[FOM] From Compactness to Completeness
John Burgess
jburgess at Princeton.EDU
Thu Dec 23 15:29:49 EST 2010
The facts to which Wikipedia is presumably alluding are the
following: (1) To prove completeness in the form of the statement
that any consistent set T of first-order sentences has model, one
needs, if T is uncountable, the axiom of choice, or if one is
careful, a weak version of it, the Boolean prime ideal theorem. (2)
Compactness follows immediately from completeness, without use of
choice. (3) It is a fairly easy application of compactness to prove
the Boolean prime ideal theorem. Thus all three statements are
equivalent over ZF set theory without choice.
John Burgess
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