Sets, An Introduction

Introduction

Having studied sets every September from 6th grade through 12th, I thought I knew what a set was.  It's like a bag with stuff in it, right?  Then I went to graduate school and learned otherwise.

Actually there are no bags; there are only rules for containment.  The even numbers form a set because they conform to a rule, i.e. divisible by 2.  You can picture them in a bag, or not, as you wish.  However, the bag metaphore gets in the way at times, because there are situations where a set can contain itself, or two sets can contain each other.

What sort of items can be clumped together into sets?  Lions?  Tigers?  Bears?  Surely the integers.  Well not really.  It is possible to start with the empty set and build the integers, and all of mathematics from there.  The empty set is denoted ∅, and it has no members.  Think of the empty set as 0, the set containing the empty set as 1, the set containing 0 and 1 as 2, the set containing 0 1 and 2 as 3, and so on.  So you see, we don't really need any items, or bags, at all.  The only thing that is real is the empty set and the rules for containment.

If two sets contain the same elements they are the same set.  This reenforces the notion that the rule is the set.  If two sets both contain the even integers, they are the same set.

We write x ∈ s when x belongs to s.  We call x an element, i.e. an element of s, but it's really just another set, because that's all we have is sets, and sets of sets, etc.

There are some rules that do not have sets associated with them.  Bertrand Russell (biography) came up with the following paradox.  Let t ∈ s iff t ∉ t.  In other words, s is the set of sets that do not contain themselves.  Now s contains itself iff it does not contain itself.  This is a contradiction, hence there is no set associated with this rule.  We can talk about the sets that don't contain themselves, and that's a perfectly good "collection" or "class" of sets, but we can't put a bag around it and call it a set.