Skolem paradox

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Leopold Löwenheim (1878–1948) in 1915 and Thoralf Skolem (1887–1963) in 1920 showed that any denumerable set of sentences that has a model has a denumerably infinite model. The theory of real numbers can be axiomatized as a theory with a denumerable set of sentences. Yet in that theory it is provable that the set of reals is larger than denumerably infinite, so ordinary or ‘standard’ models need more than denumerably infinite numbers of elements. Skolem thus showed that the theory admits of ‘non-standard’ models, or ones which are not isomorphic with the intended interpretation (see Cantor's theorem). The paradox is often regarded as relatively superficial, since the interpretation assigned to the sentences of the theory when they are given a denumerable model is not their ‘intuitive’ interpretation, according to which they imply the non-denumerable nature of the set of reals. But this raises the question of what fixes the intuitive interpretation. If the intuitive interpretation is fixed by conditions that can be expressed in a denumerable number of sentences, those may be added to the original list, and we have a theory still subject to the same result.

Subjects: Philosophy.

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