User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-11227630-20180208013330/@comment-1605058-20180208111447

There is a technical issue at hand here, namely, what exactly (and formally) do we mean by a formula of the sort "α exists"? After all, we can't just plug in the ordinal α into a formula.

The most general definition would be to take any formula \(\varphi(x)\) (like "\(x\) is recursively inaccessible") and take "there is an ordinal \(\alpha\) such that \(\varphi(\alpha)\)". But then we can take for \(\varphi(x)\) things like "\(x=0\) and there is a standard model of ZFC". Adding this to KP we will get a theory with PTO larger than ZFC, but the ordinal whose existence we assert will be equal to zero, so technically zero is the answer to your question.

Of course, we want to exclude such trivialities, but it's by no means an easy task - at a formal level, what is the difference between statements "\(x\) is recursively inaccessible" and "\(x=0\) and there is a standard model of ZFC"? Having answered that, it might also be an issue of what it means for two ordinals described in such ways to be smaller; perhaps the two statements are incompatible, so it can't be that both statements hold.

Two finishing remarks: in the question about "KP vs ZFC catching point" this issue didn't arise, because KP + X and ZFC + X never have the same PTO, for any formula X, regardless of whether or not it expresses existence of an ordinal. Also, under any reasonable interpretation of your question, the answer will not be the PTO of ZFC - that ordinal can be proven to exist in KP + "there is an admissible ordinal", much weaker than ZFC.