# B1.2 Set Theory - Material for the year 2019-2020

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There are no formal prerequisites, but familiarity with some basic mathematical objects and notions such as: the rational and real number fields; the idea of surjective, injective and bijective functions, inverse functions, order relations; the notion of a continuous function of a real variable, sequences, series, and convergence, and the definitions of basic abstract structures such as fields, vector spaces, and groups (all covered in Mathematics I and II in Prelims) will be helpful at points.

16 lectures

### Assessment type:

- Written Examination

Introduce sets and their properties as a unified way of treating mathematical structures. Emphasise the difference between an intuitive collection and a formal set. Define (infinite) cardinal and ordinal numbers and investigate their properties. Frame the Axiom of Choice and its equivalent forms and study their implications.

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 the Well-Ordering Principle.

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. Basic cardinal arithmetic.

Well-orders. Comparability of well-orders. Ordinal numbers. Transfinite induction; transfinite recursion [informal treatment only]. Ordinal arithmetic.

The Axiom of Choice, Zorn's Lemma, the Well-ordering Principle; comparability of cardinals. Equivalence of WO, CC, AC and ZL. Cardinal numbers.

- D. Goldrei,
*Classic Set Theory*(Chapman and Hall, 1996). - H. B. Enderton,
*Elements of Set Theory*(Academic Press, 1978).

- R. Cori and D. Lascar,
*Mathematical Logic: A Course with Exercises (Part II)*(Oxford University Press, 2001), section 7.1-7.5. - R. Rucker,
*Infinity and the Mind: The Science and Philosophy of the Infinite*(Princeton University Press, 1995). An accessible introduction to set theory. - J. W. Dauben,
*Georg Cantor: His Mathematics and Philosophy of the Infinite*(Princeton University Press, 1990). For some background, you may find JW Dauben's biography of Cantor interesting. - 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. - W. Sierpinski,
*Cardinal and Ordinal Numbers*(Polish Scientific Publishers, 1965). More about the arithmetic of transfinite numbers. - J. Stillwell,
*Roads to Infinity*(CRC Press, 2010).