We really don't.
> that BB(748) is independent of ZFC
> there are different models that have different values of BB(748)
> ZFC is insufficient to determine the value of BB(748)
These three statements are equivalent.
f(n)=X is independent of ZFC means there are different models of ZFC that have different values of f(n). It's a very trivial theorem[0]. If you don't like it, I can't convince you otherwise.
> that doesn’t actually change anything
Changing the model will not change how any machine works in our physical, mechanical universe. However, it does change the value of BB(748).
I understand your line of thinking: There is only one mechanical universe, which is the one where we exist. We can build Turing machines in this universe. BB(n) depends on Turning machines. Since there is only one single universe, there is only one single value of BB(n).
It's a perfectly fine mental model for most cases. This was exactly how I thought when the first time I heard about BB(n). But it's not the kind of math than Scott Aaronson et al. are doing.
Bar keeps running in our mechanical universe. But it can also halt in some non-standard number of steps. This weird, absurd-sounding proposition works because non-standard numbers simply don't map to anything in mechanical universe. They're purely abstract objects living in ZFC+~Con(ZFC).
[0]: Given f(n)=X is independent of ZFC. Which means f(n)=X and ~(f(n)=X) are both consistent relative to ZFC. Therefore, if there is any model of ZFC, there is a model M1 that entails ZFC+(f(n)=X), and a model M2 that entails ZFC+~(f(n)=X). The value of f(n) cannot be the same in M1 and M2.
