[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

More information about the FOM mailing list