Can anyone point out some good reference to understand how Paul Cohen proved that the continuum hypothesis is independent of ZFC? I know he used the so called forcing technique to construct two different models of ZFC, but I don't quite understand how.



Independence of the continuum hypothesis on ZFC
Raymond Smullyan and Melvin Fitting wrote a long (but very readable) monograph, called "Set theory and the continuum problem" (Oxford Logic Guides, 34. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xiv+288 pp. ISBN: 0-19-852395-5) which starts from the very beginning, introducing the von Neumann-Bernays-Godel "class-set" formalism, establishing all the basic properties, etc. culminating in a complete, self-contained exposition of Cohen's result and the basics of forcing.


There are three parts. The first part is foundational. The second is an exposition of Godel's relative consistency result, which says that the continuum hypothesis is consistent with NBG (or ZFC if you prefer). The third is about forcing, and Cohen's result.


In very, very broad outline, the point of Godel's argument is to exhibit a "model" of class-set theory with as few sets as possible: the only sets are those that are absolutely required to be there by some application of the axioms. Each set therefore carries with it a formula which requires it to exist. There are so few objects in this model that the continuum hypothesis is seen to be true because every element of the continuum that is required to exist is required by some explicit "reason", and the reasons can be enumerated.


The point of forcing is to show that one can build a new model in which there are many more objects, by adding new objects when there is no explicit reason why they can't exist (i.e. they are "forced" to exist) and keeping careful track of how many such objects you can add without reaching a contradiction. Certain tools (eg "compactness") are required to be able to add infinitely many new objects in this way.


