Simple (even toy) examples for uses of Ordinals?

Question

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).

Answer 1

You might find something useful in this post by Tim Gowers: http://www.dpmms.cam.ac.uk/~wtg10/ordinals.html. Especially his first example, with (countable) ordinals introduced as a convenient notation for indexing an increasing sequence of bounded increasing sequences (and so on in many levels perhaps), was quite illuminating for me.

That is, if $a_n \nearrow a$, and $a < b_n \nearrow b$, and $b < c_n \nearrow c$, etc., we will have the notational problem of running out of letters after a while. But we can instead write $a_{\omega}$ instead of $a$, and $a_{\omega+n}$ instead of $b_n$, and $a_{2\omega}$ instead of $b$, and $a_{2\omega+n}$ instead of $c_n$, etc., and thus index all the numbers using a single symbol $a$ with ordinals attached as subscripts. Even countably many sequences will not be a problem, since then we just denote the limit of the sequence $(a_{n\omega})_{n=1}^{\infty}$ by $a_{\omega^2}$. And so on...

Answer 2

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 $y$.

Answer 3

I think that the most pedagogical way of looking at ordinals is the task: How do you write the (sufficiently simple) expressions for arbitrarily large - usually extremely large - integers by addition, multiplication, and exponentiation involving a few small numbers such as $1,2,3,4,5$ and a few others as well as one million that you represent by $\omega=1,000,000$?

Well, you may write $\omega$ itself but you may also write $\omega+\omega$ or $\omega^\omega+2\omega^3+5$ or any ordinal. If $\omega$ is as small as a million, you could get wrong identities such as $[(1+1+1+1+1)(1+1)]^{1+1+1+1+1+1}=\omega$ but if $\omega$ is really large, those identities disappear and any formal prescription using $\omega$ is different.

The key feature that makes ordinals with an abstract $\omega$ worth considering is that the ordering - which of two expressions $A,B$ viewed as functions of $\omega$ is greater - doesn't depend on $\omega$ if $\omega$ is really large. So one may define the ordering $A<B$ or $A>B$ as the limit of the relationship between the evaluations of the functions of $\omega$ for which you substitute a positive integer number sent to infinity.