I have a some-what vague question. It seems to me that there are two main ways in which ultrafilters (on a set) can be used. One is in topology. The notion of an ultrafilter converging to a point is very useful since, in particular, knowing the limit points of every ultrafilter on a space is equivalent to knowing its topology. The other use is in logic (a subject about which I admittedly know very little). For instance, ultraproducts (and more generally ultralimits) can be used to construct non-standard models etc. I'm just curious about any connections that exist between these two uses of ultrafilters. For example, is there any logical interpretation of ultrafilter convergence on a topological space, etc? Is there a connection to the internal logic of its topos of sheaves for example? I'm really a beginner with this stuff, so any interesting connections, even trivial ones, would be most interesting to me. If this question is too open-ended, feel free to change this to community wiki.
[Math] Connections between ultrafilters in topology and logic
gn.general-topologylo.logicultrafilters
Best Answer
Some of the connections between topology and logic via ultrafilters have been around for quite a while.
Łoś's theorem from 1955 is the first place where ultraproducts appear in logic, as far as I know, although the ultraproduct construction is older (probably due to Hewitt). A very elegant proof of the compactness theorem of first-order logic using ultraproducts is due to Morel, Scott, and Tarski. It shows that compactness in logic is really compactness of an appropriate topological space. This pretty and inspiring connection can be extended much further.
A very nice starting point to learn about these connections is the series of papers by Xavier Caicedo (no relation):
The last paper shows how many "limit" constructions in intuitionistic logic (Kripke models), set theory (forcing) and elsewhere are examples of the same phenomenon.
This topological approach to logic is mainly guided by the ultraproduct construction. Daniele Mundici has also written about this.
Paolo Lipparini has studied variants of compactness that also turn out to have connections to logic via properties of ultrafilters, and lead to very interesting problems that seem to require Shelah's pcf theory; this line of work seems to have originated in set-theoretic topology, and R. Stephenson wrote a good survey (25 years old now) of the then state of the art as far as the topological side of things, see his article in the Handbook of Set-Theoretic Topology.
Finally, set theory is nowadays where both ultrafilters in general and the ultraproduct construction in particular are mostly used, in connection with large cardinals and elementary embeddings. Many natural problems in set-theoretic topology have been shown to have deep connections with these cardinals via the ultrafilters they generate. (Though here the connection to logic proper is weaker.)