-6
votes
1answer
100 views

What are all kind of “metamath” good for? Can it help me here?

Those logical theories, which deals with questions that isn't really mathematics but reach mathematics more or less, often seems to be like textbooks full of definitions, plus some theorems of the ...
5
votes
1answer
228 views

Quantifiers as Adjoints in Generalized Logics

It is a well known fact that the classical universal and existential quantifiers can be seen as adjoints in certain categories. In the continuous model theory of metric structures (see ...
4
votes
1answer
171 views

Proving Tychonoff's theorem with the Compactness theorem of logic

It seems to be known that Tychonoff's Theorem for Hausdorff spaces and the Compactness theorem of first order logic are both equivalent over ZF to the ultrafilter lemma. Does anyone know a slick proof ...
5
votes
1answer
135 views

Inherited topology of logical Stone's spaces.

I'm asking here if the following construction is of any interest. I can not find any reference for that kind of thing, so either the subject is completely trivial, either I just don't have the correct ...
4
votes
1answer
83 views

Proof of Compactness Theorem

I'm going through Enderton's Mathematical Logic text and have encountered a problem that I'm having trouble solving. After searching this website I've found that another user had the same problem (you ...
7
votes
3answers
290 views

Model Theory and Topology Connections

I have studied a bit of model theory, when I say "a bit" I have studied much more than is available to a typical undergraduate in the UK (i think, certainly from what I have seen) but I am sure this ...
2
votes
0answers
151 views

Can we find a nice definition of Congruence in Topology?

According to my knowledge, quotient structure is a original structure divided by a congruence. However, quotient topology space is defined this way. Quotient_topology In this way, $\sim$ is only said ...
6
votes
1answer
316 views

Topological spaces as model-theoretic structures — definitions?

How do model theorists treat topological spaces as structures, i.e., what are the options for the domain, relations, and operations? I've never done any topology, so I have only the definition to go ...
8
votes
2answers
277 views

Why is the class of topological spaces not axiomatizable?

This is a follow-up question to this one, where in the answer it is explained how topological spaces may very well be described in a purely first-order manner. Furthermore, the set of first-order ...
12
votes
1answer
742 views

(Why) is topology nonfirstorderizable?

Is it the right point of view to say, that topology is nonfirstorderizable (only) because the union of arbitrarily many open sets has to be open? And if "arbitrarily many" was relaxed to "finitely ...