Curious facts about ordinal numbers

Question

I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule doesn't always hold. Also that some things that are undefined in analysis like $0^0, \infty^0, 1^\infty$ are actually defined for ordinal numbers. I know that there's been some investigations devoted to pursuing ordinal arithmetic along the lines of classical results in number theory for example.

Do you happen to know other curious facts about ordinal numbers (compared to facts in analysis or other)?

Answer 1

1) Every countable ordinal can be written in a unique, canonical way, called the Cantor Normal form. It is basically like writing the ordinal "in base omega."

2) There is a fact in number theory that has a natural proof using (necessarily) infinite ordinals. The question is does every "Goldstein sequence" eventually converge to 0. A Goldstein sequence is, basically, a process where you take a number, write it in base 2, replace all of the 2's with 3's, then subtract 1. Then do this for base 3 into 4, etc. The answer is yes they all converge, as you can see here, and the proof is rather shocking for a number theorist.

Answer 2

Every countable ordinal is order isomorphic to some closed subset of (the rationals in) the closed unit interval.

Not really related to ordinals, but it's might be interesting: $\mathbb Q$ is universal for all countable order types, i.e. any countable totally ordered set embeds into $(\mathbb Q,\leq)$.

Answer 3

Given any mapping $f : \mathbf{Ord} \to \mathbf{Ord}$ which is strictly increasing and continuous at limit ordinals (such a mapping is called a normal mapping), there are arbitrarily large ordinals $\alpha$ such that $f ( \alpha ) = \alpha$.

• For example, the mapping $\alpha \mapsto \omega^\alpha$ is normal. The least fixed point of this function is commonly denoted $\epsilon_0$, and has great importance to the proof theory of Peano arithmetic.

  (Of course, as $\alpha \mapsto \omega^\alpha$ has arbitrarily large fixed points, we can define a new mapping $$\alpha \mapsto \epsilon_\alpha = \text{the } \alpha^{\text{th}} \text{ fixed point of the above mapping}.$$ This new mapping is also normal, and thus has arbitrarily large fixed points. And by a diagonalisation process at limit ordinals we can continue this indefinitely.)

• The mapping $\alpha \mapsto \aleph_\alpha$ is also normal, and thus has arbitrarily large fixed points. That is, there are infinite cardinals $\kappa$ that have $\kappa$-many (infinite) cardinals below them. Even more, it is consistent with ZFC that $2^{\aleph_0} = | \mathbb{R} |$ is such a cardinal.