SEVEN - Or Not

Summary

After Gödel proved the consistency of the continuum hypothesis with set theory, many people expected him also to prove its independence, but it was Paul Cohen who did so. There is a story, perhaps apocryphal, as such stories sometimes are, that Cohen passed by Georg Kreisel’s office at Stanford in the spring and asked for a problem to work on. To get rid of Cohen, Kreisel gave him the independence of the continuum hypothesis, and by the next fall Cohen had solved it. There is another story, certainly apocryphal, that after Gödel heard of Cohen’s success, Gödel feared that the Institute for Advanced Studies would fire him and he would starve to death as an old man. This story may be a garbled version of Gödel’s decline after the death of his wife, who had looked after him well.

We will show that if ZF is consistent, it remains so even with the addition of a sentence saying there is a nonconstructible set of natural numbers, so V = L is independent. We will show that if ZF is consistent, it remains so with the addition of a sentence saying there is a set of sets of natural numbers that is not well-ordered, so the axiom of choice is independent. We will show that if ZF is consistent, it remains so with the addition of the denial of the continuum hypothesis, so the generalized continuum hypothesis is independent. We will make constant use of the constructible sets from the last chapter, and, as there, our exposition will follow lectures given by Hilary Putnam at Harvard in the spring of 1968.