User blog comment:DontDrinkH20/H-Boogol-Boogol: Hopefully a very big non-salad number/@comment-35470197-20180819021045/@comment-35470197-20180821011647

> To answer your first question, see my first answer.

I have already seen it.

> There are pleanty of infinite schemas which have only finitely many distinct symbols.

I know. But we are talking about the specific axiom schema for ZFC set theory, which obviously contains infinitely many distinct symbols. Indeed, I gave you explicit examples of formulae \((f_i)_{i \in \mathbb{N}}\) contained in the axiom schema, which contains infinitely many distinct symbols \((x_i)_{i \in \mathbb{N}}\). For your convention, how many distinct symbols are contained in them? Why they are not infinity?

> To answer your second question, standard semantics is well-defined on classes using second-order logic to define it. It is defined exactly like it is defined for set-sized models.

I am not talking about the semantics. I know the quantification itself is valid in the standard semantics.

What I mentioned to is the specification axiom, which ensure the well-definedness of the "smallest" natural number. For the usual first order sorted or second order set theory, you are not allowed to use such an unbounded quantification in order to define a subset through the specification axiom.

I list up the questions above:
 * 1) What is the assumption on the well-definedness of your large number? Please specify the axioms, because your argument on set theory is not valid in ZFC or NBG set theory even if you assume the consistency of it.
 * 2) How could you define the "smallest" natural number? Please specify the argument using explicit application of axioms, because your argment is invalid in ZFC or NBG set theory.
 * 3) What is the definition of a class-sized model of ZF in your context? Please specify whether it is formalised or not, because even for a definable set-sized model, there are both notions of satisfaction in a formalised sense and a meta theoretic (proof-theoretic) sense. Usually, a class-sized model makes sense only when you are considering the latter sense, but your argument seems to be based on the former sense.
 * 4) How many distinct symbols are there in \((f_i)_{i \in \mathbb{N}}\)?