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Some examples of universal statements that are unproven but likely true include the Riemann hypothesis (all non-trivial zeros of the zeta function have real part 1/2) and the Goldbach conjecture (all even numbers greater than two can be written as the sum of two primes). In our standard models C and N we can recursively enumerate possible inputs and verify the statement is true for that concrete number. We have done this for enough examples that the statement feels like it is true in much the same way we feel like gravity always causes objects to fall.

It seems to me that the "for all" aspect of these statements is what allows such statements to be considered 'likely to be true and independent' since we continuously observe that the statement holds for all the examples we've thrown at them so far. These types of statements seem to have the property that if they are true, then they are independent of some theory, but if they are false, they are theorems of that theory.

I am wondering if there are currently unsolved mathematical questions that more closely resemble the search for extraterrestrial life: 'they exist, we just have to keep searching to find them!' I think the type of statement I am looking for would be the negation of a universal sentence that, if true (in some standard model), is independent of some theory, but if false, is a theorem.

A historical example might be a statement about the existence of large cardinals, before it was proven to be independent of ZFC. However, I think staying within a standard model like R would be much more interesting -- some kind of real that would prove some statement, but we can't prove it exists because it is actually undefinable? Or would the fact that the statement exists be a sufficient definition, contradicting this whole line of thought?

Jonny
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    False universal statements ($\Pi^0_1$-statements in the logic lingo) cannot be independent. They can be unprovable (e.g. Con(T) is a $\Pi^0_1$ statement for any axiomatizable theory $T$) but if they are false then they are provably false. This is because basic arithmetic (PA or even PRA) is $\Sigma^0_1$-complete: if a $\Pi^0_1$ statement is false in the standard model then it is false in every model since the counterexample in the standard model is truly finite and it is therefore a genuine proof of the falsehood of the $\Pi^0_1$-statement. – François G. Dorais Aug 21 '14 at 01:19
  • But certainly we can go up the hierarchy? I don't believe the Riemann Hypothesis or the Goldbach Conjecture are $\Pi_1^0$ sentences, but in less formal language they look like a simple universal sentence: do all elements in this particular set have this property? – Jonny Aug 21 '14 at 01:30
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    Number theory has plenty of unproven existential claims, e.g. (the negation of) http://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture . – Terry Tao Aug 21 '14 at 01:37
  • I like these number theory ideas, but I don't think they are quite what I am looking for. If we find a natural number that proves the negation this Second Hardy-Littlewood conjecture, or other number theory claims (like the existence of an odd weird number), then that statement becomes a theorem in a Number Theory. My intuition is that an uncountable model is required to satisfy the requirement that the 'truth' of the statement is unprovable. – Jonny Aug 21 '14 at 02:17
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    (This is a silly answer.) Every Millennium problem is an existential statement (in some silly sense). If we believe that X is true, then we believe (most likely) that there is a proof of X in ZFC. – Jason Rute Aug 21 '14 at 03:27
  • @Jonny , as for requiring that the object found be a real, I might point out that a $\Sigma^0_1$ statement about the reals (in the usual meaning) is one where you can find out that the real $x$ satisfies the statement in finite time. (The set of satisfying reals is an effectively open set.) This means you only need to know $x$ up to some open interval containing $x$. Therefore, if such an $x$ exists, there is also a rational $q$ (hence definable) that satisfies the statement. (You may have a different interpretation of $\Sigma^0_1$ sentence here, but it is not clear to me what that is.) – Jason Rute Aug 21 '14 at 03:42
  • To clarify my last comment, when one says that the negation of the Riemann hypothesis is $\Sigma^0_1$, it is not because it is of the form "there exists a complex number $z$ which is a nontrivial zero not on the critical line". Being a zero of the zeta function is actually a $\Pi^0_1$ statement (I think). The reason the RH negation is $\Sigma^0_1$ is because it is equivalent to a $\Sigma^0_1$ statement. (I don't recall what that is, but I think the idea is that you only need to zoom in on an area in the complex plane so much to find a zero in that area. But this is way out of my expertise.) – Jason Rute Aug 21 '14 at 03:56
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    (Yes, the Riemann hypothesis is a $\Pi^0_1$ sentence. There are several ways to show this, see http://mathoverflow.net/questions/31846 .) – Emil Jeřábek Aug 21 '14 at 11:44
  • @JoelDavidHamkins: Thanks for pointing out the omission. I added the word false where it should have been. – François G. Dorais Aug 21 '14 at 12:29
  • @Jason -- I don't think your answer is silly at all. The notion of searching for certain proofs may exactly fit. We have an intuition that we believe to be true, and we search for it even though incompleteness tells us there is a possibility that it is not provable. But there may in fact be a proof to its independence. Just like my large cardinal example. – Jonny Aug 21 '14 at 18:01
  • To clarify, I think any $\Sigma_n^0$ statement is sufficient to categorize loosely as 'existential' in my question. But my emphasis is less formal -- I'm looking for what is effectively the opposite of the potentially impossible search for a counterexample. It should be a potentially impossible search for an example (with the additional caveats of the original question). – Jonny Aug 21 '14 at 18:36

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