Timeline for Absoluteness, reflection to ctms, and choice in outer models
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 14, 2017 at 13:52 | comment | added | Joel David Hamkins | @ElliotGlazer Yes, one can relativize the proof to any parameter. | |
| May 14, 2017 at 13:49 | comment | added | Asaf Karagila♦ | @Elliot: It's not at all surprising. The machinery needed here is far more complicated than the one needed for (1). You don't need Shoenfield, forcing or other various nontrivial facts presented here in order to prove (1) in ZF+DC. You just need Reflection, downward LS, and Mostowski collapses. This is why you can teach (1) in a basic course of axiomatic set theory, and maybe the choice-free proof in a third course after taking forcing and basic descriptive set theory as well. | |
| May 14, 2017 at 13:49 | vote | accept | Elliot Glazer | ||
| May 14, 2017 at 13:43 | comment | added | Elliot Glazer | Just a nitpick, you only proved $HC$ is correct about $\Sigma_1$ sentences, not $\Sigma_1$ formulae. Not that it really matters, since Shoenfield absoluteness can be relativized to a real (and therefore any element in $HC,$ I think), so the same proof carries through. | |
| May 14, 2017 at 13:40 | comment | added | Elliot Glazer | Wow, it's surprising to me that statement 1, which is a theorem in pretty much every textbook, is always stated with an unnecessary assumption. Maybe that's just for simplicity of proof, since this theorem is usually earlier than the sections on forcing and well-founded trees. | |
| May 14, 2017 at 13:38 | comment | added | Asaf Karagila♦ | Nice. Real nice! | |
| May 14, 2017 at 12:51 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |