User:Dfeuer/Definition:Finite Ordinal

Definition
NOTE: not the definition used in S &amp; F.

A finite ordinal is an ordinal which is strictly well-ordered under the inverse of the epsilon relation.

That is, a finite ordinal is an ordinal which is strictly well-ordered (well-founded and linearly ordered) under the epsilon relation and also under the inverse of the epsilon relation.

Theorem
The successor of a finite ordinal is a finite ordinal.

Proof
Let $n$ be a finite ordinal.

Then $n^+ = n \cup \{n\}$ is an ordinal because the successor of an ordinal is an ordinal.

Let $a$ be a non-empty subset of $n^+$.

If $n \notin a$, then $a \subseteq n$, so $a$ has a greatest element because $n$ is a finite ordinal.

If $n \in a$, then $n$ is the greatest element of $a$.

Theorem
0 is a finite ordinal.

Proof
Trivial.

Predecessor
Each finite ordinal that is not $0$ has a predecessor.

Induction
Suppose that $A$ is a class of finite ordinals, $0 \in A$, and for any finite ordinal $n$, $n \in A \implies n^+ \in A$.

Then $A$ is the class $\omega$ of all finite ordinals.

Proof
Let $P = \omega \setminus A$.

If $A \ne \omega$, then $P$ is not empty.

Thus $P$ has a least element $m$.