Synopsis for B1b: Set Theory

Level: H-level Method of Assessment: Written examination. Weight: Half-unit (OSS paper code 2B40)

Overview

To introduce sets and their properties as a unified way of treating mathematical structures, including encoding of basic mathematical objects using set theoretic language. To emphasize the difference between intuitive collections and formal sets. To introduce and discuss the notion of the infinite, the ordinals and cardinality. The Axiom of Choice and its equivalents are presented as a tool.

Learning Outcomes

Students will have a sound knowledge of set theoretic language and be able to use it to codify mathematical objects. They will have an appreciation of the notion of infinity and arithmetic of the cardinals and ordinals. They will have developed a deep understanding of the Axiom of Choice, Zorn's Lemma and well-ordering principle, and have begun to appreciate the implications.

Synopsis

What is a set? Introduction to the basic axioms of set theory. Ordered pairs, cartesian products, relations and functions. Axiom of Infinity and the construction of the natural numbers; induction and the Recursion Theorem.

Cardinality; the notions of finite and countable and uncountable sets; Cantor's Theorem on power sets. The Tarski Fixed Point Theorem. The Schröder–Bernstein Theorem.

Isomorphism of ordered sets; well-orders. Transfinite induction; transfinite recursion [informal treatment only].

Comparability of well-orders.

The Axiom of Choice, Zorn's Lemma, the Well-ordering Principle; comparability of cardinals. Equivalence of WO, CC, AC and ZL. Ordinals. Arithmetic of cardinals and ordinals; in [ZFC],

1. D. Goldrei, Classic Set Theory (Chapman and Hall, 1996).
2. W. B. Enderton, Elements of Set Theory (Academic Press, 1978).

Further Reading

1. R. Cori and D. Lascar, Mathematical Logic: A Course with Exercises (Part II) (Oxford University Press, 2001), section 7.1–7.5.
2. R. Rucker, Infinity and the Mind: The Science and Philosophy of the Infinite (Birkhäuser, 1982). An accessible introduction to set theory.
3. J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princton University Press, 1990). For some background, you may find JW Dauben's biography of Cantor interesting.
4. M. D. Potter, Set Theory and its Philosophy: A Critical Introduction (Oxford University Press, 2004). An interestingly different way of establishing Set Theory, together with some discussion of the history and philosophy of the subject.
5. G. Frege, The Foundations of Arithmetic : A Logical-Mathematical Investigation into the Concept of Number (Pearson Longman, 2007).
6. M. Schirn, The Philosophy of Mathematics Today (Clarendon, 1998). A recentish survey of the area at research level.
7. W. Sierpinski, Cardinal and Ordinal Numbers (Polish Scientific Publishers, 1965). More about the arithmetic of transfinite numbers.