Timeline for A "surnatural numbers" as a largest model of the natural numbers
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jul 5, 2023 at 22:23 | vote | accept | Mike Battaglia | ||
| Apr 6, 2022 at 1:03 | comment | added | Mike Battaglia | I didn't mean constructivity in such a strict sense - I don't think the ordinals wouldn't even be "constructive" in the sense I think you're talking about, let alone the surreals. But I see what you are saying; I guess I am just wondering if there is any way to do this that's different from the usual ultrapower-based method, I guess. | |
| Apr 5, 2022 at 21:38 | comment | added | James E Hanson | @MikeBattaglia These are very classically minded notions of constructivity, but evidence of my claim is Tennenbaum's theorem: There are no computable non-standard models of $\mathsf{PA}$. In terms of reverse math, this means that $\mathsf{RCA}_0 + \mathrm{Con}(\mathsf{PA})$ is not enough to establish the existence of a non-standard model of $\mathsf{PA}$. On the other hand, $\mathsf{WKL}_0 + \mathrm{Con}(\mathsf{PA})$ is enough to establish it. | |
| Apr 5, 2022 at 21:35 | comment | added | James E Hanson | @MikeBattaglia The existence of a non-standard model of $\mathsf{PA}$ can be established in fairly weak theories, but you do need something that would typically be considered mildly non-constructive. You need to be able to find a completion of a given consistent theory. | |
| Apr 5, 2022 at 4:27 | comment | added | Mike Battaglia | I am not sure what I was hoping for - it would be very neat if there were something clever involving "birthdays", and ordinals, and "left and right sets" or something, similarly to the surreals, but which leads to a model of (some reasonable theory of) the naturals rather than the reals. But I am now seeing that these subtleties lead to some very different results regarding constructiveness, and even existence with respect to the ambient set theory. So I would just ask: is there any way to explicitly construct some nonstandard model of at least, let's say PA, using any of these methods? | |
| Apr 5, 2022 at 4:24 | comment | added | Mike Battaglia | After reading your comment in the other post I think I realize how the wording in my question was unclear. The question I was meaning to ask is if it's possible to build either a model of PA, or a constructible model of the entire first-order theory of the naturals (e.g. true arithmetic), and hopefully to do so in some kind of "constructive" manner in the same way that the surreals are constructive. (1/2) | |
| Apr 4, 2022 at 21:54 | comment | added | James E Hanson | @MikeBattaglia Yes. The fully general construction requires the ultrafilter lemma, but constructions of specific set-saturated models don't always require it. | |
| Apr 4, 2022 at 19:37 | comment | added | Mike Battaglia | I'm not quite following - I thought you were saying above that you need the ultrafilter lemma. Is it just that you need the ultrafilter lemma for arbitrary structures, but not for the first-order theory of the reals (for which the surreals suffice even in ZF)? | |
| Apr 4, 2022 at 15:13 | comment | added | James E Hanson | @MikeBattaglia I don't believe you need any choice to show that the surreals are a set-saturated model of $\mathsf{RCF}$. (This should follow from o-minimality, which is really an arithmetic fact about $\mathsf{RCF}$.) You may need some choice to show that they are a set-universal model of $\mathsf{RCF}$ (i.e., every set-sized model of $\mathsf{RCF}$ elementarily embeds into the surreals). | |
| Apr 1, 2022 at 1:17 | comment | added | Mike Battaglia | thanks, that makes sense. I guess the thing I don't get is, you're saying that these kinds of structures entail the ultrafilter lemma, but the surreals exist even in ZF. Is perhaps the idea that, even if the surreals exist, the ultrafilter lemma is still required to prove that they have nice properties like being the monster model of the reals, or that kind of thing? | |
| Mar 31, 2022 at 21:37 | comment | added | James E Hanson | @MikeBattaglia The actual construction is a sort of iterated ultrapower (in the more general sense that set theorists use it). The key lemma is this: There is a uniform procedure that given a structure $M$ and a cardinal $\kappa$ produces an elementary extension $M' \succeq M$ such that for any ultrafilter $\mathcal{U}$ on $\kappa$, $M^{\mathcal{U}}$ embeds into $M'$ in a way that fixes $M$. So in other words, it's possible to 'do all ultrapowers of a given size at the same time.' Ultimately you just iterate this. | |
| Mar 31, 2022 at 21:35 | comment | added | James E Hanson | @MikeBattaglia First of all, the existence of $\kappa$-saturated elementary extensions of arbitrary structures entails the ultrafilter lemma, which suggests that any construction like this will necessarily involve ultrafilters. | |
| Mar 31, 2022 at 17:59 | comment | added | Mike Battaglia | Thanks @JamesHanson. When you say that a formula exists, do you mean something more tangible than, for instance, a proper class sized ultrapower of the naturals? I am kind of curious what that would look like, or if it could be related to the "birthday" structure on the surreal numbers. | |
| Mar 21, 2022 at 22:44 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| Mar 21, 2022 at 22:33 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| Mar 21, 2022 at 22:28 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| Mar 21, 2022 at 22:21 | history | undeleted | James E Hanson | ||
| Mar 21, 2022 at 22:20 | history | edited | James E Hanson | CC BY-SA 4.0 |
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| Mar 21, 2022 at 22:03 | history | deleted | James E Hanson | via Vote | |
| Mar 21, 2022 at 22:03 | history | answered | James E Hanson | CC BY-SA 4.0 |