What other unprovable theorems are there?

Question

Gödel's incompleteness theorem presents us with the possibility of having theorems that are neither provable nor disprovable in a given axiomatic set. Already we have the continuum hypothesis which has this property (at least in the axioms we deal with in modern mathematics today).

Are there other such mathematical statements that have been "found"?

I think instead of "theorem" you meant "statement" or some variation thereof. (Using "theorem" sort of implies that there is a proof: that's how we know it's a theorem.) — user642796, Feb 25 '14 at 13:06
You can see the following post and also in Wiki the para about Examples of undecidable statements — Mauro ALLEGRANZA, Feb 25 '14 at 13:13
I don't think that the continuum hypothesis is a good example of Godel-ian incompleteness. Loosely speaking, I think you need a statement that is true but unprovable (in the system). — joeA, Feb 25 '14 at 13:18
@CarstenSchultz Well, I said "loosely speaking" because this can be very technical and I will undoubtedly falter on one of those technicalities. Therefore I will just point you to the wikipedia entry and the excerpt "For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system". — joeA, Feb 25 '14 at 13:37
Another example that you might enjoy: Color each point of $\Bbb R$ with one of a countable number of different colors. Claim: there exist $a,b,c,d$ of the same color with $a+b=c+d$. Erdős showed that the claim is equivalent to the continuum hypothesis. (Peter Smith's blog) (Detailed explanation) — MJD, Feb 25 '14 at 17:19

score 3 · Accepted Answer · answered Feb 25 '14 at 13:32

Every statement which is not logically true (i.e. provable from the laws of logic, without using any further assumptions) is unprovable in some theory. The statement "$G$ is an abelian group" is unprovable from the axioms of groups, simply because there are non-abelian groups as well.

The incompleteness phenomenon is delicate. From one end it talks about the fact that a theory cannot prove its own consistency, so it gives a very particular example for a statement which is unprovable. From the other hand, it essentially says that under certain conditions the theory is not complete (so there are in fact plenty of statements which we cannot prove from it).

If we take set theory as an example, then of course $\rm{Con}\sf(ZFC)$ is unprovable, which is an example for the one end, but also $\sf CH$ is unprovable (as you remarked) which is an example of the other end. So let me take some middle ground and find statements which are "naturally looking" but in fact imply the consistency of $\sf ZFC$ and therefore catch both ends of the spectrum.

Every projective set of real numbers is Lebesgue measurable.
There exists a measure extending the Lebesgue measure in which every set is measurable (note that this measure need not be translation invariant!).
There is an infinitary logic $\mathcal L_{\kappa,\kappa}$ which satisfies the compactness theorem.
There exists a free ultrafilter which is closed under countable intersections.

And there are many, many more examples. Each of the above examples in fact implies the consistency of $\sf ZFC$ (and much more). Of course, if we end up proving that large cardinals are inconsistent then we can disprove most of these statements, but at the philosophical (and heavily biased mathematical) evidence suggests that this is not the case.

MJD: Thank you for the example. But I particularly searched for statements whose consistency strength is stronger than that of $\sf ZFC$ itself. — Asaf Karagila, Feb 25 '14 at 17:19

What other unprovable theorems are there?

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