Ars Mathematica

Hugh Woodin has two survey articles on recent work on the Continuum Hypothesis: I and II. Most mathematicians consider the continuum hypothesis as a settled question: since it is independent of ZFC, its truth is unknowable.

Set theorists, on the other hand, sometimes hold out the hope that new, intuitive axioms will be found that will provide a definite answer. Woodin thinks that we are close to finding such an axiom, and it seems to indicate that the cardinality of the reals is aleph two. (The continuum hypothesis states that it’s aleph one.)

There is a simpler example of an intuitive result that implies that the continuum hypothesis is false. Details can be found here and here.

E. Lee Lady, a mathematician at the University of Hawaii, has a terrific collection of lecture notes in algebra. He also has posted a draft manuscript of a book on torsion-free modules over Dedekind rings, which years of graduate school brainwashing will convince you are the natural generalization of the ring of integers.