Maximal Injective Mapping from Ordinals to a Set

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Theorem

Let $F$ be a mapping satisfying the following properties:


The domain of $F$ is $\operatorname{On}$, the ordinal class
For all ordinals $x$, $F \left({x}\right) = G \left({F \restriction x}\right)$.
For all ordinals $x$, if $(A \setminus \operatorname{Im} \left({x}\right) ) \ne \varnothing$, then $G \left({F \restriction x}\right) \in (A \setminus \operatorname{Im} \left({x}\right))$ where $\operatorname{Im} \left({x}\right)$ is the image of the subset $x$ under $F$.
$A$ is a set.


Then there exists an ordinal $y$ satisfying the following properties:

$\forall x \in y: \left({A \setminus \operatorname{Im} \left({x}\right)}\right) \ne \varnothing$
$\operatorname {Im} \left({y}\right) = A$
$F \restriction y$ is an injective mapping.

Note that the first third and fourth properties of $F$ are the most important. For any mapping $G$, a mapping $F$ can be constructed satisfying the first two properties using transfinite recursion.


Proof

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Set $B$ equal to the class of all ordinals $x$ such that $\left({A \setminus \operatorname{Im} \left({x}\right)}\right) \ne \varnothing$.

Assume $B = \operatorname{On}$.

Then:

\(\displaystyle B = \operatorname{On}\) \(\implies\) \(\displaystyle \forall x: F \left({x}\right) = G \left({F \restriction x}\right)\) by definition of $B$
\(\displaystyle \) \(\implies\) \(\displaystyle \forall x: G \left({F \restriction x}\right) \in \left({A \setminus \operatorname{Im}(F)}\right)\) by hypothesis

By Condition for Injective Mapping on Ordinals, $A$ is a proper class.

This contradicts the fact that $A$ is a set.

Therefore $B \subsetneq \operatorname{On}$.


Because $B$ is bounded above, $\bigcup B \in \operatorname{On}$.

By Union of Ordinals is Least Upper Bound, the union of ordinals is the least upper bound of $B$.

Setting $\bigcup B = x$:

$(1): \quad \left({A \setminus \operatorname{Im} \left({x}\right)}\right) = \varnothing \land \forall y \in x: \left({A \setminus \operatorname{Im} \left({y}\right)}\right) \ne \varnothing$

The first condition is satisfied.

In addition:

$(2): \quad A \subseteq \operatorname{Im} \left({x}\right)$

Take any $y \in \operatorname{Im} \left({x}\right)$.

Then:

\(\displaystyle y \in \operatorname{Im} \left({x}\right)\) \(\implies\) \(\displaystyle \exists z \in x: \left({y = F \left({z}\right)}\right)\) Definition of Image of Element
\(\displaystyle \) \(\implies\) \(\displaystyle \exists z: \left({y = F \left({z}\right) \land \left({A \setminus \operatorname{Im} \left({z}\right)}\right) \ne \varnothing}\right)\) Equation $(1)$
\(\displaystyle \) \(\implies\) \(\displaystyle \exists z: \left({y = F \left({z}\right) \land F \left({z}\right) \in \left({A \setminus \operatorname{Im} \left({z}\right)}\right)}\right)\) by hypothesis
\(\displaystyle \) \(\implies\) \(\displaystyle y \in A\)

This means that:

$\operatorname{Im} \left({x}\right) \subseteq A$

Combining with $(2)$:

$\operatorname{Im} \left({x}\right) = A$

$F$ is a mapping, so $\left({F \restriction x}\right)$ is a mapping.

Take any $y, z \in x$ such that $y$ and $z$ are distinct.

Without loss of generality, allow $y \in z$ (justified by Ordinal Membership Trichotomy).

\(\displaystyle y \in z \land z \in x\) \(\implies\) \(\displaystyle F \left({y}\right) \in \operatorname{Im} \left({z}\right) \land F \left({z}\right) \in \left({A \setminus \operatorname{Im} \left({z}\right)}\right)\) by hypothesis
\(\displaystyle \) \(\implies\) \(\displaystyle F \left({y}\right) \in \operatorname{Im} \left({z}\right) \land F \left({z}\right) \notin \operatorname{Im} \left({z}\right)\) Definition of Set Difference
\(\displaystyle \) \(\implies\) \(\displaystyle F \left({y}\right) \ne F \left({z}\right)\)

From this, we may conclude that $F$ is injective.

$\blacksquare$


Also see


Sources