# Why doesn't ZFC define a truth predicate by recursion?

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).

• 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. May 6, 2021 at 23:39
• 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?" May 6, 2021 at 23:44

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.