Timeline for Could the Jacobian conjecture be undecidable?
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| Apr 7, 2023 at 21:10 | comment | added | The Amplitwist | Reposting a link mentioned in a previous comment so that it appears in the "Linked" questions list: Finite dimensional real division algebras | |
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jan 6, 2015 at 22:12 | comment | added | Timothy Chow | @AdamEpstein: Examples of natural statements that are unprovable in PA are notoriously rare. I'd also like to reiterate that although it's very tempting to mentally equate "provable in PA" with "has an elementary proof", this is probably not a useful way of thinking. Proofs in PA can use advanced machinery from all areas of mathematics, and can be long, complicated, and difficult to find. Conversely, there are statements not provable in PA whose proofs (formalizable in some other system) would be regarded by most people as "elementary" or even trivial. | |
| Jan 5, 2015 at 9:28 | comment | added | Adam Epstein | Yes, I'm aware of Goodstein's Theorem. What I am after is something that could be construed as more "natural" in some mathematical sense. While "naturality" itself is clearly subjective, I do think that it should be possible to attach meaning in certain contexts. For example, various theorems in commutative algebra, algebraic geometry, etc. that trade a finiteness hypothesis for a finiteness conclusion. If sufficiently effective bounds are known, such a proof might be unwound into an arithmetic statement. As you also mention, there is the possibility of coding into PA. | |
| Jan 4, 2015 at 20:29 | comment | added | Timothy Chow | @AdamEpstein : You may already know this, but Goodstein's theorem is not provable in PA, yet it is a universal quantification over statements that are trivially decidable. | |
| Jan 4, 2015 at 20:24 | comment | added | Timothy Chow | @AdamEpstein : I don't know of examples that I think you would regard as interesting. Any $\Pi_1$ conjecture $\forall n: P(n)$ is an example of sorts, because for any fixed $n$, $P(n)$ is trivially decidable, but it sounds like you're asking for more than that. I would add that being provable in PA does not line up very well with what most people intuitively think of as an "elementary" or "algebraic" proof. It is possible to formalize a sizable amount of real and complex analysis in PA. | |
| Jan 3, 2015 at 1:24 | comment | added | Adam Epstein | A problem of "natural origin" which may be viewed as a parametric family of positively resolvable problems in some decidable theory, but whose universal quantification is not known to be provable. Suitable algebra problems with a natural number "complexity parameter" (degree, dimension, etc.) are equivalent to true $\Pi_1$ statements in the language of arithmetic. Are these provable in first-order arithmetic? Are there documented examples which are not? For example, see the above query of Steven Landsburg regarding J(3,n). | |
| Jan 2, 2015 at 15:44 | comment | added | Timothy Chow | @AdamEpstein : What do you mean by "this phenomenon"? What phenomenon? | |
| Dec 31, 2014 at 23:54 | comment | added | Adam Epstein | Are there any known examples of results from algebra that actually exhibit this phenomenon? For example, is it conceivable that the 1,2,4,8 theorem mathoverflow.net/questions/105478/… is an example? | |
| May 25, 2013 at 1:55 | vote | accept | Marty | ||
| May 17, 2013 at 17:34 | history | answered | Timothy Chow | CC BY-SA 3.0 |