Abstract
The continuum hypothesis (CH) is the claim that any subset of the real numbers is at most countable or has the same size as the set of all real numbers. By the results of Godel and Cohen this hypothesis is independent over ZFC if ZFC is consistent.
In the talk we will focus on a couple of attempts to decide CH and also GCH (generalized continuum hypothesis) in the sense of finding a natural axiom which decides CH or GCH over ZFC. We will mention Shelah's and Woodin's positions and discuss them in the context of one particular axiom: the tree property at aleph_2, a combinatorial property of cardinals, which decides CH negatively.