I’m reading through this and I’d like to define an injective function from the set of countable ordinals $\Omega$ into $\mathbb{R}$ using transfinite induction (or maybe transfinite recursion?). Clearly, $\emptyset \in \Omega$ will be defined to map to zero: $f(\emptyset) := 0$. Next one would probably make a distinction between limit ordinals and successor ordinals. […]

I was wondering about what is the general definition of a recursive mapping between any two sets. Is there a condition for a mapping to be able to be written in recursive form? Is the following claim true: A mapping can be represented recursively if and only if its domain is a set that can […]

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor’s investigations of trigonometric series?)

The recursion theorem In set theory, this is a theorem guaranteeing that recursively defined functions exist. Given a set $X$, $a \in X$ and a function $f \colon X \to X$, the theorem states that there is a unique function $F:\mathbb{N} \to X$ (where $\mathbb{N}$ denotes the set of natural numbers including zero) such that […]

Let $f:A\rightarrow A$ be a function. We can simply define $f\circ f$, $f\circ f\circ f$, etc., for each given natural number inductively. $f^{(0)}=id_A$ $\forall n\in\omega \qquad f^{(n+1)}=f\circ f^{(n)}$ How to define $f^{(\omega)}$ such that it be a function from $A$ to $A$? Is it possible to define $f^{(\alpha)}$ for arbitrary large ordinal number $\alpha$?

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