So you can define the following,: = A cardinal is an ordinal κ such that κ is not bijective with any ordinal α < κ. How do we even know that this exists? One could start with the cumulative hierarchy, up to omega then iterate, omega + 1 etc but then by the limit we will get omega + omega which is isomorphic to omega. How do we ever reach omega 1, the first infinite cardinal that is not bijective with aleph0 as in your comment? Using the the help set {(n,m)∈ω×ω : 2n+13m+1 ∈X} we can show the existence of this cardinal. It then turns out that the successor of each cardinal is bigger than the last.
How do you know \aleph_0 \leq \kappa in the first place?
So you can define the following,: = A cardinal is an ordinal κ such that κ is not bijective with any ordinal α < κ. How do we even know that this exists? One could start with the cumulative hierarchy, up to omega then iterate, omega + 1 etc but then by the limit we will get omega + omega which is isomorphic to omega. How do we ever reach omega 1, the first infinite cardinal that is not bijective with aleph0 as in your comment? Using the the help set {(n,m)∈ω×ω : 2n+13m+1 ∈X} we can show the existence of this cardinal. It then turns out that the successor of each cardinal is bigger than the last.