Algon answers What are the best elementary math problems you know?

[Algon]

Let $\alpha$ be an ordinal s.t. $Card\left(\alpha \right)>Card\left(R\right)$. Let ${\alpha }^{\omega }$ be the cardinality of functions from $\omega$ to $\alpha$ . Is $Card\left(\alpha \right)={\alpha }^{\omega }$ ? If so, prove it.

[Ege Erdil]

Nice question. I’m providing my solution below, spoiler protected so people can attempt to solve the problem on their own.

By König’s theorem

$${\aleph }_{\omega }=\sum _{n\in \omega }{\aleph }_n<\prod _{n\in \omega }{\aleph }_{\omega }={\aleph }_{\omega }^{\omega }$$

so the answer is no. This is also the smallest counterexample that ZFC should be able to decide, since GCH is undecided in ZFC and if GCH is true then ${\aleph }_{\alpha }$ will have a base two logarithm for any successor ordinal $\alpha$ and any infinite $\kappa$ that has a base two logarithm will satisfy ${\kappa }^{\omega }=\kappa$.