Are there cardinalities beyond Aleph 2?

Question

I'm familiar with Aleph Null, Aleph 1, and Aleph 2 but are there greater [uncountable] infinities beyond (examples) Natural numbers, Real numbers, and the number of curves that can pass through a point? If so, what are some tangible examples of these cardinalities: Aleph 3, 4, ...

Answer 1

The next one after $\aleph_2$ is $\aleph_3$. There's also $c$, the cardinality of the reals. In fact $c=\aleph_\alpha$ for some $\alpha$, but nobody knows which one...

Regading your new request for "tangible examples": In fact $\aleph_0$ is the cardinality of the integers, which I gather is what you mean by a tangible example. But no, $\aleph_1$ is not the carinality of the reals.

So what are these aleph things? One can prove that given a cardinal there is a smallest larger cardinal. So $\aleph_1$ is by definition the smmallest uncountale cardinal, $\aleph_2$ is by definition the smallest cardinal larger than $\aleph_1$, etc. That's as "tangible" as it gets.

Finally, regarding "Natural numbers, Real numbers, and the number of curves that can pass through a point?": Although you didn't actually say so, this sounds like you think the cardinality of that set of curves is larger then the reals. This is not so. If a "curve" is the graph of a continuous function, then the cardinality of the set of all curves, passing through a given point or not, is just $c$.

Answer 2

Yes. Just keep taking power sets of power sets ad infinitum. There is no largest cardinal.

Answer 3

To conceptualize larger cardinals, you can use power sets, for sure. If you want something larger than the cardinality of curves passing through a point, consider subsets of the set curves passing through a point.

$$|\mathcal{P}(A)|>|A|$$

Another way of looking at this approach is to consider the set of functions from a set of one cardinality to, for example, the set $\{0,1\}$. This set is equivalent to the power set of your original set, and its cardinality is greater. In symbols: $$|\left\{f|f:A\to\{0,1\}\right\}|>|A|$$