Timeline for What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
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28 events
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| Sep 13, 2023 at 20:37 | comment | added | James E Hanson | @ElliotGlazer Oh I see, I misread 'non-principal measure' as 'non-principal ultrafilter.' | |
| Sep 13, 2023 at 19:31 | comment | added | Elliot Glazer | @JamesHanson Even full HB does not imply there is an np ultrafilter on $\mathbb{N},$ see jstor.org/stable/2272118 Thm 2. Incidentally, full UFL can be decomposed into the ordering principle (OP) and compactness for ordered languages (CfO), the latter implying "HB-for-sep-normed-spaces." I've been investigating the conjecture that OP and CfO are separately (but not together) consistent with a total, isometry-invariant extension of Lebesgue measure on $\mathbb{R}^n.$ | |
| Sep 13, 2023 at 14:50 | comment | added | James E Hanson | There are going to be many intermediate principles. For instance, you could limit HB to Banach spaces of a certain density character. | |
| Sep 13, 2023 at 9:28 | comment | added | Bastam Tajik | @ElliotGlazer yes absolutely. Sorry I edited the question. But is there intermediatory level between ZF and ZF+HB? | |
| Sep 13, 2023 at 9:27 | history | edited | Bastam Tajik | CC BY-SA 4.0 |
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| Sep 13, 2023 at 4:54 | comment | added | Elliot Glazer | The chain in your PS isn't quite right. HB does not imply ADC. In fact, UFL holds and countable choice fails in the Cohen model with an infinite Dedekind finite set of reals. | |
| Sep 9, 2023 at 17:03 | vote | accept | Bastam Tajik | ||
| Sep 9, 2023 at 12:33 | answer | added | Timothy Chow | timeline score: 6 | |
| Sep 8, 2023 at 22:09 | comment | added | Ryan Budney | @JamesHanson: I suspect the author is perhaps thinking of general relativity not as a physical theory, but as some very specific mathematical model encoded in one specific language, which is unspoken. I would say that isn't really GR, but more of a "play model". | |
| Sep 8, 2023 at 20:12 | comment | added | Timothy Chow | Related: Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle? | |
| Sep 8, 2023 at 5:57 | comment | added | James E Hanson | @RyanBudney I don't understand your comment. What is the misunderstanding exactly? | |
| Sep 8, 2023 at 1:38 | comment | added | Ryan Budney | I think this question is hiding the issue of a misunderstanding of what General Relativity is, making it into a bit of a straw man argument. | |
| Sep 8, 2023 at 0:03 | comment | added | Aaron Bergman | I was thinking of the separable Banach space case, which I think uses much less than ZF+DC. As a (former) physicist, I mainly looked into this to make sure I didn’t need to worry about it and then subsequently forgot the details, so definitely not an expert. | |
| Sep 7, 2023 at 23:03 | comment | added | James E Hanson | @YemonChoi My intuition for why it should work is that the orthogonal projection is definable in the sense of continuous logic in a Hilbert space expanded with the distance predicate of the subspace. In particular this tells you that there's a very absolute formula that defines the distance to the orthogonal projection. A more analysis argument would be to establish that you can build a canonical Cauchy filter whose limit is the orthogonal projection. You need to require that Hilbert spaces are closed under limits of Cauchy filters, but without choice you should be doing that anyway. | |
| Sep 7, 2023 at 23:00 | comment | added | Yemon Choi | @JamesHanson It's not immediately clear to me how you define the orthogonal projection onto an arbitrary closed subspace of an arbitrary Hilbert space without using DC: all proofs I am aware of use approximating sequences and taking a limit (if the subspace is infinite-dimensional) | |
| Sep 7, 2023 at 21:51 | answer | added | James E Hanson | timeline score: 12 | |
| Sep 7, 2023 at 16:36 | comment | added | James E Hanson | @AaronBergman Can't you prove HB for (arbitrary) Hilbert spaces in ZF alone? Given a linear functional on a closed subspace $X$ of a Hilbert space $H$, you extend by letting the functional be $0$ for anything in $X^\perp$. | |
| Sep 7, 2023 at 15:24 | comment | added | Aaron Bergman | For what it's worth, if I ever found any physical result that depended on anything more than ZF+DC, I would be very skeptical. That gets you HB for separable Hilbert spaces, for example, which covers everything I've seen in QM, although I know you asked about GR. | |
| Sep 7, 2023 at 15:18 | comment | added | Bastam Tajik | Can we ignore the motivation that I shared, and get to the point of the question? GR within ZF or ZF+ADC | |
| Sep 7, 2023 at 15:12 | comment | added | James E Hanson | @AaronBergman Sure, but I would argue that cardinality not being a linear order in the first place could also be regarded as failing to adhere to 'common sense.' Ultimately my point is that there's always going to be counterintuitive things once you start working with infinite sets and the ire directed towards stuff like Banach-Tarski in particular is misguided. | |
| Sep 7, 2023 at 14:36 | comment | added | Aaron Bergman | Isn't that just a reflection that the notion of larger and smaller cardinality breaks down without choice? mathoverflow.net/questions/260057/… | |
| Sep 7, 2023 at 14:07 | comment | added | James E Hanson | Furthermore, I'm a pretty firm believer that any 'physically meaningful' theory needs to boil down to a computational model that can in principle actually be implemented on a computer. Under this criterion, we know by absoluteness results that no 'physically meaningful' math can actually depend on things like AC or Hahn-Banach. | |
| Sep 7, 2023 at 14:07 | comment | added | James E Hanson | 'Departure from common sense' is a pretty relative thing. If there are no non-measurable subsets of $\mathbb{R}$, then there is an equivalence relation $\sim$ on $\mathbb{R}$ such that $\mathbb{R}/\sim$ has strictly larger cardinality than $\mathbb{R}$. | |
| Sep 7, 2023 at 12:27 | history | edited | Bastam Tajik | CC BY-SA 4.0 |
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| Sep 7, 2023 at 9:21 | history | edited | gmvh | CC BY-SA 4.0 |
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| Sep 7, 2023 at 7:08 | history | edited | Bastam Tajik | CC BY-SA 4.0 |
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| S Sep 7, 2023 at 6:44 | review | First questions | |||
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| S Sep 7, 2023 at 6:44 | history | asked | Bastam Tajik | CC BY-SA 4.0 |