Timeline for What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Sep 12, 2023 at 17:17 | comment | added | Elliot Glazer | @JamesHanson Sorry my previous comment was just following up on whether Riemannian manifolds are second countable. I haven’t checked if the Prufer manifold can be paracompact. | |
| Sep 12, 2023 at 16:53 | comment | added | James E Hanson | @ElliotGlazer The Prüfer manifold doesn't have a countable dense set. It's the other standard example of a non-paracompact manifold. | |
| Sep 12, 2023 at 15:50 | comment | added | Elliot Glazer | For a connected Riemannian manifold, I think this works: fix $p,$ an $(n-1)$-sphere $S$ centered at $p,$ and a countable dense $X \subset S.$ For points $q$ of rational distance from $p$ via a geodesic through a point in $X,$ take the balls $B(q, 1/n).$ | |
| Sep 12, 2023 at 15:31 | comment | added | James E Hanson | @ElliotGlazer Do you have an argument that the Prüfer manifold can be paracompact in ZF? | |
| Sep 12, 2023 at 14:27 | comment | added | Elliot Glazer | @JamesHanson A second countable manifold is a Polish space, so Shoenfield applies and you get basically the whole ZFC theory, include embeddings into $\mathbb{R}^n.$ I don’t know if being connected metrizable or even a connected Riemannian manifold is enough to prove second countability. | |
| Sep 12, 2023 at 14:20 | comment | added | Elliot Glazer | @JamesHanson They’re inequivalent. It’s a ZF theorem that the long line is not metrizable (if an ordinal $\alpha$ has a metric, one can can transfinitely construct a canonical enumeration for each $\beta \le \alpha$). | |
| Sep 12, 2023 at 12:55 | comment | added | James E Hanson | @ElliotGlazer In the next paragraph, Wald talks about how paracompactness is equivalent to admitting a Riemannian metric and also to being second countable (for manifolds). Are these equivalences still true without choice? | |
| Sep 12, 2023 at 12:43 | comment | added | James E Hanson | @ElliotGlazer I had wondered about this but the precise phrasing in the book really makes me think Wald didn't have this nuance in mind. He says "the 'long line'...is perhaps the simplest example [of a non-paracompact manifold], although the axiom of choice is required to define it." My guess is that Wald picked up the idea that you need choice to even define $\omega_1$, which people sometimes say (and which certainly isn't true). | |
| Sep 12, 2023 at 5:47 | comment | added | Elliot Glazer | @JamesHanson But without choice, it’s consistent that the long line is paracompact (this is equivalent to $\omega_1$ being singular). I bet it’s consistent with ZF that all manifolds are paracompact. | |
| Sep 8, 2023 at 6:49 | comment | added | James E Hanson | @WillieWong Unfortunately defining the parameter-free version $L$ carefully already takes a good chunk of a graduate-level set theory course. Intuitively $L[x]$ is the 'universe of sets that are explicitly definable in terms of $x$ and the ordinals.' | |
| Sep 8, 2023 at 5:14 | comment | added | Willie Wong | A quick n00b question: in your fifth paragraph, what is the relation between $x$ and $L[x]$? You asserted that for every object $x$ of a certain type, there is a model $L[x]$ but not how the two are related. | |
| Sep 8, 2023 at 5:05 | comment | added | Willie Wong | Re: "I think that the aforementioned 'dezornification' could have been done in a completely systematic way using Shoenfield absolutness." I disagree: I am certain that none of the folks involved there knew anything about Shoenfield absoluteness, so we certainly couldn't have done it. :-) Now if you were to want to write a paper... | |
| Sep 7, 2023 at 22:04 | comment | added | Noah Schweber | I know, I was thinking about future instances. :) | |
| Sep 7, 2023 at 22:03 | comment | added | James E Hanson | I didn't ask the question. | |
| Sep 7, 2023 at 22:02 | comment | added | Noah Schweber | Remember, you only get half points if you engineer the context yourself. | |
| Sep 7, 2023 at 22:02 | comment | added | James E Hanson | I have a long and ever growing list of complaints that need very specific contexts to be relevant. | |
| Sep 7, 2023 at 22:00 | comment | added | Noah Schweber | I think I remember you telling me about this error and complaining that you'd never get a chance to say something about it, way back in Madison. :P | |
| Sep 7, 2023 at 21:56 | comment | added | James E Hanson | Since this is literally the only venue in which this is going to be relevant, I would also like to complain that Wald's textbook General Relativity contains an inaccurate statement about the axiom of choice. In Appendix A, he gives the long line as an example of a non-paracompact manifold and states that one needs the axiom of choice to construct it, but this is just wrong. $\mathsf{ZF}$ already proves that the long line exists. | |
| Sep 7, 2023 at 21:51 | history | answered | James E Hanson | CC BY-SA 4.0 |