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Paul Cohen: My Interaction with Kurt Gödel; the man and his work
Abstract
On 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. |
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