TITLE: How many cardinals are there?
QUESTION [8 upvotes]: 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!

REPLY [4 votes]: This is a matter of philosophical approach to what it means "a natural end". After all, there is no natural end to the generation of natural numbers -- but certainly they end somewhere because we know that most ordinals are uncountable.
On the same note generation of cardinals does not end as long as you have ordinals to iterate through (either in the power set operation; or in the Hartogs number operation; or limits of these constructions). In this sense there is no natural end to the process of generating new cardinals.
However if one thinks of operations performed along the class of ordinals as operations which terminate when you "run out of ordinals", and if you can run out of natural numbers there is no reason you cannot run out of ordinals as well, in such case there is in fact a natural ending to the generation of new sets and therefore of new cardinals.