[FOM] From Compactness to Completeness

[John Burgess]

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