Condition for Injective Mapping on Ordinals
Theorem
Let $F$ be a mapping satisfying the following properties:
- $(1): \quad$ The domain of $F$ is $\On$, the class of all ordinals
- $(2): \quad$ For all ordinals $x$, $\map F x = \map G {F \restriction x}$
- $(3): \quad$ For all ordinals $x$, $\map G {F \restriction x} \in \paren {A \setminus \Img x}$ where $\Img x$ is the image of $x$ under $F$.
Let $\Img F$ denote the image of $F$.
Then the following properties hold:
- $(1): \quad \Img F \subseteq A$
- $(2): \quad F$ is injective
- $(3): \quad A$ is a proper class.
Note that only the third property of $F$ is the most important.
For any function $G$, a function $F$ can be constructed satisfying the first two using the First Principle of Transfinite Recursion.
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Proof
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Let $x$ be an ordinal.
Then $\map F x = \map G {F \restriction x}$ and $\map G {F \restriction x} \in A$ by hypothesis.
Therefore, $\map F x \in A$.
This satisfies the first statement.
Take two distinct ordinals $x$ and $y$.
Without loss of generality, assume $x \in y$ (we are justified in this by Ordinal Membership is Trichotomy).
Then:
| \(\ds x \in y\) | \(\leadsto\) | \(\ds \map F x \in \Img y\) | ||||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds \map F x \in \Img y \land \map F y \notin \Img y\) | by hypothesis | |||||||||||
| \(\ds \) | \(\leadsto\) | \(\ds \map F x \ne \map F y\) |
Thus for distinct ordinals $x$ and $y$, $\map F x \ne \map F y$.
Therefore, $F$ is injective.
$F$ is injective and $F: \On \to A$.
Therefore, if $A$ is a set, then $\On$ is a set.
But by the Burali-Forti Paradox, this is impossible, so $A$ is not a set.
Therefore, $A$ is a proper class.
$\blacksquare$
Also see
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.46$