1
$\begingroup$

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?

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ Aren't all cardinals also ordinals, though? $\endgroup$ – mrp Oct 7 '15 at 19:10
  • 2
    $\begingroup$ @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 $\kappa\otimes \lambda$. $\endgroup$ – Noah Schweber Oct 7 '15 at 19:21
  • $\begingroup$ 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. $\endgroup$ – Zhen Lin Oct 7 '15 at 19:22
  • $\begingroup$ 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. $\endgroup$ – Noah Schweber Oct 7 '15 at 19:41
  • $\begingroup$ @NoahSchweber Thanks for the clarification. $\endgroup$ – mrp Oct 7 '15 at 20:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.