Timeline for A Baire subset of reals that is not Suslin measurable
Current License: CC BY-SA 4.0
28 events
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| Jan 6, 2022 at 12:29 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Jan 6, 2022 at 12:28 | comment | added | Clement Yung | @AsafKaragila I agree. I'll take note in the future. | |
| Jan 6, 2022 at 12:27 | comment | added | Asaf Karagila♦ | Okay. Just a general writing tip, when a definition is short, or is often known by several different names, one should not refer the reader elsewhere, but instead just include the definition or a clarification. | |
| Jan 6, 2022 at 12:25 | comment | added | Clement Yung | @AsafKaragila I'm just following the terminology in Todorcevic's book. | |
| Jan 6, 2022 at 12:25 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Jan 6, 2022 at 12:24 | comment | added | Asaf Karagila♦ | Wouldn't it be easier to just say "the Boolean algebra generated by the open sets"? (Or at least "the algebra of sets"?) | |
| Jan 6, 2022 at 12:23 | comment | added | Clement Yung | @AsafKaragila please see my edit. | |
| Jan 6, 2022 at 12:23 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Jan 6, 2022 at 12:18 | comment | added | Asaf Karagila♦ | What is "the field of open subsets"? | |
| Jan 6, 2022 at 7:23 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Dec 22, 2021 at 5:02 | vote | accept | Clement Yung | ||
| Dec 20, 2021 at 18:57 | answer | added | Gerald Edgar | timeline score: 7 | |
| Dec 20, 2021 at 16:19 | comment | added | Emil Jeřábek | @DaveLRenfro The OP's Suslin measurable sets are coanalytic rather than analytic. | |
| Dec 20, 2021 at 16:15 | comment | added | Dave L Renfro | as well as ZFC-consistently beyond Baire and Lebesgue measurable sets (beginning with 2nd level?), but they are also less "explicitly definable". I don't know all that much about this, but I believe one way of viewing things is that the better known projective set operation can, in some sense (in a way independent of ZFC?), leap-frog over a lot of this other stuff rather quickly. | |
| Dec 20, 2021 at 16:06 | comment | added | Dave L Renfro | Unless I'm missing something, your Suslin (measurable) sets are the same as what I (and the references I cited) call Suslin sets, which are also called analytic sets (an unfortunate term that could suggest a close connection with complex-analytic functions or with analytic sets in algebraic geometry, although the term analytic set as used here has been around since the 1920s I think). Incidentally, the hierarchy of projective sets goes well beyond the $R$-sets (continued) | |
| Dec 20, 2021 at 15:45 | comment | added | Clement Yung | @DaveLRenfro just to check, your Suslin set is the complement of a Suslin measurable set? (complement of analytic is coanalytic) | |
| Dec 20, 2021 at 15:21 | comment | added | Dave L Renfro | in which the applications of these two processes are interlaced, with a reasonable limiting/union of previous hierarchy levels at limit ordinal levels), and the much more exotic higher levels of Kolmogorov's $R$-sets (see here also). Finally, The Ramsey sets and related sigma algebras and ideals by Jack Brown (1990) might be useful. | |
| Dec 20, 2021 at 15:21 | comment | added | Dave L Renfro | In ordinary ZFC, there is a huge gap between Suslin sets (= analytic sets) and Baire sets that is very analogous to the huge gap between Suslin sets and Lebesgue measurable sets. Indeed, there is even a huge gap between Suslin sets and sets simultaneously Baire and Lebesgue measurable. Regarding this last gap, one finds in it the $\sigma(\Sigma_1^1)$ sets $(\sigma$-algebra generated by the Suslin sets), and the $C$-sets (the collection of sets closed under $\sigma$-algebra generation and the Suslin operation -- the end result of an $\omega_1$-length hierarchy (continued) | |
| Dec 20, 2021 at 14:56 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Dec 20, 2021 at 13:40 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Dec 20, 2021 at 13:33 | answer | added | Gabe Goldberg | timeline score: 2 | |
| Dec 20, 2021 at 13:32 | comment | added | Andreas Lietz | It seems to me that this definition of Suslin measurability yields exactly the coanalytic sets (cf. Moschovakis Theorem 2B.1 & 2B.2, note that there the Suslin operation is the "complement" of the Suslin operation here). So the universal analytic set should do the trick. This all works in ZF+DC. | |
| Dec 20, 2021 at 13:32 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Dec 20, 2021 at 13:30 | comment | added | Clement Yung | @GeraldEdgar also yes, indeed the Feferman-Levy model works for my second question. Thank you. | |
| Dec 20, 2021 at 13:30 | comment | added | Clement Yung | @GeraldEdgar The only proof of Nikodym's result I can find is in Todorcevic's book, in which a lemma requires the Banach Category theorem. According to the comments in this answer, the full AC is required. I can't find the original paper for his result. | |
| Dec 20, 2021 at 13:09 | history | edited | Clement Yung | CC BY-SA 4.0 |
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| Dec 20, 2021 at 12:52 | comment | added | Gerald Edgar | Define "Baire subsets", there are two different things possible with this name. Does ZF prove Nikodym's result? Surely a countable union of countable sets is Suslin measurable? And ZF cannot disprove: "every subset of $\mathbb R$ is a countable union of countable sets". | |
| Dec 20, 2021 at 12:16 | history | asked | Clement Yung | CC BY-SA 4.0 |