Timeline for What notable theorems cannot be automatically proven without choice using Shoenfield absoluteness?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 11, 2023 at 13:15 | comment | added | Gabe Goldberg | @JoelDavidHamkins It looks like Elliot has already done a better job than I would have | |
| Sep 9, 2023 at 21:44 | comment | added | James E Hanson | @GabeGoldberg I'm guessing that relatedly one could give a model of ZF in which the reals are the union of a projectively definable sequence of countable sets. Is this right? | |
| Sep 9, 2023 at 21:14 | comment | added | Joel David Hamkins | @GabeGoldberg That sounds great! I suggest you post an answer with a fuller account. | |
| Sep 9, 2023 at 20:13 | comment | added | Gabe Goldberg | It's consistent with ZF that there is a singularization of $\omega_1$ that is $\Sigma_5$-definable over $H(\omega_1)$; this happens if one performs a symmetric collapse over $L$ making $\aleph_{\omega}$ become $\aleph_1$. ZFC proves that there is no $\Sigma_5$-definable singularization (since there is no singularization whatsoever). So ZFC is not projectively conservative over ZF, right? | |
| Sep 9, 2023 at 18:31 | comment | added | Joel David Hamkins | See also here mathoverflow.net/a/8392/1946 for the projective conservativity of ZFC over ZF+DC. | |
| Sep 9, 2023 at 18:09 | comment | added | Timothy Chow | Maybe another partial anti-answer would go something like this: If we allow ourselves to assume projective determinacy (PD), then (I believe) there are no known "natural" unprovable statements in second-order arithmetic. This would seem to narrow the space of possible candidates considerably. | |
| Sep 9, 2023 at 17:54 | comment | added | Noah Schweber | @JamesHanson Yes, it's a partial anti-answer. | |
| Sep 9, 2023 at 17:54 | comment | added | James E Hanson | So this would be an anti-answer to question 2 specifically, right? | |
| S Sep 9, 2023 at 17:52 | history | answered | Noah Schweber | CC BY-SA 4.0 | |
| S Sep 9, 2023 at 17:52 | history | made wiki | Post Made Community Wiki by Noah Schweber |