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Abstract: That two well known mathematical languages, the classical one of Aristotelian Logic and the everyday one of First Order Logic, are ``Compact'' is, for my money, one of the most striking facts of mathematical logic. What it says is that, for any collection of sentences in these languages, if every finite subcollection has a ``model'' (a mathematical structure in which all the sentences of the subcollection are true), then the whole collection must have a model.
It is surprisingly easy to see why these languages are compact. Even more surprising are the ramifications of their compactness, in all areas of mathematics, from classical analysis to modern combinatorics and, most especially, in mathematical logic, where it has some unsettling things to say about the mathematical languages we employ, and the mathematical proofs for which we search.
I will begin this talk with an historical exploration of compactness, and its connection with the corresponding topological concept, from which it gets its name; follow this with examples of the use of compactness in the ``non-standard analysis'' of Abraham Robinson and the ``probabilistic method'' of Paul Erdos; and close with a brief discussion of what compactness says about mathematical languages and proofs.