User blog comment:P進大好きbot/Whether Rayo's number is well-defined or not/@comment-25601061-20180606221005/@comment-35470197-20180609015926

> It's uncontroversial to work in a manner which is valid internal to any (countable) model of ZFC.

As I assumed in the blog post, a Platonist universe is a class model here. As you know, a set model is not a class model which is just a set. The reason why a choice of a class model works in a bad way is explained in my blog post.

So your statement is that a Platonist universe is not a class model, but a set model. Unlike a class model, there is no way to construct a set model in the theory. Therefore in the usual set theory, we formally consider statements like "if a set M is a set model, then M satisfies...".

So you may think that we can work in the same way. Of course, you can use a set model of the meta theory, in which Rayo's function lives. However, the statement that M is a set model of the meta theory does not imply that M satisfies ZFC coded in FOST, even if the axiom of the meta theory is ZFC.

Fixing a set M satisfying ZFC coded in FOST is automaticaly allowed when we deal with the coded theory, but is not when we work with the meta theory. If you allowed it, then you should allow the method like "I fix a model G of the finite group theory. Its cardinality |G| is my large number. You can never prove that your large number is larger than mine!" Fixing a non-specific model of a coded theory does not describe a specific natural number.

> There is no maximal consistent set of statements - as the theory is incomplete, there are independent statements which are as valid choices as their negations.

Every consistent set of statements is contained in a maximal one by Zorn's lemma. Please do not forget that a consistent set of statements is not necessarily recursive. A non-recursive consistent set of statements is not necessarily incomplete.

> Look up the concept "metatheory" - you seem to be trying to reinvent it.

Why do you think so? I am just using the usual notion of the meta theory in the formal logic written in elementary textbooks such as Kunen or Devine.