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Let $P$ be some large cardinal property (or indeed any first-order formula in the language of set theory, but lets focus on large cardinals for now). Does the $\omega$-consistency of $\mathsf{ZFC}+P$ follow from the consistency of $\mathsf{ZFC}+P$ (in $\mathsf{ZFC}$).

I would especially be interested in knowing if, given a model of $\mathsf{ZFC}+P$, the existence of an $\omega$-model of $\mathsf{ZFC}+P$ is guaranteed. My understanding is that the existence of an $\omega$-model is generally stronger than $\omega$-consistency.

An $\omega$-model is one without nonstandard natural numbers.

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    $\begingroup$ No, because $\omega$-consistency is preserved by adding true $\Pi^0_1$ (or even $\Sigma^0_3$) sentences. Thus, the $\omega$-consistency of ZFC + $P$ implies the consistency (or even $\omega$-consistency) of ZFC + “the consistency of ZFC + $P$”, which is not provable in ZFC + “the consistency of ZFC + $P$” itself unless it is inconsistent, by Gödel’s theorem. $\endgroup$
    – Emil Jeřábek
    Commented Oct 9, 2023 at 11:19
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    $\begingroup$ Note that Emil Jeřábek's answer also provides a negative answer to the second question of whether the consistency of some large cardinal axiom implies the existence of an $\omega$-model of that axiom, since consistency in $\omega$-logic (equivalent to the existence of an $\omega$-model via the Henkin–Orey completeness theorem) strictly implies $\omega$-consistency. $\endgroup$
    – Benedict Eastaugh
    Commented Oct 9, 2023 at 12:26

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