In symbolic logic, the compactness theorem asserts that a set of sentences is consistent if and only if any finite subset of it is consistent. Any consistent set of sentences in logic has a model, so this is equivalent to saying that set of sentences has a model if and only if any finite subset of it has a model.
This is a basic fact in logic and model theory, and has very far reaching consequences. For instance, it follows that if some sentence can be true for fields of arbitrary large characteristic, it must also be true for some field of characteristic zero. In other words, if some sentence holds for every field of characteristic zero it must hold for every field of characteristic larger than some constant - this application is called Robertson's principle
Also, it follows that any theory that has infinite model has models of arbitrary large cardinality. So, for instance, there are nonstandard models of Peano arithmetic with uncountably many natural numbers. The nonstandard analysis is another example where infinite natural numbers appear, a possibility that cannot be excluded by any axiomatization - also a consequence of the compactness theorem.