$(\beth_{\omega})^\omega=\beth_{\omega+1}$

Question

I'm trying to show that $(\beth_{\omega})^\omega=2^{\beth_\omega}$. This is an exercise in Kunen where he suggests to encode subsets of $\beth_\omega$ with functions from $\omega\rightarrow\beth_\omega$. Any help would be appreciated.

Thanks,

Cody

Answer 1

First, note that $(A\cap\beth_n)_{n<\omega}$ is a function $f_A:\omega\to\bigcup_n\mathcal P(\beth_n)$, and that the assignment $A\mapsto f_A$ is 1-1.

Then, note that $\mathcal P(\beth_n)$ is in bijection with $\beth_{n+1}$. Fix bijections for each $n$, and use them to replace $f_A$ into a function that takes ordinal values.