Ordinals vs cardinals

Ordinals. Von Neumann ordinals: each ordinal is the set of all smaller ordinals. $\omega=\{0,1,2,\ldots\}$ is the first infinite ordinal; $\omega+1$, $\omega\cdot2$, $\omega^2$, $\omega^\omega$ are further ordinals.

Cardinals. Cardinals measure cardinality; $\aleph_0=|\omega|$, $\aleph_1$ is the first uncountable cardinal.

Distinction. $\omega$ and $\omega+1$ have the same cardinality ($\aleph_0$) but different order types.

Conclusion. Ordinals refine cardinals with order structure.