Compactness theorem: Difference between revisions

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In [[symbolic logic]], the '''compactness theorem''' asserts that a set of [[first-order predicate calculus|first-order]] sentences is [[consistency|consistent]] if and only if any finite [[subset]] of it is consistent. Any consistent set of sentences in logic has a [[model theory | 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 first order sentence can be true

for [[field]]s 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.

Also, it follows that any theory that has an 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.