Paradoxes of set and property

DOI: 10.4324/9780415249126-Y024-1
Version: v1,  Published online: 1998
Retrieved September 21, 2020, from

9. Brouwer, Weyl and Skolem: attacking set theory with paradoxes

Opponents of set theory repeatedly used the paradoxes as a weapon against it. This began with Poincaré in 1906 and continued with Brouwer in 1907 (see Intuitionism). Both saw intuition as central, the axiomatization of logic as suspect, and the natural numbers as fundamental. Both engaged in later attacks on axiomatization and set theory. A decade later Hermann Weyl continued the attack on analysis and set theory, using Grelling’s and Richard’s paradoxes.

In 1923 Thoralf Skolem used a different paradox to argue that set-theoretic concepts are relative, not absolute. Skolem’s paradox states that there is a countable model for axiomatic set theory, although there exist uncountable sets. The solution distinguishes between levels. There exists some set M which is uncountable within the model, where there is no function mapping M onto the natural numbers; but outside the model there is such a function, and so, from outside, all the sets in the model are seen as countable.

Citing this article:
Moore, Gregory H.. Brouwer, Weyl and Skolem: attacking set theory with paradoxes. Paradoxes of set and property, 1998, doi:10.4324/9780415249126-Y024-1. Routledge Encyclopedia of Philosophy, Taylor and Francis,
Copyright © 1998-2020 Routledge.

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