TITLE: What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?
QUESTION [27 upvotes]: 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.

REPLY [8 votes]: Knowing the values of the Busy Beaver function is the same as knowing the truth values of $\Pi^0_1$-statements (ie statements of the form $\forall n \in \mathbb{N} \ P(n)$ for decidable properties $P$). Knowing a particular $\mathrm{BB}(m)$ means knowing all $\Pi^0_1$-statements of complexity $m$ (for a suitable complexity measure).
For any c.e. theory $T$, $\mathrm{Con}(T)$ is a $\Pi^0_1$-statement. Moreover, if $T$ is of the form "$\mathrm{ZFC}$ + some short axiom", the complexity of $\mathrm{Con}(T)$ (in the sense above) will not be much more than the complexity of $\mathrm{Con}(\mathrm{ZFC})$. Hence if we want to settle values of the Busy-Beaver function on "medium-sized" inputs, we would need to add axioms that settle a lot of consistency statements.
Proving that adding an axiom actually increases the range of provable Busy-Beaver values will be non-trivial. It does not suffice to show that it settles more $\Pi^0_1$-statements, but one needs to show that actually increases the least complexity of an undetermined $\Pi^0_1$-statement. Unfortunately, the complexity notion here is essentially Kolmogorov complexity -- which means that we cannot compute the complexity of a logical formula from that formula.