[FOM] what prohibits AC--->CH? (was:: Re: WHY DO SET THEORISTS DISLIKE CHRIS FREILING'S EVIDENCE)

[Matt Insall]

Timothy Chow wrote:
<<A less formal but maybe more striking way to put it is that since the
axiom of choice implies something as weird as the Banach-Tarski paradox,
what's to stop it from implying something as weird as the continuum
hypothesis?>>

I recall a colleague pointing out to me that someone he knew provided a
proof of the Banach-Tarski Paradox (BTP for short) without AC, so I am not
sure that discussing the fact that AC implies BTP is not a red herring here.
(However, the proof must have used the existence of a nonmeasurable set, of
course, which is apparently (assuming some strong cardinal axioms) formally
weaker than AC.)  Also, I seem to recall that ZFC proves Con(ZFC+not(CH)).
Am I mistaken?  If I am not mistaken, then the question ``what's to stop it
(AC) from implying something as weird as the continuum hypothesis?'' is
answered:  Existence of a model of ZFC provides a way of constructing a
model of ZFC in which CH fails.  Or, it can be answered less technically by
the claim that AC just does not imply CH (thereby taking a (philosophical?)
stance as to what model of ZFC one wishes to work in).



Dr. Matt Insall
Associate Professor of Mathematics
Department of Mathematics and Statistics
University of Missouri - Rolla
Rolla MO 65409-0020

insall at umr.edu
(573)341-4901