Timeline for Does the consistency of a large cardinal axiom imply the $\omega$-consistency of that axiom?
Current License: CC BY-SA 4.0
3 events
when toggle format | what | by | license | comment | |
---|---|---|---|---|---|
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 |