The empty set is both an ordinal and a cardinal. The ordinals are certain sets that are well-ordered (in von Neumann's version of the ordinals). Every well-ordered set is order-isomorphic to an ordinal. The empty set is well ordered (by epsilon, i.e., by the "element of" relation). Definition: A set is *well-ordered* if every non-empty subset has a smallest element.

The natural numbers (0, 1, 2, ...) are both ordinals and cardinals. There are lots of infinite ordinals that are not cardinals.

There are probably various treatments of this on the Web. When I took set theory in graduate school, we used "Set Theory" by Kenneth Kunen as the textbook. Kunen is a well-known text. He covers ordinals and cardinals in the first chapter. For a less formal treatment, "Naive Set Theory" by Paul Halmos is nice. Here are some of the ordinals that Halmos lists (I'll write "w" for omega): 0, 1, 2, ..., w, w+1, w+2, ..., w2, w2+1, w2+2, ..., w3, ..., w4, ..., w^2, w^2+1, ..., w^2+w, w^2+w+1, ..., w^2+w2, ... w^3, ..., w^w, ..., w^(w^w), ..., w^(w^(w^w)), ... He lists some more, but that should give you an idea.

The Romans didn't have zero. It took a long time for people to realize that having zero as a number was useful.