Timeline for Taller models of ZFC
Current License: CC BY-SA 3.0
19 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 29, 2016 at 19:48 | vote | accept | Mirco A. Mannucci | ||
| Jan 29, 2016 at 19:10 | answer | added | Ali Enayat | timeline score: 14 | |
| Jan 29, 2016 at 12:44 | comment | added | Stefan Hoffelner | @Mirco Consider the strictly increasing $\omega_1$-sequence of critical points of the iterated ultrapower maps, i.e let $\kappa_0$ be the critical point of $j_0: M \rightarrow Ult$, $\kappa_1$ the critical point of $j_1: Ult(M, U) \rightarrow Ult(Ult(M,U), U_1),... $ The $\kappa_i$'s will be elements of the direct limit. | |
| Jan 29, 2016 at 11:58 | comment | added | Mirco A. Mannucci | @StefanHoffelner : are you sure that your omega_1 ultrapower has height strictly greater than the one of M? | |
| Jan 29, 2016 at 10:37 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jan 29, 2016 at 10:30 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jan 29, 2016 at 10:29 | comment | added | Mirco A. Mannucci | @VictoriaGitman absolutely! I will add it now in the question. | |
| Jan 29, 2016 at 10:27 | comment | added | Mirco A. Mannucci | @CarlMummert here M is a transitive SET, which happens to be a model of ZFC | |
| Jan 29, 2016 at 10:25 | comment | added | Mirco A. Mannucci | @EmilJeřábek I meant using just ZFC, no more, no less | |
| Jan 29, 2016 at 1:02 | comment | added | Victoria Gitman | @Mirco: Maybe (2) should be modified to say that $M=V_{\eta_0}^N$. It seems this would capture better the notion of "making a model taller". | |
| Jan 28, 2016 at 18:31 | comment | added | Asaf Karagila♦ | @Carl: Yes, I'm just saying that the axiom "there is a proper class of $V_\alpha$'s which are models of ZFC" is weaker than the existence of an inaccessible. Of course there can be just one single worldly cardinal (cut the universe at the second worldly cardinal!). So all I'm saying is that you can get it from even milder assumptions. :-) | |
| Jan 28, 2016 at 18:18 | comment | added | Carl Mummert | @Asaf Karagila: I think you are thinking about the consistency of the statement (you are saying it is consistent relative to the existence of one inaccessible), but I was thinking about the provability of the statement. Is it consistent with ZFC that the collection of wordly cardinals is nonempty and bounded? In my earlier comment, I was trying to suggest that the axiom would follow from very mild large cardinal assumptions. | |
| Jan 28, 2016 at 18:14 | comment | added | Asaf Karagila♦ | @Carl: Just one inaccessible cardinal is enough. Since an inaccessible cardinal is the limit of worldly cardinals (those that $V_\alpha$ is a model of ZFC), so if $\kappa$ is inaccessible $V_\kappa$ satisfies that "There is a proper class of $\alpha$'s such that $V_\alpha$ is a model of ZFC". You can probably even have something weaker like $\kappa$ is a worldly limit of worldly cardinals. | |
| Jan 28, 2016 at 17:07 | comment | added | Stefan Hoffelner | Concerning your question 1 which is stated quite vaguely, maybe iterated ultrapowers would meet your requirements. If you assume a say countable model $M$ with an $M$-ultrafilter $U$ you can form the ultrapower $Ult(M,U)$ and continue. At limit stages you take the direct limit. Performing this $\omega_1$-many times and assuming that the final model is wellfounded you end up with an $\omega_1$ sized model. | |
| Jan 28, 2016 at 16:26 | comment | added | Carl Mummert | In (2), are you referring to transitive set models (so that the question is interpreted within a single class model of ZFC, so to speak) or are you asking about class models (so that the question is about the relationship between different class models of ZFC)? In the former case, doesn't (2) follow from the axiom that every set is contained in a transitive set model of ZFC, which in turn follows just from the existence of arbitrarily large inaccessible cardinals? | |
| Jan 28, 2016 at 15:06 | comment | added | Emil Jeřábek | In 1: “..without adding new axioms” – adding where? Can you clarify what does the sentence mean? | |
| Jan 28, 2016 at 13:47 | history | edited | Mirco A. Mannucci | CC BY-SA 3.0 |
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| Jan 28, 2016 at 13:38 | history | asked | Mirco A. Mannucci | CC BY-SA 3.0 |