[FOM] RE Axioms that imply AC
John Bell
jbell at uwo.ca
Sun Feb 5 14:00:04 EST 2006
In connection with Tom Forster's posting, here are some other assertions
equivalent to the prime ideal theorem (PIT: (1) every consistent first-order
theory has a model; (2) the compactness theorem for first-order logic; (3)
every distributive lattice with a top element contains a prime ideal; (4)
every commutative ring with 1 contains a prime ideal; (5) Alaoglu's theorem
that the closed unit ball of the dual of a normed space is weak*-compact;
(6) The Stone Representation Theorem for Boolean algebras.
If in (3) or (4) "prime" is replaced by "maximal", the result is equivalent
to the axiom of choice (AC).
What of assertions that are strictly between PIT and AC in logical strength?
One possible (and natural) example is the Sikorski extension theorem for
Boolean algebras (SIK): every complete Boolean algebra is injective. SIK is
equivalent to the following assertions, inter alia: (i) if A is a subalgebra
of a Boolean algebra B, there is an inclusion-maximal proper filter F in B
whose intersection with A is {1}; (ii) any continuous surjection between
compact Hausdorff spaces has an irreducible restriction to a closed subset
of its domain; (iii) for any complete Boolean algebra B, PIT holds in the
B-extension V^(B) of the universe of sets.
SIK is known (Bell 1983) to be independent of PIT (but is easily seen to
imply the latter), while it is still not known whether SIK implies AC. This
seems unlikely, but proving independence of AC from SIK looks difficult
since (by iii) it may require the construction of a symmetric model in which
AC fails but in every Boolean extension of which PIT holds.
-- John Bell
Professor John L. Bell
Department of Philosophy
University of Western Ontario
London, Ontario
Canada N6A 3K7
http://publish.uwo.ca/~jbell
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