[FOM] From Compactness to Completeness

[David Auerbach]

And John Bell's article in SEP (http://plato.stanford.edu/entries/axiom-choice/) has more details and references. 


David Auerbach                                                      auerbach at unity.ncsu.edu
Department of Philosophy and Religious Studies
NCSU
Raleigh, NC 27695-8103

On Dec 23, 2010, at 3:29 PM, John Burgess (by way of Martin Davis <eipye at pacbell.net>) wrote:

> 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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