[FOM] From Compactness to Completeness
David Auerbach
auerbach at unity.ncsu.edu
Thu Dec 23 17:15:48 EST 2010
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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