> By the way, Paul Cohen, who proved the independence of the Continuum Hypothesis, says that Godel did not express much interest in independence results (I thin

> By the way, Paul Cohen, who proved the independence of the Continuum Hypothesis, says that Godel did not express much interest in independence results (I think Godel believed in Platonism): Gödel proved the other direction of the independence of CH (and the Axiom of Choice), and had been trying to prove independence some years after that. He had actually succeeded in proving the independence of AC by 1943, and later said in an interview with Hao Wang that his methods could have probably been extended to prove the independence of CH by 1950, but he stopped working on it because he developed a distaste for the subject. In retrospect, Gödel wished that he had continued his work so that set theory progressed faster. The article The Origins of Forcing by G. H. Moore in the proceedings of the 1986 Logic Colloquium is a good discussion of the relevant history. > Godel believed that for well-defined areas of mathematics, we can add more _intuitive_ axioms to make the theory decidable: The appropriate quote from the article: "I would like to point out that this intuitive grasping of ever newer axioms that are logically independent from the earlier ones, which is necessary for the solvability of all problems even within a very limited domain, agrees in principle with the Kantian conception of mathematics. " This line of research hasn't gone very well in practice since Gödel's time. It has produced a lot of interesting set theory, but has not really impacted mathematics at large.