Transfinite Recursion Theorem/Theorem 2
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Theorem
Let $\Dom x$ denote the domain of $x$.
Let $\Img x$ denote the image of the mapping $x$.
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Let $G$ be a class of ordered pairs $\tuple {x, y}$ satisfying at least one of the following conditions:
- $(1): \quad x = \O$ and $y = a$
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- $(2): \quad \exists \beta: \Dom x = \beta^+$ and $y = \map H {\map x {\bigcup \Dom x} }$
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- $(3): \quad \Dom x$ is a limit ordinal and $y = \bigcup \Rng x$.
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Let $\map F \alpha = \map G {F \restriction \alpha}$ for all ordinals $\alpha$.
Then:
- $F$ is a mapping and the domain of $F$ is the class of all ordinals, $\On$.
- $\map F \O = a$
- $\map F {\beta^+} = \map H {\map F \beta}$
- For limit ordinals $\beta$, $\ds \map F \beta = \bigcup_{\gamma \mathop \in \beta} \map F \gamma$
- $F$ is unique.
- That is, if there is another function $A$ satisfying the above three properties, then $A = F$.
Proof
| \(\ds \map F \O\) | \(=\) | \(\ds \map G {F \restriction \O}\) | by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map G \O\) | Restriction of $\O$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds a\) | Definition of $G$ |
$\Box$
| \(\ds \map F {\beta^+}\) | \(=\) | \(\ds \map G {F \restriction \beta^+}\) | by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map H {\map {F \restriction \beta^+} {\bigcup \beta^+} }\) | Definition of $G$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map H {\map F \beta}\) | Union of successor set is the original set by Union of Ordinals is Least Upper Bound |
$\Box$
| \(\ds \map F \beta\) | \(=\) | \(\ds \map G {F \restriction \beta}\) | by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \bigcup \Img {F \restriction \beta}\) | Definition of $G$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \bigcup_{\gamma \mathop \in \beta} \map F \gamma\) |
$\Box$
We can proceed in the fourth part by Transfinite Induction.
Basis for the Induction
| \(\ds \map F \O\) | \(=\) | \(\ds a\) | First part | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map A \O\) | by hypothesis |
This proves the basis for the induction.
$\Box$
Induction Step
| \(\ds \map F \beta\) | \(=\) | \(\ds \map A \beta\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map H {\map F \beta}\) | \(=\) | \(\ds \map H {\map A \beta}\) | Substitutivity of equality | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map F {\beta^+}\) | \(=\) | \(\ds \map A {\beta^+}\) | Second part |
This proves the induction step.
$\Box$
Limit Case
| \(\ds \forall \gamma \in \beta: \, \) | \(\ds \map F \gamma\) | \(=\) | \(\ds \map A \gamma\) | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \bigcup_{\gamma \mathop \in \beta} \map F \gamma\) | \(=\) | \(\ds \bigcup_{\gamma \mathop \in \beta} \map A \gamma\) | Substitutivity of equality | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map F \beta\) | \(=\) | \(\ds \map A \beta\) | Third part |
This proves the limit case.
$\blacksquare$
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.42$