Is $\aleph_1= \aleph_0+1 $ wrong?

Question

My understanding is that any cardinality is always an integer because it expresses how many elements are in a given set. And I read that $\aleph_1$ is the next smallest cardinality that's larger than $\aleph_0$, so it seems to me that $\aleph_1 = \aleph_0+1$. However I don't see this equation anywhere on the Internet so I guess I'm wrong. Where am I wrong?

Answer 1

The wrong part is that you took what you know about finite cardinals and extend it to infinite ones.

Assuming choice $\aleph_0$ is the smallest cardinal that is greater than all the finite cardinal, the first limit ordinal. To advance to $\aleph_1$ we need to find a set such that there is no injective from $A$ to $\Bbb N$, but $\aleph_0+1=|\Bbb N\cup\{0\}|=|\Bbb N|=\aleph_0$.

Even more: for $\kappa,\nu$ cardinals such that one of them is infinite we have $\kappa+\nu=\kappa\cdot\nu=\max\{\kappa,\nu\}$