Axiomatic set theory

Set theory is a branch of mathematics created principally by the German mathematician Georg Cantor at the end of the 19th century. Initially controversial, set theory has come to play the role of a foundational theory in modern mathematics, in the sense of a theory invoked to justify assumptions made in mathematics concerning the existence of mathematical objects (such as numbers or functions) and their properties.

Objections to set theory

Since its inception, there have been some mathematicians who have

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objected to using set theory as a foundation for mathematics, claiming that it is just a game which includes elements of fantasy. Notably, Henri Poincaré is supposed to have said "set theory is a disease from which mathematics will one day recover", (this quotation is part of the folklore of mathematics; the original source is unknown) and Errett Bishop dismissed set theory as God's mathematics, which we should leave for God to do.

Related Topics:
Objected to using set theory as a foundation for mathematics - Henri Poincaré - Folklore - Errett Bishop

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The most frequent objection to set theory is the constructivist view that mathematics is loosely related to computation and that naive set theory is being formalised with the addition of noncomputational elements.

Related Topics:
Constructivist - Naive set theory

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Topos theory has been proposed as an alternative to traditional axiomatic set theory. Topos theory includes various alternatives to set theory such as constructivism, fuzzy set theory, finite set theory, and computable set theory.

Related Topics:
Topos theory - Constructivism - Fuzzy set theory - Computable

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