[mturmon]

Some of this was familiar, but I also saw this:

> In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian group A with Ext^1(A, Z) = 0 a free abelian group?") is independent of ZFC.

(This is the same Shelah who proved what is usually called "Sauer's Lemma" on set separability, upon which the "VC dimension" and the resulting VC learning theory are based.)

Anyway, this was surprising because I didn't know there were any theorems outside of set theory that had been known and studied, and then later turned out to be independent of ZFC.

Apparently, the surprise I felt is just because I wasn't paying attention -- there are a couple of other problems in the page that seem to have a similar flavor. I'm not enough of a mathematician to appreciate to what extent they are interesting problems that arise independently of counterexamples.

[edited to add: the comment of @triska nearby addresses exactly this point!]

[ginnungagap]

Another famous example of that kind is the existence of outer automorphisms of the Calkin algebra, which is a simple question about a naturally occuring object which turned out to be independent of ZFC by work of Farah and Weaver.

Topology and set theoretic topology is full of those statements, are there Suslin lines? Are there S-spaces? Is the product of two ccc spaces also ccc? The list goes on

Another one from analysis is a very strong form of Fubini's theorem that was shown independent by Friedman.

There's surely more but those are the ones I could remember right now!