[FOM] independence problem

[John Bell]

 I was wondering if anybody on the list might be able to help me with the
following problem.

I have long been interested in the question of the strength of the Sikorski
Extension Theorem (SET) - every complete Boolean algebra is injective. When
Luxemburg first investigated the issue in the 1960s he left open the
question of whether either of the implications Axiom of Choice (AC) => SET
=> Boolean Prime Ideal Theorem (BPI) can be reversed over ZF, but
conjectured that neither could be. A while ago I showed (The Strength of the
Sikorski Extension Theorem for Boolean Algebras, Journal of Symbolic Logic,
48, 1983) that the second of these implications was irreversible, but
establishing the irreversibility of the first (and I remain convinced it is
irreversible) has proved entirely beyond my powers. I was hoping that
someone might be able to settle this question.

In the paper mentioned above I showed that the truth of SET in a model M of
ZF is equivalent to the assertion that BPI holds in every Boolean extension
of M. Accordingly, in order to show that AC is independent of SET, one would
have to construct a model M of ZF in which AC fails but BPI holds, not just
in M, but in all of its Boolean extensions. As I said, I don't know how to
go about doing this. Any thoughts on the matter?


John Bell


Professor John L. Bell
Department of Philosophy
University of Western Ontario
London, Ontario
Canada N6A 3K7

http://publish.uwo.ca/~jbell