Timeline for An axiomatic approach to the multiverse of sets
Current License: CC BY-SA 4.0
43 events
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| S Jul 26 at 0:05 | history | bounty ended | CommunityBot | ||
| S Jul 26 at 0:05 | history | notice removed | CommunityBot | ||
| Jul 18 at 15:16 | history | edited | Alec Rhea | CC BY-SA 4.0 |
edited terminology according to Joel's recommendation
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| Jul 18 at 14:49 | comment | added | Alec Rhea | @JoelDavidHamkins Thank you for the terminology correction, I’ll edit accordingly. Yes, I think that would be a model of the multiverse in $T_\emptyset$; if things work as intended this is a feature, showing that $T_\emptyset$ places minimal requirements on what the multiverse ‘needs’ to contain, allowing us build up a multiverse containing ‘as much as possible’ from this base version which is’t required to contain anything except the empty set. | |
| Jul 18 at 0:50 | comment | added | Joel David Hamkins | It seems that $V=\{\varnothing\}$ is a standard transitive universe. Is there anything preventing the "multiverse" $\mathcal{M}$ from having this $V$ as its only member? | |
| Jul 17 at 23:30 | comment | added | Joel David Hamkins | What you call "complete" is usually called supertransitive. That is, a transitive set that contains all the subsets of its elements. | |
| Jul 17 at 22:30 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| S Jul 17 at 22:20 | history | bounty started | Alec Rhea | ||
| S Jul 17 at 22:20 | history | notice added | Alec Rhea | Draw attention | |
| Jul 17 at 22:19 | history | edited | Alec Rhea | CC BY-SA 4.0 |
answered comments, about to re-bounty question
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| S Jun 30, 2023 at 17:05 | history | bounty ended | CommunityBot | ||
| S Jun 30, 2023 at 17:05 | history | notice removed | CommunityBot | ||
| Jun 22, 2023 at 16:58 | comment | added | James E Hanson | On the other hand if the predicates $\phi$ aren't entirely first-order, you open up a can of worms regarding the possibility of two extensions being individually consistent without being mutually consistent. Say for example, $\phi$ says "$V$'s copy of $2^{\mathbb{R}}$ is the real $2^{\mathbb{R}}$ and CH holds" and $\psi$ says "$V$'s copy of $2^{\mathbb{R}}$ is the real $2^{\mathbb{R}}$ and CH fails." In this case you can't have models of both $\phi$ and $\psi$ existing at the same time. (This is the well-known issue that second-order logical entailment is not absolute.) | |
| Jun 22, 2023 at 16:58 | comment | added | James E Hanson | I'm having a bit of difficulty seeing how axiom 8 is going to work. First of all, it's not really clear how you can formalize the notion of an object's existence being instantiated by an axiom without putting in explicit machinery to track this (e.g., Skolem functions). Also, if $V \preceq V+\phi$, then $V$ and $V+\phi$ have the same first-order theory. This means that if $\phi$ is a first-order property of the $V$ in question, you can't have both $V+\phi$ and $V+\neg \phi$ existing as elementary extensions. | |
| Jun 22, 2023 at 16:34 | comment | added | Mikhail Katz | Hi Alec, Could you possibly elaborate on how your proposed axiomatisation is different from Hamkins's, and what you are trying to accomplish in non-technical terms? | |
| S Jun 22, 2023 at 15:59 | history | bounty started | Alec Rhea | ||
| S Jun 22, 2023 at 15:59 | history | notice added | Alec Rhea | Draw attention | |
| Jun 8, 2022 at 2:53 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Jun 2, 2022 at 20:49 | history | edited | Alec Rhea | CC BY-SA 4.0 |
correction
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| Jun 2, 2022 at 20:24 | history | edited | Alec Rhea | CC BY-SA 4.0 |
added elementary submodel axioms
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| Jun 2, 2022 at 20:17 | history | edited | Alec Rhea | CC BY-SA 4.0 |
fixed typo
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| Jun 2, 2022 at 19:58 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Jun 2, 2022 at 19:40 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| S Jun 1, 2022 at 10:07 | history | bounty ended | CommunityBot | ||
| S Jun 1, 2022 at 10:07 | history | notice removed | CommunityBot | ||
| May 31, 2022 at 2:04 | history | edited | Alec Rhea | CC BY-SA 4.0 |
fixed missing bracket
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| May 31, 2022 at 1:09 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 31, 2022 at 0:53 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 31, 2022 at 0:40 | history | edited | Alec Rhea | CC BY-SA 4.0 |
changed axioms to allow for correct notion of modeling, fixed links in accordance with guidelines at https://meta.mathoverflow.net/questions/5346/citing-papers-in-questions-answers
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| May 29, 2022 at 3:49 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 29, 2022 at 3:08 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 29, 2022 at 3:01 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 29, 2022 at 2:54 | history | edited | Alec Rhea | CC BY-SA 4.0 |
changed proposed axioms statement, fixed links as suggested by Emil here: https://meta.mathoverflow.net/questions/5346/citing-papers-in-questions-answers
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| May 25, 2022 at 18:32 | comment | added | Alec Rhea | @PaceNielsen When trying to edit just now, I realized that 'the multiverse' I am referring to is the one that Hamkins asserts set theorists implicitly work in when considering forcing arguments and the like. For me to say that I am axiomatizing 'a multiverse' makes the question pointless in a certain sense; basically whatever I consider to be 'a multiverse' would be an affirmative answer, but that's a boring question. I suppose what I mean by asking for 'the multiverse' is the one implicitly used by set theorists, and wether or not these axioms fully and faithfully describe that multiverse. | |
| May 25, 2022 at 18:25 | comment | added | Alec Rhea | @PaceNielsen Good point; implicitly I am hoping that the 'correct' notion of multiverse will admit a metatheorem to the effect that any 'multi-multiverse' we define is already equiconsistent with the 'correct' theory of the multiverse. But I have no idea if such a metatheorem holds for any definition, let alone this one, so I take your point and I'll edit accordingly. | |
| May 24, 2022 at 18:48 | comment | added | Pace Nielsen | Alec, your post (and title) consistently uses the phrase "the multiverse". I see the mathematical value of having an individual constant symbol, but why do you call it "the multiverse" instead of "a multiverse"? (I ask because it strikes me as a little strange to allow for pluralism at the level of "class of all sets", but not at the next level.) | |
| S May 24, 2022 at 8:56 | history | bounty started | Alec Rhea | ||
| S May 24, 2022 at 8:56 | history | notice added | Alec Rhea | Draw attention | |
| May 23, 2022 at 5:31 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 22, 2022 at 6:48 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 21, 2022 at 15:48 | answer | added | Alexander Praehauser | timeline score: 4 | |
| May 21, 2022 at 12:28 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| May 21, 2022 at 2:05 | history | asked | Alec Rhea | CC BY-SA 4.0 |