Compactness theorem: Difference between revisions

Revision as of 15:21, 28 November 2002

Revision as of 23:25, 28 November 2002

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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 [[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 - 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.