What's the largest number

Question

Originally this question started as 'what is the largest number' using $\aleph_0$ as a start, and continuing using concepts such as ${\aleph_0}^{\aleph_0}$, and Knuth's Tower notation $\uparrow$, so something like ${\aleph_0}^{\uparrow\uparrow\dots^{\aleph_0}}$, and I was wondering if anyone could come up with anything bigger?

Only in researching the problem I found a definition for $\aleph_{\omega}$, which states that for any $n, 2^{\aleph_0}=\aleph_n$, which doesn't make sense to me. In the previous section at Wikipedia - 'Continuum hypothesis' it is stated that $2^{\aleph_0}=\aleph_1$.

So:

1. What is the representation of the largest number possible?
2. What is the value of $2^{\aleph_0}$?

Answer 1

There is no largest "number" (i.e., cardinality) possible. Cantor showed that for any cardinal $\kappa$, $2^\kappa > \kappa$, where $2^\kappa$ is the power set of $\kappa$ (the set of all subsets of $\kappa$).

The value of $2^{\aleph_0}$ is not fixed by the usual axioms of set theory. That is, in standard set theory (Zermelo–Fraenkel with choice, ZFC for short) you cannot prove that $2^{\aleph_0} = \kappa$ for any cardinal $\kappa$. You also cannot prove that $2^{\aleph_0} \neq \aleph_n$ for any integer $n \geq 1$, that is, it could be the case that $2^{\aleph_0} = \aleph_n$. But you can prove that $2^{\aleph_0} \neq \aleph_\omega$, that is, this case can be ruled out. See the answer to this question for a complete description of the cardinals $\kappa$ that $2^{\aleph_0}$ could equal to (these are the cardinals with uncountable cofinality). For more elementary background, read on the continuum hypothesis in one of the many available sources.