MA3H3 Set Theory Notes | Assignment Help | Syllabus
Country: United Kingdom
University: The University Of Warwick
Modern mathematics is replete with set theoretical ideas and expressions. As a result, set theory is often referred to as a mathematical basis. The term ‘foundation’ may have a variety of connotations in this context. Set theoretical language is a very helpful tool for defining and constructing mathematical things on a practical level. On a more theoretical level, the concept of a foundation has philosophical implications in terms of knowledge’s reducibility to agreed-upon fundamental principles.
A short overview of naïve set theory will kick off the program. Axiomatic set theory is motivated by the paradoxes that result from unrestricted set creation (Russell, Cantor, Burali-Forti). The Zermelo-Fraenkel system will be discussed in detail, with emphasis on the exact specification of axioms and axiom schemata, the function of proper classes, and the cumulative hierarchy image of the set theoretical world. This framework will cover transfinite induction and recursion, cardinal and ordinal numbers, and the real number system. The Choice Axiom, as well as its many counterparts and implications, will be explored, as will a number of other concepts known to be independent of Zermelo-Fraenkel set theory, such as the Continuum Hypothesis and the existence of Inaccessible Cardinals.
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