I have often heard of various statements being independent from the axioms of set theory (typically ZFC). Some examples include

- The continuum hypothesis is probably the most famous
- The independence of the axiom of choice from plain ZF
- My professor told me that the following theorem is independent from the standard axioms:
*Theorem*: If $U$ is a regular set then $U\times [0,1]$ is regular.

I'm wondering what a proof of such a statement would look like. What context do you do the proof in? What kind of theoretical framework do you have to build up before you can answer such questions?

In addition to answers I would also be interested in resources that would let me find out more about these ideas. I'm looking for books/web sites that start a relatively elementary level but still build up to dealing with some of the examples I mentioned above.