an old trompe-l'oeil I arranged while TA-ing a metalogic course, which often dealt with the limits of first order logic and in particular first order definability. That included definability -in- a model (perhaps of certain objects or relations in it) and definability -of- classes of models with certain properties.

It's easier to anticipate how to show that something is definable. You just define it. Showing something is not definable is more involved. You might show that any definition would get it wrong, or that if we could define it, we could do other things we already know we cannot do. You can also do it by describing how certain kinds of deterministic games would be won or lost.

We cannot, for instance, write a formula of first order logic that is true on a model only when that model is finite. Nor even a set of formulas. First order logic has a poor grip on Finiteness. Curiously, it has a better grip on Infiniteness, which can be defined with a set of formulas. Of course that set must itself be infinite.

I thought the limitations of (first order) logic were more interesting than the applications.

lace lichen, colored pencil dr suess shapes, clown from goodwill, and text taken from a tampon box. And one of their homework assignments adorned with quails.

#lacelichen #clowns #compactness #logic #games #finite #infinite #trompeloeil #photography #definability

- a space is compact if and only if every family of closed subsets having the finite intersection property has non-empty intersection.
This formulation of #compactness is used in some proofs of #Tychonoff's theorem and the #uncountability of the real numbers

#ultrafilter can be viewed as a "perfect locating scheme" where each subset E of space X can be used in deciding whether "what is looked for" might lie in E.
Thus #compactness :"no location scheme can end up with nothing" , "always something will be found".

#Compactness - The state or quality of being compact; close union of parts; density.