Comment by edanm

> An axiomatic system can be consistent, but wrong.

But then its unsound, isn't it? Isn't our background assumption that ZFC is consistent and sound? It can't prove its own consistency, but we are assuming that under standard models, it is sound.

> For example, if ZFC is consistent, then T = ZFC+~Con(ZFC) would be consistent as well.

It would be consistent if ZFC didn't also prove ZFC+Con(ZFC), but then it would indeed be unsound.

> Similarly, if ZFC is indeed consistent, then T is wrong about which Turing machines halt. Therefore it would have a wrong value of BB(748) (and many other BB(n)).

No, if it's sound, it just doesn't have a proof of the form "BB(748)=K" for any K.

> However, since ZFC can't prove its own consistency, it can't prove that value is wrong. That's why there are different values of BB(748). Those values are not necessarily equally correct, it's just that ZFC isn't strong enough to prove which one is wrong.

No, ZFC is just not strong enough to prove any of these.