Property of Increasing Mapping on Ordinals

Theorem

Let $\On$ denote the class of all ordinals.

Let $K_{II}$ denote the class of all limit ordinals.

Let $F$ be a mapping defined on $\On$ satisfying the following conditions:

\(\text {(1)}: \quad\)

\(\ds \forall \alpha \in \On: \, \)

\(\ds \map F \alpha\)

\(\subseteq\)

\(\ds \map F {\alpha^+}\)

\(\text {(2)}: \quad\)

\(\ds \forall \lambda \in K_{II}: \forall \alpha \in \On: \, \)

\(\ds \alpha\)

\(<\)

\(\ds \lambda\)

\(\, \ds \implies \, \)

\(\ds \map F \alpha\)

\(\le\)

\(\ds \map F \lambda\)

Then:

\(\ds \forall \alpha, \beta \in \On: \, \)

\(\ds \alpha\)

\(\le\)

\(\ds \beta\)

\(\, \ds \implies \, \)

\(\ds \map F \alpha\)

\(\le\)

\(\ds \map F \beta\)

Proof

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Sources

• 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Exercise $5.4$