Simple (even toy) examples for uses of Ordinals? I want to describe Ordinals using as much low-level mathematics as possible, but I need examples in order to explain the general idea. I want to show how certain mathematical objects are constructed using transfinite recursion, but can't think of anything simple and yet not artificial looking. The simplest natural example I have are Borel sets, which can be defined via transfinite recursion, but I think it's already too much (another example are Conway's Surreal numbers, but that again may already be too much).

Some accessible applications transfinite induction could be the following (depending on what the audience already knows):


*

*Defining the addition, multiplication (or even exponentiation) of ordinal numbers by  transfinite recursion and then showing some of their basic properties. (Probably most of the claims for addition and multiplication can be proved easier in a non-inductive way.)

*$a.a=a$ holds for every cardinal $a\ge\aleph_0$. E.g. Cieselski: Set theory for the working mathematician, Theorem 5.2.4, p.69. Using the result that any two cardinals are comparable, this implies $a.b=a+b=\max\{a,b\}$. See e.g. here

*The proof that Axiom of Choice implies Zorn's lemma. (This implication is undestood as a theorem in ZF - in all other bullets we work in ZFC.)

*Proof of Steinitz theorem - every field has an algebraically closed extension. E.g. Antoine Chambert-Loir: A field guide to algebra, Theorem 2.3.3, proof is given on p.39-p.40.

*Some constructions of interesting subsets of plane are given in Cieselski's book, e.g. Theorem 6.1.1 in which a set $A\subseteq\mathbb R\times\mathbb R$ is constructed such that $A_x=\{y\in\mathbb R; (x,y)\in A\}$ is singleton for each $x$ and $A^y=\{x\in\mathbb R; (x,y)\in A\}$ is dense in $\mathbb R$ for every $y$.
