Completeness and Iteration in Modern Set Theory

Set theory entered the modern era through the work of Gödel and Cohen. This work provided set-theorists with the necessary tools to analyse a large number of mathematical problems which are unsolvable using only the traditional axiom system ZFC for set theory. Through these methods, together with their subsequent generalisation into the context of large cardinals, settheorists have had great success in determining the axiomatic strength of a wide range of ZFC-undecidable statements, not only within set theory but also within other areas of mathematics.