Order Isomorphism between Ordinals and Proper Class/Corollary
Corollary to Order Isomorphism between Ordinals and Proper Class
Let $A$ be a proper class of ordinals.
We will take ordering on $A$ to be $\in$.
Set $G$ equal to the class of all ordered pairs $\tuple {x, y}$ such that:
- $y \in A \setminus \Img x$
- $\paren {A \setminus \Img x} \cap A_y = \O$
Define $F$ by the First Principle of Transfinite Recursion to construct a mapping $F$ such that:
- The domain of $F$ is $\On$.
- For all ordinals $x$, $\map F x = \map G {F \restriction x}$.
Then $F: \On \to A$ is an order isomorphism between $\struct {\On, \in}$ and $\struct {A, \in}$.
Proof
$A$ is a proper class of ordinals.
It is strictly well-ordered by $\in$.
Moreover, every initial segment of $A$ is a set, since the initial segment of the ordinal is simply the ordinal itself.
Therefore, we may apply Order Isomorphism between Ordinals and Proper Class to achieve the desired isomorphism.
$\blacksquare$
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This theorem shows that every proper class of ordinals can be put in a unique order-isomorphism with the set of all ordinals.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.50$