Models of extensions of ZFC, within $L_{\omega_1^{CK}}$?

Question

Is it the case that every consistent recurively axiomatized extension of $\sf ZFC$ has a model in a level of the constructible universe below $L_{\omega_1^{CK}}$? If not, then what is the least consistency strength of such an extension?

Answer 1

It depends on what sort of model you're looking for - $L_{\omega_1^{CK}}$ is either unspeakably more or vastly less than is necessary.

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If you just want any old model, then the answer is yes - massively so. The model construction process in the proof of the completeness theorem is fully definable, and so if $T$ is a consistent theory which is a definable subset of $L_{\omega+\alpha}$ then $T$ has a model in $L_{\omega+\alpha+1}$. In particular, $\mathsf{ZFC}$ has a model in $L_{\omega+1}$. (In fact, we can do much better than this via the low basis theorem, but that only works for countable theories and isn't necessary for the question as posed.)

Note that the strength of the theory is totally irrelevant; we have a single process (the Henkin process) which builds a model of a given theory if such a model exists in the first place, so the only thing that matters is the complexity of the theory itself.

On the other hand, if you're looking for well-founded models, then $L_{\omega_1^{CK}}$ isn't nearly enough: it doesn't even have a well-founded model of Kripke-Platek set theory with infinity ($\mathsf{KP\omega}$). Indeed, it's not even enough to build an $\omega$-model of $\mathsf{KP\omega}$. However, it is worth noting that we have an analogue of the low basis theorem for $\omega$-models, the Gandy basis theorem which implies in particular that every computably axiomatizable extension of $\mathsf{ZFC}$ which has an $\omega$-model has an $\omega$-model which is "low for hyperjump." That still doesn't get us down to $L_{\omega_1^{CK}}$, but it's quite a useful observation.

(Roughly, the reason $L_{\omega_1^{CK}}$ doesn't contain an $\omega$-model of $\mathsf{KP\omega}$ is the following. If $M$ is an $\omega$-model of $\mathsf{KP\omega}$ then the well-founded part of $M$ has height $\ge\omega_1^{CK}$, and in fact $M$ is an end extension of $L_{\omega_1^{CK}}$. Now if $M\in L_{\omega_1^{CK}}$ we would have $M\in L_\alpha$ for some $\alpha<\omega_1^{CK}$. But we also have $L_{\alpha+1}\in M$ in an appropriate sense, and so we get a contradiction after a quick diagonalization.)