Timeline for Is there any version of the Banach-Tarski paradox in ZF?
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17 events
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| Aug 22, 2021 at 12:13 | comment | added | Esa Pulkkinen | @David Roberts Yes, thank you. | |
| Aug 22, 2021 at 11:27 | comment | added | David Roberts♦ | @Esa yes, they exist by the definition of a topology (the open sets are the data of a topology, the closed sets are their complements). No AC is needed. The new points in the closure are already elements of the given space. It seems you may be thinking of a completion, which is not what is happening here. | |
| Aug 22, 2021 at 10:14 | comment | added | Esa Pulkkinen | @Timothy Chow When you have an open set and construct its closure, where do the additional limit points come from? I mean typically they represent "points at infinity" of sequences of elements taken from the open sets. When you consider cl S as intersection of all closed sets containing S, can the "all closed sets" (including their limit points) be assumed to exist without invoking AC? | |
| Aug 21, 2021 at 13:06 | comment | added | Timothy Chow | @EsaPulkkinen What is wrong with the elementary definition of the closure of a set as the intersection of all closed sets containing it? | |
| Aug 21, 2021 at 7:32 | comment | added | Esa Pulkkinen | Hmm. I wonder if this means AC is hidden in the definition of closure. I mean, closure operator can be split to epimorphism and section [1] which form a Galois connection (the closure operator can be recovered by composition). But the section itself is normally proven to exist by the AC which states that every epi has section. So assuming closure exists assumes AC? Or we must assume AC to split the closure to epi and section? [1] Lawvere, Rosebrugh:Sets for Mathematics | |
| Aug 21, 2021 at 5:58 | comment | added | David Roberts♦ | The paper explicitly clarifies constructive to mean without the use of AC. Not saying the theorem is not actually possible in intuitionistic logic, though, just that the paper doesn't seem to be claiming it works in that setting. | |
| Aug 21, 2021 at 0:32 | comment | added | paul garrett | This fact is quite amazing to me! Thanks for bringing it into this forum! :) | |
| Aug 20, 2021 at 23:59 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Aug 20, 2021 at 22:07 | comment | added | user334725 | At the end of Section 3 in the longer paper, they note one needs to: fix an enumeration of the given countable group of homeomorphisms, fix an enumeration of a countable dense subset of the given separable metric space X (thereby giving a countable base for its topology, via rational radii), and give a constructive proof of the Baire category theorem. | |
| Aug 20, 2021 at 21:36 | comment | added | Paul Blain Levy | Reading again, I think Feferman was right and it really is constructive, but a more careful eye is needed than mine. Some versions of constructivism have multiple not-provably-isomorphic versions of $\mathbb{R}$, so the question may be subtle. | |
| Aug 20, 2021 at 21:09 | comment | added | Paul Blain Levy | I have removed my constructivity claim. | |
| Aug 20, 2021 at 21:09 | comment | added | Paul Blain Levy | In this paper by Feferman, the following is stated: "Not only does the proof not make use of AC, it is completely constructive in the data." However, as noted by @FrançoisG.Dorais, the authors Dougherty and Foreman seem to use "constructive" to mean just avoidance of AC, so perhaps Feferman was mistaken? This needs to be investigated. | |
| Aug 20, 2021 at 21:02 | history | edited | Paul Blain Levy | CC BY-SA 4.0 |
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| Aug 20, 2021 at 20:38 | comment | added | user334725 | The given citation is to the PNAS paper, and a fuller description is given (by the same authors) in: ams.org/journals/jams/1994-07-01/S0894-0347-1994-1227475-8/… one cannot explain away their paradoxical nature by blaming it on [AC]. Instead, one can note that the open sets resulting from our construction have boundaries of positive measure. When the open sets are packed into a small space, these boundaries overlap greatly; when the open sets are rearranged within a larger set, the boundary overlap is less extreme, so the total measure of the closures is greater. | |
| Aug 20, 2021 at 19:50 | comment | added | François G. Dorais | The "constructive" adjective most often includes not using the law of excluded middle. So "classically constructive" would be a more precise adjective in this case. (Uses of "constructive" are generally relative to the author and authors with more a restrictive notion of "constructive" tend to use it more often, so this is a touchy subject. On MO, and also in more general settings,a search often leads to "most often used in this context", so you can see this is sometimes problematic regardless of what you or anyone else thinks is most appropriate.) [FWIW: I am not a constructivist.] | |
| Aug 20, 2021 at 14:40 | history | edited | Paul Blain Levy | CC BY-SA 4.0 |
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| Aug 20, 2021 at 14:32 | history | answered | Paul Blain Levy | CC BY-SA 4.0 |