2 - Elements of set theory

Summary

We recapitulate the axiomatic foundations on which this book is based in Section 2.1. We then recall a few points about finiteness and countability in Section 2.2 and some basics of order theory in Section 2.3, and discuss the Axiom of Choice and some of its consequences in Section 2.4. These points will be needed often in the rest of this book. If you prefer to read about topology right away, and feel confident enough, please proceed directly to Chapter 3.

Foundations

We shall rest on ordinary set theory. While the latter has been synonymous with Zermelo–Fraenkel (ZF) set theory with the Axiom of Choice (ZFC) for some time, we shall use von Neumann–Gödel–Bernays (VBG) set theory instead (Mendelson, 1997).

There is not much difference between these theories: VBG is a conservative extension of ZFC. That VBG is an extension means that any theorem of ZFC is also a theorem of VBG. That it is conservative means that any theorem of VBG that one can express in the language of ZFC is also provable in ZFC.

The main difference between VBG and ZFC is that the former allows one to talk about collections that are too big to be sets. This is required, in all rigor, in the definition of (big) graphs and categories of Section 4.12. VBG allows us to talk about, say, the collection V of all sets, although V cannot itself be a set. This is the essence of Russell's paradox: assume there were a set V of all sets.