KGFM 19: Cohen’s interactions with Gödel

The next paper in KGFM is a short talk by the late Paul Cohen, ‘My Interaction with Kurt Gödel: The Man and His Work’. The title is full of promise, but there seems relatively little new here. For Cohen had previously written with great lucidity a quite fascinating paper ‘The Discovery of Forcing‘ and he already touches there on his interactions with Gödel:

A rumor had circulated, very well known in all circles of logicians, that Gödel had actually partially solved the [independence] problem, specifically as I heard it, for AC and only for the theory of types (years later, after my own proof of the independence of CH, AC, etc., I asked Gödel directly about this and he confirmed that he had found such a method, specifically contradicted the idea that type theory was involved, but would tell me absolutely nothing of what he had done). … It seems that from 1941 to 1946 he devoted himself to attempts to prove the independence [of AC and CH]. In 1967 in a letter he wrote that he had indeed obtained some results in 1942 but could only reconstruct the proof of the independence of the axiom of constructibility, not that of AC, and in type theory (contradicting what he had told me in 1966).

In this present paper, Cohen can shed no more real light on this unclear situation. But still,  what he writes is perhaps interesting enough to quote. So, Cohen first repeats again the basic story, though with a comment that chimes with other accounts of Gödel’s philosophical disposition:

I visited Princeton again for several months and had many meetings with Gödel. I brought up the question of whether, as rumor had it, he had proved the independence of the axiom of choice. He replied that he had, evidently by a method related to my own, but he gave me no precise idea or explanation of why his method evidently failed to succeed with the CH. His main interest seemed to lie in discussing the truth or falsity of these questions, not merely their undecidability. He struck me as having an almost unshakable belief in this realist position that I found difficult to share. His ideas were grounded in a deep philosophical belief as to what the human mind could achieve.

And then at the end of the talk, Cohen sums up his assessment like this:

Did Gödel have unpublished methods for the CH? This is a tantalizing question. Let me state some incontrovertible facts. First, much effort was spent analysing Gödel’s notes and papers, and no idea has emerged about what kinds of methods he might have used. Second, I did ask him point-blank whether he had proved the independence of CH, and he said no, but that he had had success with the axiom of choice. I asked him what his methods were, and he said only that they resembled my own; he seemed extremely reluctant to give any further information.

My conclusion is that Gödel did not complete any serious work on this topic that he thought was correct. In our discussions, the word model almost never occurred. Therefore I assume that he was looking for a syntactical analysis that was in the spirit of his definition of constructibility. His total lack of interest in a model-theoretic approach quite astounded me. Thus, when I mentioned to him my discovery of the minimal model also found by John Shepherdson, he indicated that this was clear and, indirectly, that he knew of it. However, he did not mention the implication that no purely inner model could be found. Given that I also believe he was strongly wedded to the syntactical approach, this would have been of great interest. My conclusion, perhaps uncharitable, is that he totally ignored questions of models and was perhaps only subconsciously aware of the minimal model.

That hints at an interesting diagnosis of Gödel’s failure to prove the independence results he wanted.

This entry was posted in Phil. of maths. Bookmark the permalink.

Leave a Reply

Your email address will not be published. Required fields are marked *