User blog comment:Pellucidar12/Proof length function/@comment-1605058-20161226131313/@comment-1605058-20161228104118

Your comment describes two very different functions. One of these functions is (an analogue of) the one used by Yudkowsky - "the largest output of a Turing machine which can be proven in ZFC to halt in n symbols".

The other is much stronger function - "the largest number defined by a formula \(\varphi\), such that ZFC proves \(\varphi\) defines a unique number with a proof of length n". This function is not computable. For example, ZFC can prove that values of BB function, or even an oracle BB or, say, the xi function, are well-defined, so they are definable in a manner above. Indeed, this function is probably much closer to Rayo's function than any computable function.

By the way, the complications you mention - models here aren't really the issue; it's not like in different models ZFC proves different things, provability doesn't have to do with a model. But there is a different issue at hand - soundness (which here ought to mean: if ZFC proves some TM halts, then this TM really halts). If we assume that ZFC is \(\Sigma_1\)-sound (which is widely believed to be true, and is a bit stronger than consistency of ZFC), then the Turing machines which ZFC to proves to halt will really halt, and existence of various models is no issue for Yudkowsky's number to be well-defined.

To be fair, this also applies to Peano arithmetic - if PA wasn't \(\Sigma_1\)-sound (which, again, it is widely believed to be, and in fact can be proven in ZFC), then it might prove some machine halts, while really it doesn't (PA + "PA is inconsistent" does that).