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Hometransfinite recursion
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transfinite recursion
Transfinite recursion, roughly speaking, is a statement about the ability to define a function recursively using transfinite induction. In its most general and intuitive form, it says
Theorem 1.
Let $G$ be a (class) function on $V$, the class of all sets. Then there exists a unique (class) function $F$ on $\mathbf{On}$, the class of ordinals, such that
$F(\alpha)=G(F\alpha)$ 
where $F\alpha$ is the function whose domain is $\operatorname{seg}(\alpha):=\{\beta\mid\beta<\alpha\}$ and whose values coincide with $F$ on every $\beta\in\operatorname{seg}(\alpha)$. In other words, $F\alpha$ is the restriction of $F$ to $\operatorname{\alpha}$.
Notice that the theorem above is not provable in ZF set theory, as $G$ and $F$ are both classes, not sets. In order to prove this statement, one way of getting around this difficulty is to convert both $G$ and $F$ into formulas, and modify the statement, as follows:
Let $\varphi(x,y)$ be a formula such that
$\forall x\exists y\forall z(\varphi(x,z)\leftrightarrow z=y).$ 
Think of $G=\{(x,y)\mid\varphi(x,y)\}$. Then there is a unique formula $\psi(\alpha,z)$ (think of $F$ as $\{(\alpha,z)\mid\psi(\alpha,z)\}$) such that the following two sentences are derivable using ZF axioms:
1. 2. $\forall x\forall y\Big(\mathbf{On}(x)\wedge\big(\psi(x,y)\leftrightarrow% \exists f(A\wedge B\wedge C\wedge D)\big)\Big)$, where

$A$ is the formula β$f$ is a functionβ,

$B$ is the formula β$\operatorname{dom}(f)=x$β,

$C$ is the formula $\forall z\big(z\in x\wedge\varphi(fz,f(z))\big)$, and

$D$ is the formula $\varphi(f,y)$.

A stronger form of the transfinite recursion theorem says:
Theorem 2.
Let $\varphi(x,y)$ be any formula (in the language of set theory). Then the following is a theorem: assume that $\varphi$ satisfies property that, for every $x$, there is a unique $y$ such that $\varphi(x,y)$. If $A$ be a wellordered set (wellordered by $\leq$), then there is a unique function $f$ defined on $A$ such that
$\varphi(f\operatorname{seg}(s),f(s))$ 
for every $s\in A$. Here, $\operatorname{seg}(s):=\{t\in A\mid t<s\}$, the initial segment of $s$ in $A$.
The above theorem is actually a collection of theorems, or known as a theorem schema, where each theorem corresponds to a formula. The other difference between this and the previous theorem is that this theorem is provable in ZF, because the domain of the function $f$ is now a set.
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