Foundations of Set Theory

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Set Theory and its Place in the Foundations of Mathematics: A New Look at an Old Question

Lecturer: Dr Adam Epstein. Content : Set theoretical concepts and formulations are pervasive in modern mathematics.

For this reason it is often said that set theory provides a foundation for mathematics. On a practical level, set theoretical language is a highly useful tool for the definition and construction of mathematical objects. On a more theoretical level, the very notion of a foundation has definite philosophical overtones, in connection with the reducibility of knowledge to agreed first principles.

The module will commence with a brief review of naive set theory. Unrestricted set formation leads to various paradoxes Russell, Cantor, Burali-Forti , thereby motivating axiomatic set theory. The Zermelo-Fraenkel system will be introduced, with attention to the precise formulation of axioms and axiom schemata, the role played by proper classes, and the cumulative hierarchy picture of the set theoretical universe.

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Transfinite induction and recursion, cardinal and ordinal numbers, and the real number system will all be developed within this framework. MSC: 03B15 03B30 03E30 03F25 1 Introduction The idea of founding mathematics on a theory of types was rst proposed by Russell [20] foreshadowed already in [19] , and subsequently implemented by Whitehead and Russell [26].

Axiom of Foundation

The formal systems presented in these works were later simplied and cast into their modern shape by Ramsey [18]. Documents: Advanced Search Include Citations. Citations: 3 - 1 self.

Abstract The systems K of transnite cumulative types up to are extended to systems K 1 that include a natural innitary inference rule, the so-called limit rule.

Foundations of Set Theory Foundations of Set Theory
Foundations of Set Theory Foundations of Set Theory
Foundations of Set Theory Foundations of Set Theory
Foundations of Set Theory Foundations of Set Theory
Foundations of Set Theory Foundations of Set Theory

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