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.

 

[blue_raver22]

The compactness theorem in logic and the concept of compactness in topology are indeed closely related. Both involve the idea of finite subcollections covering a larger set. In topology, this means that a finite number of open sets can cover a space, while in logic, it means that a finite number of sentences can satisfy a larger sentence.

One way to think about this connection is through the concept of "compactness" itself. In topology, a space is considered compact if every open cover has a finite subcover. This means that the space is "compact" in the sense that it can be covered by a finite number of open sets. Similarly, in logic, a sentence is considered satisfiable if every finite subset of sentences is satisfiable. This means that the sentence is "compact" in the sense that it can be satisfied by a finite number of sentences.

In the example provided, the Cantor space is used to demonstrate this connection. The Cantor space is a topological space that can be thought of as an infinite product of {0,1} with the discrete topology. This means that any string of 0's and 1's can be thought of as a clopen (closed and open) subset of the Cantor space. By using the compactness of the Cantor space, it is possible to show that any sentence can be satisfied by a finite number of sentences, or in other words, that it is compact in the logical sense.

However, as mentioned, there may be some gaps in this argument and it is important to carefully consider the details in order to fully understand the connection between compactness in topology and in logic. Additionally, this is just one example of how these concepts are related and there may be other ways to approach this connection.