# 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 ordinal class
$(2): \quad$ 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 transfinite recursion.

## Proof

This page is beyond the scope of ZFC, and should not be used in anything other than the theory in which it resides.

If you believe that the contents of this page can be reworked to allow ZFC, then you can discuss it at the talk page.

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.

$\blacksquare$