How many cardinals are there? I'm trying to do the following exercise: > EXERCISE 9(X): Is there a natural end to this process of forming new infinite cardinals? We recommend this exercise instead of counting sheep when you have trouble falling asleep. (This is from W. Just and M. Weese, _Discovering Modern Set Theory, vol.1_ , p.34.) By _this process_ they mean $|\mathbb N| < |\mathcal P(\mathbb N)| < |\mathcal P (\mathcal P (\mathbb N))| < \dots$. My first response to this was "Obviously there is no end to it." but then the exercise is supposed to be challenging ("X-rated") so this must be wrong and there is an end to it. But when exactly? How many cardinals are there? What would be a "natural end"? Thank you for your help!

I don’t know what they have in mind, but the process as described goes only $\omega$ steps. To go further, you have take the union of what you already have and start over. That is, the process of repeatedly taking the power set will get you $\beth_0,\beth_1,\beth_2,\dots$ and hence $\beth_n$ for each $n\in\omega$, but it won’t get you beyond those. If you take the union of all those power sets, you get something whose cardinality is $\beth_\omega$, and you can start powering up again.