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2023•arXiv (Cornell University)Open access

Very basic set theory

Doeko Homan

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Abstract

Ernst Zermelo's axiomatization of set theory (1908) did not exclude `a set that is a member of itself'. We call a set that is a member of itself `an individual'. In this article we prove the elimination of Russell's paradox is equivalent to "For every set S, an individual is a member of S or a set (but not an individual) is not a member of S". This shows there is place in set theory for individuals. And we show the set theory with individuals has its philosophical foundation in Ludwig Wittgenstein's Tractatus Logico-Philosophicus.

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Ernst Zermelo's axiomatization of set theory (1908) did not exclude `a set that is a member of itself'. We call a set that is a member of itself `an individual'. In this article we prove the elimination of Russell's paradox is equivalent to "For every set S, an individual is a member of S or a set (but not an individual) is not a member of S". This shows there is place in set theory for individuals. And we show the set theory with individuals has its philosophical foundation in Ludwig Wittgenstein's Tractatus Logico-Philosophicus.

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Available abstract

Ernst Zermelo's axiomatization of set theory (1908) did not exclude `a set that is a member of itself'. We call a set that is a member of itself `an individual'. In this article we prove the elimination of Russell's paradox is equivalent to "For every set S, an individual is a member of S or a set (but not an individual) is not a member of S". This shows there is place in set theory for individuals. And we show the set theory with individuals has its philosophical foundation in Ludwig Wittgenstein's Tractatus Logico-Philosophicus.

Key concepts: Universal set, Set (abstract data type), Set theory, Infinite set, Mathematical economics, Mathematics, Epistemology, Computer science

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