Timeline for If a result is apparently provable with AC, is actually independent of ZF?
Current License: CC BY-SA 2.5
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 18, 2011 at 19:21 | vote | accept | Gabriel Furstenheim | ||
| Feb 5, 2011 at 22:38 | answer | added | Andreas Blass | timeline score: 5 | |
| Feb 5, 2011 at 19:44 | answer | added | Jim Belk | timeline score: 7 | |
| Feb 5, 2011 at 19:33 | answer | added | Not Mike | timeline score: 2 | |
| Feb 5, 2011 at 18:00 | answer | added | Gerald Edgar | timeline score: 7 | |
| Feb 5, 2011 at 15:42 | comment | added | Gabriel Furstenheim | @Qiaochu The examples I'm looking for are like the Cantor-Bernstein theorem. | |
| Feb 5, 2011 at 15:34 | comment | added | Qiaochu Yuan | I don't think the question is well-defined as stated. You can trivially insert "by the Axiom of Choice, ..." into the proof of any statement and then remove it. I think you meant to say "if a result apparently depends on AC, does it necessarily?" and then there are a million counterexamples. | |
| Feb 5, 2011 at 15:01 | comment | added | Asaf Karagila♦ | The Cantor-Bernstein theorem was first proved using AC but is provable without it as well. | |
| Feb 5, 2011 at 14:39 | comment | added | darij grinberg | Yes, but few of their applications do. If you read some French, try out this: hlombardi.free.fr/liens/constr.html (particularly "Algèbre Commutative"). There are lots of results of the kind "any sufficiently elementary theorem provable in ZFC is provable in ZF or even constructively", where "sufficiently elementary" means things like "1st order formula", "geometric formula" and the likes. Some of these results, ironically, require ZFC themselves, while others don't. Alas, I am not an expert in this field ("program extraction from classical proofs"). | |
| Feb 5, 2011 at 14:23 | comment | added | Gabriel Furstenheim | @darij At least the basic results in algebra need AC: the different definitions of noetherian aren't equivalent without DC. ( (mathoverflow.net/questions/53523/maximal-ideal-and-zorns-lemma). | |
| Feb 5, 2011 at 14:06 | comment | added | darij grinberg | I don't understand your question. 98% of theorems in advanced algebra that are usually proven with the AC don't depend on the AC. | |
| Feb 5, 2011 at 13:52 | history | asked | Gabriel Furstenheim | CC BY-SA 2.5 |