Sets

In Chapter 4 we introduced the terms

set

and

element of a set

. In order to have some flexibility and to avoid the slightly awkward phrase

set of sets

, we will use the word

collection

as a synonym for set. In particular, we will usually speak of a collection of sets, which is just a set of sets. Just as you worked with points and lines in geometry without having a rigorous definition of those terms, we will ask you to use your intuition as you work with the terms

set

and

element.

It’s important to note that you will need to exercise care when you use these words. Mathematics describes very carefully and exactly what you can do with

sets

and when you can use the words

element of a set

. In particular, the construction of “the set of all sets” is forbidden, as this would lead to contradictions. In order to get around such contradictions, mathematicians have developed axioms. These are listed in the Appendix as the Zermelo–Fraenkel system together with the axiom of choice (ZFC, for short). At this point, we will introduce our subject in a less formal way, leaving a more axiomatic treatment for a later course in set theory.