Timeline for Is there any version of the Banach-Tarski paradox in ZF?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 31, 2021 at 13:52 | comment | added | Dabed | A somewhat related twitter thread | |
| S Aug 23, 2021 at 12:25 | history | suggested | psmears | CC BY-SA 4.0 |
Improve wording and grammar
|
| Aug 23, 2021 at 11:53 | review | Suggested edits | |||
| S Aug 23, 2021 at 12:25 | |||||
| Aug 21, 2021 at 2:11 | comment | added | Robert Furber | The fact that there is no finitely-additive SO(3)-invariant measure on S2 with the discrete σ-algebra has been proven in ZF + Hahn-Banach by Foreman and Wehrung: hal.archives-ouvertes.fr/hal-00004713 Their argument transports the lack of invariant measure from F2 to S2. It can then be used to transport F2's explicit paradoxical decomposition across: matwbn.icm.edu.pl/ksiazki/fm/fm138/fm13813.pdf | |
| Aug 20, 2021 at 23:56 | comment | added | David Roberts♦ | Given that the BT result deals with objects no bigger than $2^\mathfrak{c}$, certainly AC could be removed in favour of only needing it for very small sets at most. The question is what kind of weak choice principle is needed to establish the required results about how the free subgroup on two generators of SO(3) leads to the required decomposition. Incidentally, there is a thorough elementary discussion in this project report arxiv.org/abs/2108.05714 | |
| Aug 20, 2021 at 22:17 | history | became hot network question | |||
| Aug 20, 2021 at 22:03 | comment | added | Sam Hopkins | In light of the very nice answer of Paul, I retract my insinuation that this question might be too basic. But I will leave the link to the Solovay model, which is still certainly relevant for some considerations... | |
| Aug 20, 2021 at 17:45 | comment | added | YCor | "which is independent of AC": I guess you mean "which is a theorem of ZF". The formulation that P is independent of AC might (?) mean that both ZFC+P and ZFC+(not P) are consistent. | |
| Aug 20, 2021 at 15:23 | comment | added | Asaf Karagila♦ | To add on to what @Sam wrote; many theorems in analysis either don't use choice, or their specific uses in classical analysis don't use choice (e.g. Baire Category Theorem is equivalent to Dependent Choice; but for separable spaces it is provable in ZF). Asking about development of the whole of analysis in ZF or ZF+DC is tantamount to going through Rudin (or some other book on analysis) and checking each theorem to see what you can or cannot do wit or without choice. Some places do that, to some extent (e.g. Schechter's book), but generally it's too broad of a question for a Q&A website. | |
| Aug 20, 2021 at 14:32 | answer | added | Paul Blain Levy | timeline score: 57 | |
| Aug 20, 2021 at 14:23 | history | edited | mahdi meisami | CC BY-SA 4.0 |
edited title
|
| Aug 20, 2021 at 14:18 | comment | added | Sam Hopkins | Depends on what you mean exactly. There are models of ZF in which every set of real numbers is measurable (en.wikipedia.org/wiki/Solovay_model), which would prevent a lot of the paradoxes you might have in mind. This question is probably too basic for MO, though. | |
| Aug 20, 2021 at 14:16 | history | asked | mahdi meisami | CC BY-SA 4.0 |