Every well-ordered set is isomorphic to a unique ordinal. The example you gave is a well-ordered set isomorphic to the ordinal

. The ordinal

is not a cardinal. It's cardinality is

or as you probably prefer to call it

. So

because these are nonisomorphic well-ordered sets. But the cardinality of

is

because there is a bijection between

and

.

From the set theoretic point of view every cardinal is an ordinal. A cardinal is just an ordinal that is not equinumerous with any smaller ordinal (A and B are equinumerous if there is a bijection between them).

Given an arbitrary set, assuming the axiom of choice, this set has a unique cardinality. The cardinality of the set is a cardinal (thus it is also an ordinal). Note that to "see" cardinality you just need a bijection, whereas to see which ordinal you are isomorphic to requires an

*order-preserving* bijection.

As one more example (to try to help clarify my use of notation), every countable ordinal (and in fact every countable set) has cardinality

, but only the ordinal

is actually equal to

. thus it is mathematically correct to use

in place of

. The difference between them is only the way you think about them. People tend to think about

as the size of the natural numbers, and as

as the ordering of the natural numbers, but in fact they are the same (at least when using the standard set-theoretical definitions).

Note: I realize that you probably understand most of this already - I'm just trying to give you enough information to justify my use of notation.