Timeline for What if Current Foundations of Mathematics are Inconsistent?
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 19, 2020 at 22:14 | comment | added | user76284 | "Unlikely" under what probability distribution? | |
| Oct 15, 2010 at 12:10 | comment | added | Todd Trimble | @Andy: thanks! That's very interesting. | |
| Oct 6, 2010 at 23:36 | comment | added | Ryan Budney | @Peter LeFanu Lumsdaine: of course you'd have to remove an axiom from ZFC and replace it with something weaker. | |
| Oct 6, 2010 at 19:20 | comment | added | Andy Putman | @Todd : It's actually a theorem that in a precise sense, almost all group presentations are word hyperbolic (see en.wikipedia.org/wiki/Hyperbolic_group). In particular, almost all algorithmic problems are solvable within a "random group presentation". | |
| Oct 4, 2010 at 17:17 | comment | added | Peter LeFanu Lumsdaine | I'm also intrigued: what does Freedman intend by "patch [the inconsistent theory] with a new axiom"? Strengthening a too-weak theory is easy, and can indeed be patched — at the crudest level, you can just add an axiom doing whatever you want. But weakening an inconsistent theory is harder: all the existing axioms work together in complicated ways, and taking any one axiom out usually makes it break down (it certainly does with ZFC), so you have to rewrite at least parts of the theory from scratch. Hence things like constructive set theories, dependent type theory, etc.. | |
| Oct 4, 2010 at 12:11 | comment | added | Todd Trimble | @Harrison: interesting question(s). My first wonder is whether it's even (intuitively) true, that most theories are inconsistent; I can't make up my mind. One could ask similar but perhaps simpler questions like, "are most group presentations presentations of the trivial group?" Here I think my instinct leans more towards saying that "most" group presentations, if not of the trivial group, are undecidable. | |
| Oct 4, 2010 at 11:46 | comment | added | Jim Conant | @Pietro: Yes, I meant contradiction! | |
| Oct 4, 2010 at 4:23 | comment | added | Harrison Brown | I'm curious: Is there a rigorous sense in which it can be said that "most" theories are inconsistent? (I'd imagine the answer here to be yes.) But it might be worth asking if there's some sort of phase transition, such that almost all inconsistent theories have a relatively short contradiction... | |
| Oct 3, 2010 at 20:09 | comment | added | Todd Trimble | Compare Pierre Cartier, as quoted by David Ruelle in Chance and Chaos: "The axioms of set theory are inconsistent, but the proof of inconsistency is too long for our physical universe." | |
| Oct 3, 2010 at 19:15 | comment | added | Pietro Majer | Agree. Actually I think you mean antinomy or contradiction, rather than paradox (a harmless thing). | |
| Oct 3, 2010 at 16:30 | comment | added | Ryan Budney | To add to that, if ZFC was found to be inconsistent I doubt the inconsistency would be as interesting as something like Russell's paradox. If it were, I suppose that would be quite informative. | |
| Oct 3, 2010 at 16:24 | comment | added | Qiaochu Yuan | Couldn't it also be the case that the minimal-length paradox is so long that human mathematics will always be able to avoid it? | |
| Oct 3, 2010 at 13:50 | history | answered | Jim Conant | CC BY-SA 2.5 |