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YongTao Chen
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    01/12/2014
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FCST Fundamentals of Classical Set Theory

  • Mr. Chen YongTao
    Teacher
Course subject : Mathematical Analysis
Language : English (US)
Since the time of the Pythagoreans, mathematicians and philosophers had been concerned with infinities including continuum. It was Bernard Bolzano who began a systematic study of infinity. Then Georg Cantor initiated an adequate theory, fruitful with important results, which is referred to as Classical Set Theory.

Set theory has provided a standard foundation for mathematics, in a sense that every mathematical object can be defined in terms of the single notion of a “set”, and all of mathematics can be formalized within the axiom system ZFC of Set Theory. In other words, essentially all mathematical theories deal with sets in one way or another. Accordingly, Set Theory is a versatile tool used to build mathematical models of various phenomena. Hilbert said: "No one shall be able to drive us from the paradise which Cantor created for us". Indeed, the influence of Set Theory has pervaded the whole of mathematics.

Present course is intended to serve as an introduction to and preparation for a futher course such as the theory of ordinals and cardinals.

The goal of this compact course is to introduce the audiences to the fundamental concepts surrounding classical set theory as quickly as possible, which paves the way for learning in-depth topics such as the theory of ordinals and cardinals. The subject is treated in informal manner that makes the course different from axiomatic set theory. Present course is intended for two audiences -- students who are interested in studying classical set theory for its own sake, and students in other areas which involves the applications of classical set theory. I trust that present course will help readers at various levels of mathematical education to begin their explorations in set theory.

The course consists of 6 lectures, each of which contains two periods of lessons. The specific content of present course is indicated by the titles of the lectures, which are as follows:

  1. Sets, Subsets, Power sets;

  2. Set Operations;

  3. Cartesian products, Relations, Partial orderings;

  4. Well orderings, Equivalence relations, Partitions;

  5. Functions and Associated Concepts;

  6. Cardinality and Isomorphism.

A set of review problems is provided at the end of the course. In the spirit of teachings in reality, in each week are delivered two lectures, and the lecturing on the entire course is thus completed in 3 weeks. Certainly, participants may go through the course at their own paces.

No background in mathematical logic is required, however participants are expected to be familiar with methods of proof that are developed in abstract mathematics courses such as analysis and number theory; in other words, participants are expected to be comfortable with mathematical thinking and rigorous arguments.

  1. H. B. Enderton, Elements of Set Theory, Academic Press, New York 1977. 
  2. T. Jech, Set Theory, The Third Millennium Edition, revised and expanded, Springer-Verlag, Berlin 2003. 
  3. K. Ciesielski, Set Theory for the Working Mathematician, Cambridge University Press, Cambridge, 1997.
  4. M. Potter, Set Theory and its Philosophy, Oxford University Press, Oxford, 2004.