I'm just starting to a little bit here and there about large cardinals. I have a basic question, namely, how do we know that every cardinality is realized by an ordinal?
One possible definition of an ordinal is a transitive set on which $\in$ is trichotomous. (I'm not sure which definition is most natural for non-well-founded set theories).
Cardinality can be defined in terms of the existence of injections (which gives us a relation $\le$). By picking the least ordinal with a given cardinality, we get the von Neumann cardinal assignment.
How do we know, however, that every cardinality has at least one ordinal that realizes it?
Also, are there any set theories where that is not true?