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.



Independence from Set Theory Axioms
It might help to understand look at how the fifth postulate is proved independent of Euclid's other axioms: One constructs a model, such as the Poincare disc, where the axioms can be given new interpretations. So the word LINE now means "arc perpindicular to the boundary of the disc", the word CONGRUENT now means "related by a Mobius transformation fixing the boundary of the disc" and so forth. One then checks that, if you take each axiom and replace the capitalized words by the quoted strings, the axiom remains true, except for the fifth postulate, which becomes false. 


Similarly, to show that AC is independent of ZF, one works inside ZFC and builds a model where the axioms of ZF, suitably reinterpreted, stay true but where choice is false. The technical tool used to build that model is called forcing. I don't really understand it, but Timothy Chow's introduction is the closest I have come to doing so.


