[FOM] From Compactness to Completeness
T.Forster@dpmms.cam.ac.uk
T.Forster at dpmms.cam.ac.uk
Fri Dec 24 15:25:06 EST 2010
I think if the language is countable you don'tneed KL, surely?
Merry Christmas
tf
On Dec 24 2010, G. Aldo Antonelli wrote:
>On 12/24/10 9:00 AM, John Burgess 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.
>
>When I first learned this as an undergraduate (in the old country, as
>they say) the proof of completeness proceeded through compactness and
>what we called 'weak completeness' i.e. the statement that any validity
>is provable.
>
>The usual Henkin proof of compactness requires a weak version of choice,
>as Burgess points out. In turn, compactness implies that if T entails A,
>then a finite T' already does, so T'-->A is valid, hence provable.
>
>Interestingly perhaps, "weak completeness" also requires a combinatorial
>principle, viz., König's lemma, in order to extract a counter-model from
>a non-terminating truth tree.
>
>I have some recollection that this strategy was credited to H Hermes,
>"Enumerability, Decidability, Computability" (1965) but I have no access
>to the book now.
>
>
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