User blog comment:DontDrinkH20/H-Boogol-Boogol: Hopefully a very big non-salad number/@comment-35470197-20180819021045/@comment-35470197-20180820013010

Thank you for the reply.

> L(T)

For example, let \(f_i\) with \(i \in \mathbb{N}\) denote the formula \(\forall x_0 \cdots \forall x_i, x_{i+1} = x_{i+1}\) with the free occurrence of \(x_{i+1}\) in ZFC set theory. Then the axiom of ZFC set theory contains the specification formula \begin{eqnarray} \forall x_{i+2}, \exists x_{i+3}, \forall x_{i+1}}, (x_{i+1} \in x_{i+3}) \leftrihtarrow (x_{i+1} \in x_{i+2} \wedge f_i), \end{eqnarray*} which contains \(i+3\) distinct term symbols. Therefore \(L(\textrm{ZFC})\) itself contains infinitely many distinct symbols.

Maybe you are implicitly removing for the axiom schema such formulae which are obviously equivalent to other short formulae. However, by the well-known fact that ZFC set theory is not finitely axiomaised under the assumption of the consistency, there is no reasonable way to remove all of such trivial examples without changing the strength of the axiom schema. That is why I wrote that you need a finite segment.

> which is defined as (cal)H_a(b) across all class-sized models of ZF:

How could you refer to all class-sized model in the definition of the smallest number? In order to define the smallest one, you need to define "the subset of natural numbers satisfying the property" by the specification formula. On the other hand, the property contains an unbounded quantifier of classes, which is invalid in the specification axiom schema in ZFC or NBG.

Also, how do you formalise the notion of "class-sized model"? A subclass "satisfying" ZF makes sense only when you consider the proof-theoretic satisfaction of a definable class. Then it is much weaker than the set-sized model argument (based on formalised truth but not proof-theoretic satisfaction), which is implicitly used in Rayo's number.

For example, could you explicitly tell me what you intend as the definition of the meta-theoretic (not formalised) formula "a subclass \(X \subset V\) is a class-sized model of ZF with respect to the restriction of \(\in\)"?