TITLE: What is the difference between cardinals and alephs?
QUESTION [13 upvotes]: It is stated that by Zermelo’s theorem, every cardinal is an aleph. But what is the difference between cardinals and alephs? I thought that alephs were just a way to denote cardinals (just for notation), but then this theory doesn't make sense.

REPLY [11 votes]: Well. The finite ordinals are also cardinals, but they are not $\aleph$ numbers.
But without the axiom of choice, there can be sets which cannot be well-ordered, and therefore their cardinality cannot be "measured" using $\aleph$ numbers. In that case we fallback to Scott cardinals, which allow us to define cardinals even in the absence of choice, for any set.