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.

  • $\begingroup$ Yes. And you can replace ZFC with ZF. Just use the Lowenheim-Skolem construction in ZF+S, the same as you do in ZFC. $\endgroup$ – DanielWainfleet Aug 26 '16 at 16:17
  • $\begingroup$ 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. $\endgroup$ – GME Aug 26 '16 at 19:10
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    $\begingroup$ Don't you need AC to obtain a countable $c$? $\endgroup$ – Pedro Sánchez Terraf Aug 27 '16 at 4:34
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    $\begingroup$ @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. $\endgroup$ – GME Aug 27 '16 at 10:43
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    $\begingroup$ @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. $\endgroup$ – 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.

  • $\begingroup$ Thank you for the response, Pedro. It allowed me to answer my question in its entirety. $\endgroup$ – Transcendental Sep 9 '16 at 5:21
  • $\begingroup$ @Transcendental You're welcome. $\endgroup$ – 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 $$ \mathsf{ZFC} + S \vdash (\exists c) \! \left( “\text{$ c $ is countable and transitive}” \land \left( \bigwedge_{i = 1}^{n} \phi_{i}^{c} \right) \right). $$


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