Ordinals

Overview

Assume $<$ is a strict well ordering on $A$ and define $\gamma (x,y)$ as the formula "$y=\text{ran}x$". The transfinite recursion theorem then presents us with a unique function with domain $A$ such that for any $t\in A$,

Then $\alpha =\text{ran}(E)$ is called the epsilon-image (denoted $\in$-image) of the well-ordered structure $⟨A,<⟩$. The name "$\in$-image" derives from the fact that $\alpha$ is well ordered by epsilon.

Transitive Sets

A set $A$ is said to be transitive iff every member of a member of $A$ is itself a member of $A$. We can equivalently express this using any of the following formulations:

• $x\in a\in A⟹x\in A$
• $\bigcup A\subseteq A$
• $a\in A⟹a\subseteq A$
• $A\subseteq \mathcal{P}A$

A transitive class is a class $C$ in which every member of a member of $C$ is itself a member of $C$.

Transitive Closures

Let $C$ be a fixed set. Apply transfinite recursion to $\omega$ using for $\gamma (x,y)$ the formula

Let $F$ be the $\gamma$-constructed function on $\omega$. Then $\bar{C}=\bigcup \text{ran}F$ is the transitive closure of $C$. $\bar{C}$ is a transitive set and $C\subseteq \bar{C}$.

Ordinal Numbers

Let $<$ be a strict well ordering on a set $A$. The ordinal number of $⟨A,<⟩$ is its $\in$-image. An ordinal number is a set that is the ordinal number of some well-ordered structure.

An $\in$-image (and therefore an ordinal number) is a transitive set. Let $\alpha$ be any transitive set that is well ordered by epsilon. Then $\alpha$ is an ordinal number. In fact, $\alpha$ is the $\in$-image of $⟨\alpha ,{\in }_{\alpha }⟩$.

The class of all ordinal numbers is a transitive class well ordered by epsilon.

Burali-Forti Theorem

There is no set to which every ordinal number belongs.

Hartogs's Theorem

For any set $A$, there is an ordinal not dominated by $A$. The Hartogs number of $A$ is the least ordinal $\alpha$ such that $\alpha ⪯̸A$. This ordinal is always a cardinal.

Limit Ordinals

Ordinals are categorized in one of three ways:

1. Zero, the ordinal $0$.
2. The successor ordinals which are those of form ${\alpha }^{+}$ for some ordinal $\alpha$.
3. The limit ordinals which constitute all remaining ordinals.

Initial Ordinals

An initial ordinal is an ordinal number that is not equinumerous to any smaller ordinal number. The initial numbers and cardinal numbers are therefore exactly the same thing.

Rank

For any ordinal number $\delta$, use transfinite recursion to define function $F_{\delta }$ with domain $\delta$ such that, for all $\alpha \in \delta$,

For ordinal number $\alpha$, define $V_{\alpha }$ as the set $F_{\delta }(\alpha )$ for some $\delta$ greater than $\alpha$. Therefore,

This can be written more compactly as:

1. $V_0=\varnothing$.
2. $V_{{\alpha }^{+}}=\mathcal{P}{V}_{\alpha }$ for any ordinal number $\alpha$.
3. $V_{\lambda }=\bigcup _{\beta \in \lambda }{V}_{\beta }$ for any limit ordinal $\lambda$.

This cumulative hierarchy of stes $V_{\alpha }$ indexed by the class of ordinal numbers is known as the von Neumann universe. A set $A$ is grounded if $A\subseteq V_{\alpha }$ for some ordinal number $\alpha$. The rank of a grounded set, denoted $\text{rank}A$, is the least such ordinal $\alpha$.

Regularity

Let $A$ be a set. We say $A$ is regular if $A$ is empty or $A$ has a member $m\in A$ such that $m\cap A=\varnothing$.

A set is grounded if and only if it is regular. This latter assertion, applied to the universe of all sets, is known as the regularity axiom.