Countable Ordinals

In an intro topology class we briefly brought up ordinal numbers during a conversation of transfinite induction. I believe I understand how the ordinal numbers work, at least up to $\omega^\omega$.

It seems that a set of cardinality $\omega^\omega$ must be uncountable, since $\mathbb{N}^\omega$ is uncountable, but on wikipedia it says that $\omega^\omega$ as an ordinal is countable (as are many ordinals after it). I guess that past a certain point it does not make complete sense to relate the cardinal and ordinal numbers of a set, but is there an intuitive example of a countable set with ordinal number $\omega^\omega$? If not how would one go about proving that it was countable?

Cardinal exponentiation and ordinal exponentiation are fundamentally different things; it's unfortunate that they use the same notation. Some old books write "${}^\beta\alpha$" for the cardinal exponentiation "$\alpha^\beta$," but that's very much not in fashion. (See https://en.wikipedia.org/wiki/Ordinal_arithmetic#Exponentiation and https://en.wikipedia.org/wiki/Cardinal_number#Cardinal_exponentiation.)

• In cardinal exponentiation, "$\kappa^\mu$" is the cardinality of the set of all functions from $\mu$ to $\kappa$.

• Ordinal exponentiation is defined inductively: $\alpha^0=1$, $\alpha^{\beta+1}=\alpha^\beta\cdot \alpha$, and $\alpha^\lambda=\sup_{\beta<\lambda}\alpha^\beta$ for $\lambda$ a limit ordinal. In particular, the ordinal exponentiation $2^\omega$ is just $\omega$ - which makes it very frustrating that the symbol "$2^\omega$" is, universally, used to refer to the set of functions from $\omega$ to 2!

In general, you have to read context carefully.

• Aren't all cardinals also ordinals, though? – mrp Oct 7 '15 at 19:10
• @mrp Yes, but cardinal exponentiation and ordinal exponentiation are two very different things. For clarity, write them as "$*$" and "$\otimes$" - the point is, even if $\kappa$ and $\lambda$ are cardinals, $\kappa*\lambda$ will be very different from $\kappa\otimes \lambda$. – Noah Schweber Oct 7 '15 at 19:21
• There is a similar issue with addition, but less troublesome – the sum of $\kappa$ and $\lambda$ as ordinals may not be the same as their sum as cardinals. – Zhen Lin Oct 7 '15 at 19:22
• A bit of pre-emptive clarification: It's less troublesome because cardinal addition is rarely something we care about, whereas both cardinal and ordinal exponentiation are important. – Noah Schweber Oct 7 '15 at 19:41
• @NoahSchweber Thanks for the clarification. – mrp Oct 7 '15 at 20:09