From: renfrod@central.edu (Dave L. Renfro)
Subject: "How bad can 2^(Aleph_0) be?" history question
Date: 27 May 2001 22:59:47 -0400
Newsgroups: sci.math

Paul Cohen proved that 2^(Aleph_0) =/= Aleph_1 is consistent
with ZFC in 1963. [Kanamori, "The Higher Infinite", p. 114, says
that Cohen circulated notes for the proof in April 1963.]

I am interested in who, when, and where (seminar presentation,
published paper, etc.) proved the stronger result that it is
consistent with ZFC that 2^(Aleph_0) can be any specified cardinal
whose cofinality is greater than Aleph_0. I know that stronger
results were proved by Richard Soloay and William Easton, but I'm
only interested in the possibilities for 2^(Aleph_0).

I think maybe this was proved independently by Cohen and Solovay.
In abstract 63T-395 on page 595 of the Oct. 1963 Notices of the
American Mathematical Society (dated Sept. 3, 1963), Solovay writes
"(The first was discovered independently by Cohen.)" in reference
to the fact that 2^(Aleph_0) can be any cardinal with cofinality
greater than Aleph_0. That Cohen proved this appears to be
confirmed by Lemma 16 on page 49 of his paper "Independence results
in set theory", which appears in John W. Addison, Leon Henkin, and
Alfred Tarski, "The Theory of Models", Proceedings of the 1963
International Symposium at Berkeley, North-Holland, 1965.

However, Cohen mentions a result of Solovay's [Not the result that
I'm interested in, but rather that 2^(Aleph_n) = Aleph_g(n) can
be satisfied simultaneously for all positive integers n in some
model for ZFC, where g is any specified strictly increasing function
 from the positive integers into the positive integers.] as well
as a result *later* announced by Easton. Because of this, I would
guess that Cohen wrote this paper well after his Berkeley talk,
and so not all of the results mentioned in it may have been
presented at the Berkeley conference (in particular, Cohen's
Lemma 16 that I mentioned above).

Can someone tell me what dates this conference took place?
I forgot to write this down when I was photocopying this stuff
yesterday and the library I was at is a 160 mile round-trip drive.
Also, does anyone know whether or not Cohen had actually proved the
result I'm interested in--that 2^(Aleph_0) can be any cardinal
with uncountable cofinality--when this conference took place?
If not, when did Cohen obtain it?

Dave L. Renfro