Compactness theorem: Difference between revisions

Revision as of 10:50, 28 November 2002

Revision as of 15:21, 28 November 2002

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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.