# Chapter 1 - Set Theory

## 1.3 Ordinal Sets and Ordinal Numbers

### Results

**Isomorphism**
Two ordered sets $M$ and $M'$ are said to be

*isomorphic* if there exists an isomorphism
between them, i.e a bijective function $f:M\rightarrow M'$ such that $a\leq b$ for $a,b\in M$ iff
$f(a)\leq f(b)$ for $f(a), f(b)\in M'$.

Definition 1

An ordered set $M$ is said to be

**well-ordered** if every nonempty subset $A$ of $M$ has
a

**smallest** (or 'first')

**element**, i.e. an element $\mu$ such that $\mu < a$ for all
$a\in A$.

Definition 2

The order type of a well-ordered set is called an

**ordinal number** or simply an

**ordinal**.
If the set is infinite, the ordinal is said to be

**transfinite**.

Theorem 1

The ordered sum of a finite number of well-ordered sets $M_1, M_2, \ldots, M_n$ is
itself a well-ordered set.

Corollary

The sum of a finite number of ordinal numbes is itself an ordinal number.

Theorem 2

The ordered product of two well-ordered sets $M_1$ and $M_2$ is itself a well-ordered set.

Corollary 1

The ordered product of a finite number of well-ordered sets is itself a well-ordered set.

Corollary 2

The ordered product of a finite number of ordinal numbers is itself an ordinal number.

Theorem 3

Two given ordinal numbers $\alpha$ and $\beta$ satisfy one and only one of the relations
$$
\alpha < \beta,\qquad
\alpha = \beta,\qquad
\alpha > \beta.
$$

Theorem 4

Let $A$ and $B$ be well-ordered sets. Then either $A$ is equivalent to $B$ or one of the sets is
of greater power than the other, i.e. the powers of $A$ and $B$ are comparable.

Well-ordering Theorem

Every set can be well-ordered.

Axiom of choice

Given any set $M$, there is a "choice function" $f$ such that $f(A)$ is an element of $A$ for
every nonempty subset $A\subset M$.

**Note**: Skipping these exercises for now.