Sets - An Informal View

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In this section we will get a somewhat vague description of how sets are obtained. This is done hoping to illuminate the motivation behind some of the things we will do.

First we have to gather together all those things that are not themselves sets but that we want to have as members of sets - call such things as atoms. For instance, if we want to speak of the set of all two-headed coins, then we must include all such coins in our collection of atom.

Let A be the set of all atoms; it is the first set in our description. We now proceed to build up a hierarchy

of sets.

At the bottom level we take , the set of atoms. The next level will also contain all sets of atoms:

The third level contains everything that is in a lower level, plus all sets of things from lower levels:

In general,

But this hierarchy does not include enough sets. For example, , etc., but we do not yet have the infinite set

To remedy this lack, we take the infinite union

and then we can continue, . In general for any

and this goes on "forever".

A better explanations of the "forever" idea will be discuss in the future section but for the time being, the subscripts we are using are the so-called "ordinal numbers". The ordinal numbers begin with 0,1,2,....; then there is the infinite number then and then goes on "forever".

The fundamental principle is the following:

every set appears somewhere in this hierarchy. That is, for every set a there is some with . That is what the sets are; they are the members of the levels of our hierarchy.

There is one way in which we can simplify our picture.

We did introduce the idea of atom, but we have no idea what exactly is in the atoms. The fact of the matter is that the atoms serve no mathematically necessary purpose, so we can banish them; we take A = . But in doing so, we lose the ability to form sets of flowers or sets of people. But this is no cause for concern; we do not need set theory to talk about people and we do not need people in our set theory.

What we want is to have sets of numbers, e.g., .

Numbers do not appear at first glance to be sets. But we shall discover, we can find sets that serve perfectly well as numbers.

Our theory, then, will ignore all objects that are not sets. Instead we will concentrate just on "pure" sets that can be constructed without the use of external objects.

For instance, any member of one of our sets will itself be a set, and each of its members, if any, will be a set, and so forth. We will only stop when we reach the

With , we simply have:

Exercise 6:

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We have stated that . Prove this at least for .

Solution:

^{Disclaimer: this is a summary of section 1.2 from the book "Elements of Set Theory" by Herbert B. Enderton, the content apart from rephrasing is identical, most of the equations are from the book and the same examples are treated. All of the equation images were screenshot from generated latex form using typora.}

1. Elements of Set Theory by Herbert B. Enderton

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