# busy beaver – Are the outcomes of the maximum shifts function fixed regardless of our choice of axiomatic system?

It is known that there is a $$748$$-state Turing machine that halts if and only if $$mathsf{ZF}$$ is inconsistent. So by Gödel’s second incompleteness, $$mathsf{ZF}$$ cannot find what $$S(748)$$ exactly is, where $$S$$ is the maximum shifts function (Also known as the “Frantic Frog”).

I’m rather confused by this fact. As $$S$$ is well-defined, that should mean that regardless of what axiomatic system we use, the exact value of $$S(748)$$ always stays the same. We just need a stronger axiomatic system to find $$S(748)$$, like $$mathsf{ZFC}$$, $$mathsf{ZFC+CH}$$, or $$mathsf{ZFC+(V=L)}$$.

If $$mathsf{ZFC}$$ and $$mathsf{ZF¬C}$$ entailed $$S(748)$$ to be different numbers, since Axiom of Choice is independent to $$mathsf{ZF}$$, it would be concluded that $$mathsf{ZF}$$ was inconsistent at the first place. So far, is my understanding correct?