Paul Cohen: My Interaction with Kurt Gödel; the man and his work
AbstractOn this centenary of Kurt Gödel, it is most appropriate to celebrate the man who more than any other transformed logic from its philosphical past to be a vibrant part of mathematics. His contributions are so well known and the recognition he has received so plentiful, that it might be extraneous to recite them here. I have chosen to relate how , first his work, and then his personal interaction, affected me so strongly.
It was my great fortune, and privilege, to be the person who fulfilled the expectations of Gödel, in showing that the Continuum Hypothesis , as well as other questions in Set Theory are independent of the usual axioms of Set Theory. In his article What is Cantor's continuum problem, he foresaw this possibility, and reading it greatly inspired me to work on the problem. His point of departure in discussing these questions always seemed to be grounded in philosophical discussions, and in the work of previous researchers. In my case, I felt it was necessary to start afresh, and to treat it as an "ordinary" problem of mathematics, perhaps similarly to how Cantor and later Hilbert regarded it. None of my work, however, would have been possible without the momentous discovery of the "Constructible Universe", achieved by Gödel around 1938. At first I had not actually understood these results, finding the expposition of his monograph rather difficult, but as I thought more deeply, I realized that all my attempts were intimately allied to his methods, and I eventually mastered them completely. Suddenly, I had a tool with which I could make my own somewhat vague ideas more precise. Thanks to his discovery, I saw that the new universes of set theory I wished to construct consissted of sets which were constructible from a limited number of new sets which I would introduce in a very particular way.
Soon after proving my first results, I had a great desire to meet Gödel and to do him and myself the great honor of personally explaining them to him. I met him at Princeton with a preprint, which gave all the essential details. Very graciously, he read it, and after just a few days, pronounced my proof correct. We had a series of intense discussions, and I asked if he would communicate the paper to the Proceedings of the National Academy, where his own results were first announced. He graciously consented to do this, and we had a lively correspondence about the best way to present it, and he suggested several revisions. With this acknowledgment, and with his later overly kind remarks about my discovery, I felt that we shared a great bond , in that we had successfully discovered , each in his own way, new fundamental methods in Set Theory.
I visited Princeton again, for several months, and had many meetings with him. 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 gave me no precise idea, nor why his method evidently failed to succeed with the Continuum hypothesis. His main interest seemed to lie in discussing the "truth" or "falsity" of these questions, not merely in their undecidability. He struck me as having an almost unshakable belief in this "realist" position, which I found difficult to share. His ideas were grounded in a deep philosophical belief as to what the human mind could achieve. I greatly admired this faith in the power and beauty of Western Culture, as he put it, and would have liked to understand more deeply what were the sources of his strongly held beliefs. Through our discussions, I came closer to his point of view, although I never shared completely his "realist" point of view, that all questions of Set Theory were in the final analysis, either true or false. In my lecture, I shall explain both the similarities and divergences between our points of view, and what may have been the avenues which he explored in his own, never published, researches.