Timeline for Measurability and Axiom of choice
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 14, 2015 at 20:19 | vote | accept | Matthias Ludewig | ||
| Jul 14, 2015 at 20:01 | history | edited | Benoît Kloeckner | CC BY-SA 3.0 |
Changed a "Borel" by a "Lebesgue".
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| Jul 14, 2015 at 12:19 | answer | added | Joel David Hamkins | timeline score: 45 | |
| Jul 14, 2015 at 12:18 | history | edited | Matthias Ludewig | CC BY-SA 3.0 |
added 3 characters in body
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| Jul 14, 2015 at 12:12 | answer | added | Gerald Edgar | timeline score: 6 | |
| Jul 14, 2015 at 12:04 | comment | added | Andrés E. Caicedo | (To see that some restrictions are needed, either by requiring the definition to be "sufficiently simple", or by having to assume a theory beyond $\mathsf{ZFC}$, consider that well-orderings of $\mathbb R$, seen as subsets of $\mathbb R^2$, are not measurable. This is a classical result due to Sierpinski. On the other hand, it is consistent with $\mathsf{ZFC}$ that $V=L$, which implies that there is a (simply, though not quite "very" simply) definable well-ordering of the reals.) | |
| Jul 14, 2015 at 12:00 | comment | added | Simon Henry | Isn't the general form of this argument just wrong ?: it only proves that it is not possible to prove in ZF that the function is non-measurable, it does not prove that it is actually possible to prove that it is measurable, and I don't even think it shows that ZFC will not prove that the function is non measurable. | |
| Jul 14, 2015 at 11:59 | comment | added | Andrés E. Caicedo | Yes, this can be made rigorous. If the definition is simple enough (measured in descriptive set theoretic terms), this follows from classical theorems on so-called analytic sets. For much more generous notions of definability, this is a theorem, but not of $\mathsf{ZFC}$ alone. Rather, of the theory resulting from extending $\mathsf{ZFC}$ with appropriate large cardinals. For an example of what I mean here, see this paper; stronger results are possible. | |
| Jul 14, 2015 at 11:42 | history | edited | Asaf Karagila♦ |
edited tags
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| Jul 14, 2015 at 11:39 | history | asked | Matthias Ludewig | CC BY-SA 3.0 |