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This is a bit of a philosophical question. Due to Gödel, we know that there are undecidable statements in ZFC set theory. But why is it that most statements that mathematicians tend to study in practice are decidable? Is it something to do with the fact that "humanly interesting" mathematical statements are special in some way? To put the question another way, why is it so hard to find "natural" examples of undecidable statements? To date, I don't think anyone has found a natural undecidable statement about finitary objects that everyone recognizes to be undecidable. Anyway, I would love to read some text on this topic.

Tankut Beygu
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user107952
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  • There are classical statements about arithmetic objects that are known to be unprovable (either way) in first-order Peano arithmetic but true in $\mathbb{N}$ with the axioms of full set theory: I’m thinking of Goodstein and Paris-Harrington. – Aphelli May 29 '21 at 22:53
  • Is it meaningful to call a single statement decidable? My understanding is that only a set of statements can be called decidable, based on whether there is a decision procedure for membership in said set. – Mark May 29 '21 at 22:55
  • @Mark what about the continuum hypothesis? – wormram May 29 '21 at 23:00
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    @Mark: I think we're using "decidable" here in the sense of "can be proved or disproved from the axioms", – Nate Eldredge May 29 '21 at 23:01
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    I might argue it's the other way around: the ZFC axioms were developed in hopes that they would decide most humanly interesting statements, and we continue to use them because they are reasonably successful in that regard. If there were an excessive number of "interesting" undecidable statements in ZFC, we would probably by now have added more axioms to resolve them. – Nate Eldredge May 29 '21 at 23:02
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    There is a specific diophantine equation that can be written down on a page of paper that has no solution and cannot be proved to have no solution in ZFC. I would encourage you to look into the negative result of Hilbert's Tenth Problem, from which a number of other interesting results follow (such as Richardson's Theorem). You may also be interested in Laver Tables, about which there may be a theorem requiring large cardinal axioms: https://johncarlosbaez.wordpress.com/2016/05/06/shelves-and-the-infinite/ – TomKern May 29 '21 at 23:22
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    I suggest you look into the Paris-Harrington theorem for a natural statement that can be formulated but not proved in PA. Harvey Friedman and others have done a great deal of work in finding natural problems that are provable, but not in ZFC or interesting subsystems thereof. You can find manuscripts of Friedman's work on this here. – Rob Arthan May 29 '21 at 23:26
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    @NateEldredge I think, though, that there's still a surprising feature here: the ZFC axioms are relatively easy to motivate. If the mathematical community found certain extremely complicated Diophantine equations particularly interesting for whatever reason, I'm not sure there would be a "natural" set of axioms which would decide them. So there's still something arguably surprising here: that there is a natural(ish) axiom system which seems to decide "most" statements we care about. – Noah Schweber May 30 '21 at 00:55
  • FYI, your question is a meta-mathematical analogue of some of the discussions (answers and comments) in Situations where “naturally occurring” mathematical objects behave very differently from “typical” ones (mathoverflow). (moments later) I just noticed that this question is from over 3 years ago. An edit a few minutes ago brought it up to the front page. – Dave L. Renfro Jan 16 '25 at 08:16

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There is obviously a big difference between an open problem and an undecidable statement. However, the point is that we don't know what it is. Every open conjecture is potentially undecidable, even though this may seem implausible to the specialists working on it. The Whitehead Problem used to be considered a natural conjecture, until Shelah proved its independence. The big gap between open conjectures and undecidable statements may in principle be due to the fact that the independence techniques necessary have not been developed yet.

So any meaningful discussion would concern open problems (rather than undecidable ones, since we don't know the difference). There are certainly plenty of those: just in the context of the Riemann zeta function, there are numerous questions of various strength that are completely open. Just to mention a cute independence result (that has certainly not attracted the attention of the Riemann hypothesis): consider the ultrapower construction of the hyperreals as ${}^\ast \mathbb R= \mathbb R^{\mathbb N}/\mathcal U$, where $\mathcal U$ is a nonprincipal ultrafilter. Then null sequences (tending to zero) naturally represent infinitesimals in ${}^\ast \mathbb R$.

Can one find a $\mathcal U$ such that conversely, every infinitesimal is the equivalence class of a null sequence? The question turns out to be independent of ZFC! At any rate, it would seem that in fact it is not hard to come up with natural open questions - which are potentially undecidable as far as we currently know.

Mikhail Katz
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