Compactness in Topology and in Logic

[Bacle]

Hi, All:

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?



.

 

[Preno]

Well, the only gap is that you appear to be talking about propositional logic, not first-order logic. The completeness therem for classical first-order logic is more complicated to prove.

For propositional logic, to each set of sentences, assign the set of valuations which make all of the sentences true. Then the finite intersection property of the (Cantor) space of valuations is simply another way of stating the compactness of the logic.

 

[Bacle]

Thanks, Preno; can you think of an argument along these lines to prove compactness
in FOL? I know (please critique/suggest) that in FOL, a valuation making all sentences true (if this is possible; I guess if the sentences are not contradictory) is a model/interpretation for the collection of sentences, and not just an assignment in a truth table. So, given a collection of sentences, we need to "interpret" the bound sentences by using Tarski's definition of truth . Any comments?

 

[Bacle]

Never mind, Preno, I saw an argument that has to see with Stone's theorem on representability of Boolean algebras. I will read it and post it here soon.