TITLE: What are Aleph numbers intuitively?
QUESTION [13 upvotes]: I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?

REPLY [4 votes]: The cardinal number $\aleph_1$ is defined as the cardinality of the set of all countable ordinals, i.e. ordinals of cardinality $\le\aleph_0$.  If you believe that that set of countable ordinals exists, then you've got it.