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I am asking about the answer given by @aws on the page Why do class-sized models escape the completeness theorem?, and whether and why what aws says is correct. In the answer aws sketches a fake "proof" that ZFC proves its own consistency, by defining a class function $F$ that determines truth given a string/godel number associated to an input formula. They say that this proof goes wrong because "We want $F(\forall x \phi(x))$ to be one if $F(\phi(a))=1$ for every $a\in V$ and zero otherwise. However, we can't state this directly in the language of set theory," and "because the formulas can have parameters from V, there is a proper class of them, and in particular the inductive clauses for quantifiers would be most problematic." However, I don't see why it isn't perfectly fine to define $$ F(\ulcorner\forall x \phi \urcorner,[a])= \begin{cases} 1 & \forall x (F(\ulcorner\phi \urcorner, [a^i_x])=1)\\ 0 & \textrm{else} \end{cases} $$ and then use recursion to find a class function/truth formula $F$, proving the consistency of ZFC.

Am I making a mistake? Or am I right and the problem with this construction lies elsewhere (like maybe in the fact that resulting class function $F$ will have free parameters).

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    $\begingroup$ Try writing down in detail how you'll "use recursion" to define an $F$ satisfying that recursive equation. $\mathsf{ZFC}$ lets you carry out some recursive constructions, but not all - and this is one which it does not permit. $\endgroup$
    – Noah Schweber
    May 6, 2021 at 23:39
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    $\begingroup$ I explained this in the second part of my answer to mathoverflow.net/questions/87238 ; start reading at the 7th sentence, "What goes wrong if you try to define, in the language of ZFC, this notion of truth (or satisfaction) in the universe?" $\endgroup$
    – Andreas Blass
    May 6, 2021 at 23:44

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Thanks to @NoahSchweber and @AndreasBlass for their comments. Following Andreas's mathoverflow post, the truth of $\forall x \phi(x)$ depends on $\phi(x)$ for class-many values $x$.

Suppose we are using godel numbering to index formulas by elements of $\omega.$ If we were to try to define the class function giving truth $F$ recursively by the equation given in my question, we would have to construct a relation $R$ on the set $\omega\times V^{\omega}$ of formula-valuation pairs such that evaluating $F$ at an element of $\omega\times V^{\omega}$ is only dependent on $F$ at previous values under the relation $R,$ and such that $R$ is set-like, meaning that the collection of predecessors of any element under $R$ forms a set. However, because the the truth of a formula with unrestricted quantification is dependent on the truth of class-many valuations, such a relation $R$ cannot possibly be set-like.

I would disagree with aws when they say "we can't state this directly in the language of set theory," but otherwise I think they are right.

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