Examples of unproven but likely true existential sentence (in the sense of incompleteness)

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?

• 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 1:19
• 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 1:30
• Number theory has plenty of unproven existential claims, e.g. (the negation of) en.wikipedia.org/wiki/… . – Terry Tao Aug 21 '14 at 1:37
• (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 3:27
• (Yes, the Riemann hypothesis is a $\Pi^0_1$ sentence. There are several ways to show this, see mathoverflow.net/questions/31846 .) – Emil Jeřábek Aug 21 '14 at 11:44