Transfinite Recursion

Theorem

Uniqueness

Let $f$ be a mapping with a domain $y$ where $y$ is an ordinal.

Let $f$ satisfy the condition that:

$\forall x \in y: f \left({x}\right) = G \left({f \restriction x}\right)$

where $f \restriction x$ denotes the restriction of $f$ to $x$.

Let $g$ be a mapping with a domain $z$ where $z$ is an ordinal.

Let $g$ satisfy the condition that:

$\forall x \in z: g \left({x}\right) = G \left({g \restriction x}\right)$

Let $y \subseteq z$.

Then:

$\forall x \in y: f \left({x}\right) = g \left({x}\right)$

First Principle of Transfinite Recursion

Let $G$ be a mapping.

Let $K$ be a class of mappings $f$ that satisfy:

the domain of $f$ is some ordinal $y$

$\forall x \in y: f \left({x}\right) = G \left({f {\restriction_x} }\right)$

where $f {\restriction_x}$ denotes the restriction of $f$ to $x$.

Let $F = \bigcup K$, the union of $K$.

Then:

$(1): \quad F$ is a mapping with domain $\operatorname{On}$

$(2): \quad \forall x \in \operatorname{On}: F \left({x}\right) = G \left({F {\restriction_x} }\right)$

$(3): \quad F$ is unique. That is, if another mapping $A$ has the above two properties, then $A = F$.

Corollary

Let $x$ be an ordinal.

Let $G$ be a mapping

There exists a unique mapping $f$ that satisfies the following properties:

The domain of $f$ is $x$

$\forall y \in x: f \left({y}\right) = G \left({f \restriction y}\right)$

Second Principle of Transfinite Recursion

Let $\operatorname{Dom} \left({x}\right)$ denote the domain of $x$.

Let $\operatorname{Im} \left({x}\right)$ denote the image of the mapping $x$.

Let $G$ be a class of ordered pairs $\left({x, y}\right)$ satisfying at least one of the following conditions:

$(1): \quad x = \varnothing$ and $y = a$

$(2): \quad \exists \beta: \operatorname{Dom} \left({x}\right) = \beta^+$ and $y = H \left({x \left({\bigcup \operatorname{Dom} \left({x}\right)}\right)}\right)$

$(3): \quad \operatorname{Dom} \left({x}\right)$ is a limit ordinal and $y = \bigcup \operatorname{Rng} \left({x}\right)$.

Let $F \left({\alpha}\right) = G \left({F \restriction \alpha}\right)$ for all ordinals $\alpha$.

Then:

$F$ is a mapping and the domain of $F$ is the ordinals, $\operatorname{On}$.

$F \left({\varnothing}\right) = a$

$F \left({\beta^+}\right) = H \left({F \left({\beta}\right)}\right)$

For limit ordinals $\beta$, $\displaystyle F \left({\beta}\right) = \bigcup_{\gamma \mathop \in \beta} F \left({\gamma}\right)$

$F$ is unique.

That is, if there is another function $A$ satisfying the above three properties, then $A = F$.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 7.40, \ \S7.41, \ \S7.42$