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Is there any problem in mathematics that is proved not independent of ZFC but the problem itself is not proved yet?

Asaf Karagila
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Minghao Liu
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  • Look also at the "Linked" of the duplicate question, where you will find many other related questions. – Asaf Karagila Jul 16 '15 at 12:09
  • @AsafKaragila sorry, updated my question. seems the old question is misleading – Minghao Liu Jul 16 '15 at 12:18
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    Given an integer $n$, the question of whether $n$ is prime is not independent of ZFC, but there are many such $n$ for which the question of primality has not been settled. – Gerry Myerson Jul 16 '15 at 12:21
  • @GerryMyerson I am enlightened! So should I just close the problem or you make it an answer... – Minghao Liu Jul 16 '15 at 12:23
  • For what it's worth, I do think this is a duplicate, of another question where the solutions where proposed in line of what @Gerry pointed out here. – Asaf Karagila Jul 16 '15 at 12:23
  • @GerryMyerson also we know that the primality is decidable, is there any problem satisfies the question and also is not decidable? – Minghao Liu Jul 16 '15 at 12:27
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    What do you mean by "decidable" here? – Asaf Karagila Jul 16 '15 at 12:29
  • @GerryMyerson although in this case there exist algorithms to answer these questions, so it's just a matter of time for a given $n$ and it doesn't require any special methods of proof (they can greatly reduce the time though) – ljfa Jul 16 '15 at 12:37
  • @ljfa: So, you know what is the smallest prime after Graham's number? – Asaf Karagila Jul 16 '15 at 12:41
  • @AsafKaragila well no, but you'd just need to wait long enough... but yeah you're right as in it will never be solved this way in practice – ljfa Jul 16 '15 at 12:42
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    @MinghaoLiu: Whenever you know that a given claim can be either proved or disproved by ZFC, there will always be an algorithmic way to figure out which is the case: Just search for a proof or a disproof in parallel, and sooner or later one of these searches will complete. – hmakholm left over Monica Jul 16 '15 at 13:57
  • "Proved not independent of ZFC" is unintelligible. – Rob Arthan Jul 16 '15 at 21:31
  • @RobArthan It may be oddly phrased, but it's not unintelligible: "proved not independent" just means "we have proved it is decidable in . . ." – Noah Schweber Jul 16 '15 at 23:31

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I'll take the question to mean,

Is there any statement $S$ such that we can prove that $S$ is not independent of ZFC, but we have not yet been able to prove $S$, nor to refute $S$?

Any statement $S$ that can be settled by a finite computation, but by a computation that no one has carried out yet (or by a computation so large that no one is able to carry it out), will do. For example, the question of whether the Ramsey number $R(5,5)$ is 43. For another example, according to the most recent correspondence I have from Sam Wagstaff, who keeps track of this kind of thing, no one has been able to fully factor $2^{1207}-1$ (and when/as/if that number is factored, some other number will take its place).

Gerry Myerson
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