n)$. So in a way, this "real world" value changes depending on our philosophical stance.

My question then is: what sorts of axioms might we accept to strengthen as much as possible the values of $BB(n)$ which we can prove? And more generally, is there a way we can evaluate whether certain statements independent of ZFC are "true" based on their implications for the "real world" value of $BB(n)$ (if there is even a "true" value for this number).

EDIT: useful reading for those not familiar with the phenomenon of independence of $BB(n)$ from ZFC:

- Adam Yedidia, Scott Aaronson,
*A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory*, Complex Systems**25**(4) (2016), journal, arXiv:1605.04343

EDIT: As requested, an example of a "real world" consequence of a set theoretical axiom would be how Con(ZFC) would prove the Turing machine in the above paper doesn’t halt, whereas the negation of Con(ZFC) proves it halts in finite time.