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.