Timeline for How much Replacement does this axiom provide?
Current License: CC BY-SA 4.0
22 events
| when toggle format | what | by | license | comment | |
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| Jul 27, 2021 at 16:53 | vote | accept | Tom Leinster | ||
| Jun 16, 2021 at 8:28 | comment | added | Asaf Karagila♦ | @FrançoisG.Dorais: Interesting, I always had the impression it was a middle stage. But are you sure that it was developed to show properness? Both axioms seem to have risen in the late 1970s. | |
| Jun 16, 2021 at 3:52 | comment | added | François G. Dorais | @AsafKaragila I believe "Axiom A" was designed (by Baumgartner) as an easy to prove property that implies properness and follows from both ccc and $\sigma$-closed, so it does not precede properness in the historical sense. | |
| May 15, 2019 at 10:04 | review | Suggested edits | |||
| May 15, 2019 at 11:36 | |||||
| May 15, 2019 at 8:25 | history | edited | Tom Leinster | CC BY-SA 4.0 |
added 363 characters in body
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| May 13, 2019 at 12:06 | comment | added | Asaf Karagila♦ | Here's a conjecture. Since having a power set is a $\Sigma_2$ formula, and it's "sorta kinda universal" (in the sense that if you reflect that one, then you're pretty unique, to some extent), it might be that the Cantorian Axiom is equivalent to $\Sigma_2$-Replacement. It might still be weaker, though. I really don't know enough about ETCS (and its identical twin, Bounded Zermelo) to say as much. | |
| May 13, 2019 at 11:45 | answer | added | Joel David Hamkins | timeline score: 6 | |
| May 13, 2019 at 11:25 | comment | added | Tom Leinster | @DavidRoberts, thanks, that's a very apposite quotation! Now I see what Asaf meant about being Cantorian. | |
| May 13, 2019 at 10:17 | comment | added | David Roberts♦ | Yes, this sounds like Cantor's "second principle of generation": if there is some determinate succession of defined whole real numbers, [ie ordinals] among which there exists no greatest, on the basis of this second principle of generation a new number is obtained which is regarded as the limit of those numbers, i.e. is defined as the next greater number than all of them. | |
| May 13, 2019 at 9:48 | comment | added | Asaf Karagila♦ | (Which by the way would be a good name for this axiom.) | |
| May 13, 2019 at 9:46 | comment | added | Asaf Karagila♦ | Very Cantorian of you, Tom! | |
| May 13, 2019 at 9:45 | comment | added | Tom Leinster | @DavidRoberts: I was thinking about the fact that ETCS doesn't establish the existence of $\aleph_\alpha$ for infinite $\alpha$, and wondered what would happen if you just threw in an axiom saying that they did exist. | |
| May 13, 2019 at 9:40 | comment | added | Tom Leinster | Thanks for the various comments @AsafKaragila. You're right, "Axiom A" was just an ad hoc name (which I wasn't planning to use again), and if I'd known it already meant something else, I'd have chosen another one. Too late now! | |
| May 13, 2019 at 7:50 | comment | added | David Roberts♦ | Joel Hamkins points out on Twitter that this axiom is essentially asking that $\aleph_\alpha$ exists for all ordinals $\alpha$ (and like others here that this doesn't hold for certain stages of the von Neumann hierarchy corresponding to beth fixed points). I'm curious to know just why this axiom, and why now? :-) | |
| May 13, 2019 at 5:25 | comment | added | Asaf Karagila♦ | (Oh, and I know that Axiom A is just an ad hoc name, but in case you decide to put this into further use, please note that set theory already have an Axiom A, which is the property which preceded properness in forcing. So maybe another name is preferable?) | |
| May 13, 2019 at 4:58 | comment | added | Asaf Karagila♦ | @David: I know that. I was just pointing this out, in case the thought goes through someone's head. | |
| May 13, 2019 at 2:52 | history | became hot network question | |||
| May 13, 2019 at 2:11 | answer | added | B2C | timeline score: 8 | |
| May 13, 2019 at 1:19 | comment | added | David Roberts♦ | @AsafKaragila yes, a topos has power objects, so we are ok there :-) | |
| May 12, 2019 at 23:03 | comment | added | Asaf Karagila♦ | Also, note that Power set is quite essentially here. Otherwise $H(\omega_1)$, the set of hereditarily countable sets, satisfies ZFC-Power set (including full Replacement), but all infinite sets are equipotent there. | |
| May 12, 2019 at 23:00 | comment | added | Asaf Karagila♦ | One easy way to see that this is weaker than Replacement is to take any $\beth$-fixed point, $\delta$, and look at $V_\delta$ as a model of Zermelo set theory (which is stronger than ETCS) where $\aleph_\alpha$ exists for every ordinal $\alpha$, and every well-ordered set is isomorphic to a von Neumann ordinal. So the question is, essentially, how much Replacement holds for the least such $\delta$... | |
| May 12, 2019 at 22:23 | history | asked | Tom Leinster | CC BY-SA 4.0 |