Generalizing the Reflection Principle of $\mathsf{ZFC}$

The Reflection Principle states that for any finite list $\phi_{1},\ldots,\phi_{n}$ of axioms of $\mathsf{ZFC}$, we have $$\mathsf{ZFC} \vdash (\exists c) \! \left( \text{“ c  is countable and transitive”} \land \left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right) \right),$$ where $\phi_{i}^{c}$ denotes the relativization of $\phi_{i}$ to $c$ for each $i \in \{ 1,\ldots,n \}$.

What I would like to know is, if $S$ is a set of sentences in the first-order language of set-theory, and if $\mathsf{ZFC} + S$ is consistent, then is the following assertion also true?

For any finite list $\phi_{1},\ldots,\phi_{n}$ of sentences taken from $\mathsf{ZFC} + S$, we have $$\mathsf{ZFC} + S \vdash (\exists c) \! \left( \text{“ c  is countable and transitive”} \land \left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right) \right).$$

Thank you very much for your help.

• Yes. And you can replace ZFC with ZF. Just use the Lowenheim-Skolem construction in ZF+S, the same as you do in ZFC. – DanielWainfleet Aug 26 '16 at 16:17
• It suffices to note that ZF actually proves the more general conditional claim for any $\phi_0,...,\phi_n\in\mathcal L_\in$: if $\phi_0 \wedge,...,\wedge\phi_n$, then there is a $c$ such that.... Trivially, any extension of ZF also proves that claim. – GME Aug 26 '16 at 19:10
• Don't you need AC to obtain a countable $c$? – Pedro Sánchez Terraf Aug 27 '16 at 4:34
• @PedroSánchezTerraf Sorry, my bad. For my claim, I think you're right: the conditional claim for countable transitive models implies dependant choice for $\Delta_1^{ZF}$ relations (without parameters). Anyway, my response works for ZFC, which is what the OP was after. – GME Aug 27 '16 at 10:43
• @GME: The downwards LS is equivalent to DC over ZF. Not just for $\Delta_1$ relations. And the usual proof is to find some $V_\theta$ satisfying the axioms and take a countable elementary submodel of that. Which will, again, lead you to a full DC situation. – Asaf Karagila Aug 29 '16 at 6:22

$\newcommand{\ZF}{\mathrm{ZF}}$I would like to state a version of the Reflection Principle under $\ZF$ that (at least to me) makes the point a little clearer. Informally, it says that the truth value each every formula (not just sentences) gets reflected in many transitive sets.

For this, recall the cumulative hierarchy $V_\alpha$ given by iterated powersets. The precise statement of this version says:

Theorem. ($\ZF$) For each formula $\phi(x_1,\dots,x_n)$, sets $a_1,\dots,a_n$ and each ordinal $\beta$ with $\bar a = a_1,\dots,a_n\in V_\beta$, there exists $\alpha>\beta$ such that $$V\models\phi(\bar a) \iff V_\alpha\models \phi(\bar a).$$

Here, as usual, $V$ is the universe of sets. Actually, one can show that for each $\phi$, there is a club of ordinals $\alpha$ for which the conclusion holds.

In the case of $\phi$ a sentence that holds in $V$ (that is, any consequence of the axioms you put to the left of $\vdash$), you obtain immediately your conclusion$^1$ (without the “countable” adjective). Some care is needed here since intuitively $V_\alpha\models\phi$ is “equivalent” to $\phi^{V_\alpha}$ but the last expression is not a formula. But perhaps you can fill the gaps.

Finally, using the downward Löwenheim-Skolem-Tarski theorem you may obtain a countable elementary submodel (choose some Skolem functions, take the closure, using that countable union of countable sets is countable verify that it is indeed countable). This is a submodel of a well founded model, hence well founded, and you can collapse it to a transitive model.

I hope this helps; actually, the idea marked with $(^1)$ contains the meat of my answer, since that's how I managed to understand the point of your question.

• Thank you for the response, Pedro. It allowed me to answer my question in its entirety. – Transcendental Sep 9 '16 at 5:21
• @Transcendental You're welcome. – Pedro Sánchez Terraf Sep 9 '16 at 17:18

I just thought that I should convert Pedro’s response into a fully direct answer to my question.

Theorem Scheme 7.4 in Nik Weaver’s book Forcing for Mathematicians states the following:

For any sentence $\phi$ in the first-order language of set theory (henceforth called “LOST”), we have $$\mathsf{ZFC} \vdash (\exists c) (“\text{ c  is countable and transitive}” \land (\phi \iff \phi^{c})).$$

This is very similar to the version of the Reflection Principle stated by Pedro.

Let $S$ be a set of LOST-sentences such that $\mathsf{ZFC} + S$ is consistent, and let $(\phi_{i})_{i \in [n]}$ be a finite list of sentences taken from $\mathsf{ZFC} + S$. Basic logic gives us $$\mathsf{ZFC} + S \vdash \bigwedge_{i = 1}^{n} \phi_{i}. \qquad (\clubsuit)$$ By the quoted result, we have $$\mathsf{ZFC} \vdash (\exists c) \! \left( “\text{ c  is countable and transitive}” \land \left( \bigwedge_{i = 1}^{n} \phi_{i} \iff \left( \bigwedge_{i = 1}^{n} \phi_{i} \right)^{c} \right) \right).$$ This is equivalent to $$\mathsf{ZFC} \vdash (\exists c) \! \left( “\text{ c  is countable and transitive}” \land \left( \bigwedge_{i = 1}^{n} \phi_{i} \iff \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right) \right).$$ As $\mathsf{ZFC} + S$ is clearly at least as strong as $\mathsf{ZFC}$, it follows that $$\mathsf{ZFC} + S \vdash (\exists c) \! \left( “\text{ c  is countable and transitive}” \land \left( \bigwedge_{i = 1}^{n} \phi_{i} \iff \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right) \right). \qquad (\spadesuit)$$ Finally, applying modus ponens to $(\clubsuit)$ and $(\spadesuit)$, we obtain $$\mathsf{ZFC} + S \vdash (\exists c) \! \left( “\text{ c  is countable and transitive}” \land \left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right) \right).$$