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Oct 9, 2023 at 12:26 comment added Benedict Eastaugh 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.
Oct 9, 2023 at 11:19 comment added Emil Jeřábek 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.
Oct 9, 2023 at 10:28 history asked Calliope Ryan-Smith CC BY-SA 4.0