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Section B.19 The Zermelo-Fraenkel Axioms of Set Theory

In the first part of this appendix, we put number systems on a firm foundation, but in the process, we used an intuitive understanding of sets. Not surprisingly, this approach is fraught with danger. As was first discovered more than 100 years ago, there are major conceptual hurdles in formulating consistent systems of axioms for set theory. And it is very easy to make statements that sound “obvious” but are not.

Here is one very famous example. Let \(X\) and \(Y\) be sets and consider the following two statements:

  1. There exists an injection \(f:X\rightarrow Y\text{.}\)

  2. There exists a surjection \(g:Y\rightarrow X\text{.}\)

If \(X\) and \(Y\) are finite sets, these statements are equivalent, and it is perhaps natural to surmise that the same is true when \(X\) and \(Y\) are infinite. But that is not the case.

Here is the system of axioms popularly known as ZFC, which is an abbreviation for Zermelo-Fraenkel plus the Axiom of Choice. In this system, the notion of set and the membership operator \(\in\) are undefined. However, if \(A\) and \(B\) are sets, then exactly one of the following statements is true: (i) \(A\in B\) is true; (ii) \(A\in B\) is false. When \(A\in B\) is false, we write \(A\notin B\text{.}\) Also, there is an equivalence relation \(=\) defined on sets.

A good source of additional (free) information on set theory is the collection of Wikipedia articles. Do a web search and look up the following topics and people:

  1. Zermelo-Fraenkel set theory.

  2. Axiom of Choice.

  3. Peano postulates.

  4. Georg Cantor, Augustus De Morgan, George Boole, Bertrand Russell and Kurt Gödel.