Classes

The chapter covers notes on classes. The materials assume having read the previous sections on relations.

Objects and Collections
Russell's Paradox
Classes

Objects and Collections

Recall that we defined a set as follows:

set. A set is a collection of objects, where each object in the set is called a member or element of the set. If an object ${x}$ is an element of the set ${X,}$ we write:

x \in X

Generally, the word object refers to some mathematical object. E.g., the number ${1}$ or ${\pi.}$ We can think of a collection of these objects as essentially a pile of the objects; a box where we place all of the objects.

Collections, as we can likely tell by now, are also objects. This leads to a few big problems. Let's examine some of them.

Russell's Paradox

The first big problem is Russell's paradox, named after the British philosopher Bertrand Russell (1872-1970). When Russell first heard of Cantor's notion of sets, he joined an ambitious project to ground all of mathematics in set theory. To Russell, and many other logicians, it seemed as if set theory is what would ground all of mathematics. For example, the equation:

x + 1 = 0

is just another way of expressing the set:

\set{x \in \reals : x = -1}

or simply:

\set{-1}

Likewise, the statement:

${x}$ is a natural number between 0 and 3, inclusive.

Could be expressed as:

\set{x \in \nat: 0 \leq x \leq 3}

or simply:

\set{0,1,2,3}

Thus, it seemd as if we could describe any object in mathematics in the form:

\set{\sf{x}:\sf{x}\text{ has property }\sf{P}}

or, more concisely:

\set{\sf{x} : \sf{P(x)}}

At some point, however, Russell ran into a problem: Sets are also objects. This means that given a set ${X,}$ there's nothing stopping ${X}$ from being a member of itself:

X = \set{X : \text{${X}$ is a set and ${X}$ has at least one element}}

Staring at this statement, Russell defined another set (called Russell's set):

russell's set. Russell's set ${\Rr}$ is defined as the set

\Rr = \set{\Rr : \text{${\Rr}$ is a set and ${\Rr}$ is not an element of itself}}.

Russell then asked a question: Is ${\Rr}$ an element of itself? Well, there are two cases: (1) Either ${\Rr}$ is an element of itself, or (2) ${\Rr}$ is not an element of itself. Let's consider these two cases.

Case (1). If ${\Rr}$ is an element of itself, then ${\Rr}$ is in ${\Rr}.$ But that can't be true, because ${\Rr}$ is not an element of itself by definition. So, case 2 has to be true. Let's look at that case.

Case (2). If ${\Rr}$ is not an element of itself, then great! We satisfy the definition. But there's a problem. Because of set theory's axiom of extension, the set ${\Rr}$ is, by definition, the set of all elements that are:

sets, and
not elements of themselves.

And since ${\Rr}$ is a set, and not an element of itself, it must be a member of ${\Rr.}$ Which means that ${\Rr}$ is an element itself ... Wait, we just said that's false! This conundrum is called Russell's paradox.

russell's paradox. Let ${x}$ be a set. Then:

x \in x \iff x \notin x

Classes

What Russell's paradox demonstrates the following conclusion:

All sets are collections, but not all collections are sets.

One such set is Russell's set ${\Rr.}$ Because of these sets, the set theorists introduced a new kind of collection, called the class.

class. A class is a collection of sets, where each set can be described in a given language.

For now, we'll simply think of classes as collections of sets (not that the word collection is very important — we can't define it as a "set of sets"). We'll also adopt the convention that all sets are classes (but not all classes are sets). Thus, whenever we use the word class, we mean a collection of sets.