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I am trying to understand better the similarity between the compactness theorem

in logic--every first-order sentence is satisfiable (has a model) iff every finite subset

of sentences is satisfiable, and the property of compactness : a topological space X is

said to be compact iff (def.) every cover C of X by open sets has a finite subcover, i.e.,

a subcollection C' of the cover C that also covers X , i.e., the union of the elements of

the cover contains X. But, by De Morgan, compactness is equivalent to the finite

intersection property, i.e., every finite subcollection (of the elements of a cover) has a

non-empty intersection (the subcollections will be sentences, and the intersection

has to see with the finite subcollection having a model ).

Anyway, so the topological space we consider is that of the infinite product

of {0,1} (discrete topology ) -- 0 and 1 will be the values we will be giving to the free

variables in a sentence--with the product indexed by I:=[0,1] in the Reals. We then

get the Cantor Space . Then any

string of 0's , 1's is a valuation, is a clopen subset of the Cantor Space. By compactness,

of the Cantor space (and some hand-waving) we get satisfiability.

I think this is "spiritually correct", but I think there are some gaps. Any Ideas?

.

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# Compactness in Topology and in Logic

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