Fundamentals of Classical Set Theory

Instructor

YongTao Chen

YongTao Chen has a MSc in pure mathematics, and has been an university math teacher for over two decades. His teaching focuses on Advanced Calculus, Linear Algebra, Abstract Algebra, Real Analysis, and Elementary Number Theory. He also taught Mathematical Logic, Introductory Probability, and Category Theory. Since the year 2011, he has been teaching online mathematical courses with enthusiasm.

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Overview

About the course

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 formalised 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.

The goal of this compact course is to introduce the audiences to the fundamental concepts surrounding classical set theory as quickly as possible. Present course can serve as an introduction to and preparation for 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 participants at various levels of mathematical education to begin their explorations in set theory.

Prerequisites

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. Besides, the willingness to spend at least 6 hours on the course every week is necessary for a successful learning.

Course Format

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. Every lecture is followed by a problem set used for home work. A set of review problems is provided at the end of the course so as to test the participants' understanding of the whole 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.

Course content

  • 1st Week-Lecture 1: Sets, Subsets, Power sets

  • 1st Week-Lecture 2: Set Operations

  • 2nd Week-Lecture 3: Cartesian products, Relations, Partial orderings

  • 2nd Week-Lecture 4: Well orderings, Equivalence relations, Partitions

  • 3rd Week-Lecture 5: Functions and Associated Concepts

  • 3rd Week-Lecture 6: Cardinality and Isomorphism

  • Review Problems

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