Originally Posted by

**DrSteve** Every well-ordered set is isomorphic to a unique ordinal. The example you gave is a well-ordered set isomorphic to the ordinal $\displaystyle \omega +1$. The ordinal $\displaystyle \omega + 1$ is not a cardinal. It's cardinality is $\displaystyle \omega$ or as you probably prefer to call it $\displaystyle \aleph_0$. So $\displaystyle \omega +1 \ne \omega$ because these are nonisomorphic well-ordered sets. But the cardinality of $\displaystyle \omega + 1$ is $\displaystyle \omega$ because there is a bijection between $\displaystyle \omega +1$ and $\displaystyle \omega$.

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 $\displaystyle \aleph_0$, but only the ordinal $\displaystyle \omega$ is actually equal to $\displaystyle \aleph_0$. thus it is mathematically correct to use $\displaystyle \omega$ in place of $\displaystyle \aleph_0$. The difference between them is only the way you think about them. People tend to think about $\displaystyle \aleph_0$ as the size of the natural numbers, and as $\displaystyle \omega$ 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.