# Order Isomorphism between Ordinals and Proper Class/Lemma

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## Lemma for Order Isomorphism between Ordinals and Proper Class

Suppose the following conditions are met:

Let $A$ be a class.

We allow $A$ to be a proper class or a set.

Let $\struct {A, \prec}$ be a strict well-ordering.

Let every $\prec$-initial segment be a set, not a proper class.

Let $\Img x$ denote the image of a subclass $x$.

Let $G$ equal the class of all ordered pairs $\tuple {x, y}$ satisfying:

- $y \in A \setminus \Img x$
- The initial segment $A_y$ of $\struct {A, \prec}$ is a subset of $\Img x$

Let $F$ be a mapping with a domain of $\On$.

Let $F$ also satisfy:

- $\map F x = \map G {F \restriction x}$

Then:

- $G$ is a mapping
- $\map G x \in A \setminus \Img x \iff A \setminus \Img x \ne \O$

Note that only the first four conditions need hold: we may construct classes $F$ and $G$ satisfying the other conditions using the First Principle of Transfinite Recursion.

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## Proof

\(\ds \tuple {x, y}\) | \(\in\) | \(\ds G\) | ||||||||||||

\(\, \ds \land \, \) | \(\ds \tuple {x, z}\) | \(\in\) | \(\ds G\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(\in\) | \(\ds A \setminus \Img x\) | Definition of $G$ | ||||||||||

\(\, \ds \land \, \) | \(\ds z\) | \(\in\) | \(\ds A \setminus \Img x\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(\notin\) | \(\ds A_z\) | $A_y$ is disjoint with $A \setminus \Img x$. Same with $A_z$. | ||||||||||

\(\, \ds \land \, \) | \(\ds y\) | \(\notin\) | \(\ds A_y\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(\nprec\) | \(\ds z\) | Definition of Initial Segment | ||||||||||

\(\, \ds \land \, \) | \(\ds z\) | \(\nprec\) | \(\ds y\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds z\) | $\prec$ is a strict well-ordering |

Therefore, we may conclude, that $G$ is a single-valued relation and therefore a mapping.

For the second part:

\(\ds A \setminus \Img x\) | \(\ne\) | \(\ds \O\) | ||||||||||||

\(\text {(1)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \exists y \in A \setminus \Img x: \, \) | \(\ds \paren {A \cap A_y} \setminus \Img x\) | \(=\) | \(\ds \O\) | Proper Well-Ordering Determines Smallest Elements | ||||||||

\(\ds \leadsto \ \ \) | \(\ds \map G x\) | \(=\) | \(\ds y\) | Conditions are satisfied for $\tuple {x, y} \in G$. Follows from first part. | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map G x\) | \(\in\) | \(\ds A \setminus \Img x\) | equation $(1)$, $y \in A \setminus \Img x$ |

Furthermore:

- $\map G x \in A \setminus \Img x \implies A \setminus \Img x \ne \O$ by the definition of non-empty.

$\blacksquare$

## Also see

- Transfinite Recursion Theorem
- Condition for Injective Mapping on Ordinals
- Maximal Injective Mapping from Ordinals to a Set

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 7.48$