User blog comment:Nayuta Ito/Is this bigger than Rayo's number?/@comment-27513631-20180504234554/@comment-35470197-20180528040718

I noticed that Emlightedned was referring to Platonist's universe of the coded ZFC set theory, but not the ZFC set theory given as the meta theory.

If it is correct, then my large number is NOT bigger (or even not smaller) than ones defined by using truth in Platonist's universe for the coded theory, because the latter ones are NOT well-defined in the meta theory, I think. We can never compare a well-defined number with a not defined number. I am sorry if I am wrong.

Indeed, since Platonist's universe for the coded theory is not described in the meta theory, there is no way to define ``the minimum of a natural number greater than any natural number satisfying P'' for a given property P in the meta theory even if one believes the existence of an abstract Platonist's universe as long as it is not declared in a specific way in the meta theory.

The axiom of separation in the coded theory does not work in order to define ``the minimum'' in Platonist's universe because P is a meta property. That in the meta theory does not either, because Platonist's universe for the coded theory is not a universe of the meta theory itself. Say, Platonist's universe for the coded theory might contain a natural number which is not derived from a meta natural number defined by applying successors.

So using Platonist's universe in this way (not for the meta theory, but for the coded theory) does not give a natural number in the meta theory. Of course, Platonist's universe may be helpful if one wants to defined a natural number in the coded theory, which is not necessarily derived from a meta theory. But this is not the case.

I note that for a given set M in the meta theory, the satisfaction of M for given formulae in the meta theory such as coded ZFC axioms can be referred in the meta theory, because it can be formally coded using a formal language L_M of ZFC directly constructed by using M itself. But for the abstract existence of Platonist's universe for coded theory described over a fixed language such as FOST, what can we refer to in the meta theory?

Maybe I am misunderstanding what a ``large number'' and Platonist's universe precisely mean in Googology wiki. If so, I am sorry very much. Is there a precise rule to define a ``large number'' in Googology wiki? I appreciate any precise reference of a rule here given by anyone. Thank you.