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The Lévy reflection theorem says that for every finite collection $\Phi$ of first-order formulas there are arbitrarily large ordinals $\alpha$ such that, for every $\phi \in \Phi$ and every tuple of parameters $a_1,\dots ,a_n\in V_{\alpha }$,Equivalently, the ordinals simultaneously reflecting all formulas in $\Phi$ form a closed unbounded class.

Solved by gpt-5.6-sol high.

Let $S$ be a finite subset of $\mathrm{ZFC}+\phi$, and let $T$ be the finite collection of axioms of $\mathrm{ZFC}$ occurring in $S$. Choose the finite fragment $T^{\ast }$ supplied by the hypothesis. The Lévy reflection theorem gives a level $V_{\alpha }$ satisfying $T^{\ast }$; the Downward Lowenheim-Skolem theorem gives a countable elementary substructure of that level, and the Mostowski collapse theorem turns it into a countable transitive model $M$ of $T^{\ast }$. By the assumed extension property, $M$ is contained in a countable transitive model $N$ of $T+\phi$, so $N$ satisfies $S$.

This argument is formalizable over $\mathrm{ZFC}$ for each finite $T$. Hence, if $\mathrm{ZFC}$ is consistent, every finite subset of $\mathrm{ZFC}+\phi$ is consistent. The compactness theorem now gives

Solved by gpt-5.6-sol high.

Let $a,p\in M[G]$, and suppose a first-order formula $\psi (x,y,p)$ defines exactly one $y$ for every $x\in a$. Apply the Lévy reflection theorem to the formulas needed to express this assertion, choosing an ordinal $\alpha$ with $a,p\in V_{\alpha }^{M[G]}$ such thatfor every $x\in a$. Therefore every required value lies in the set $V_{\alpha }^{M[G]}$.

The already established axiom schema of separation forms the setFunctionality makes $b$ exactly the range of the definable function on $a$. This proves every instance of the Axiom schema of replacement in the generic extension $M[G]$.

Solved by gpt-5.6-sol high.