Infinite Numbers

We use numbers for two purposes: ordering things (first, second, third, ...) and determining quantity (half a dozen). The first kind of number is called an ordinal number and the second kind is called a cardinal number. That seems a bit silly for finite numbers, because the two kinds are in one-to-one correspondence, but for infinite numbers they come apart.

Ordinal numbers:

$\[0,1,2,\dots ,\omega ,\omega +1,\omega +2,\dots ,\omega +\omega ,\dots \omega +\omega +\omega ,\dots , 4\omega ,\dots ,\omega \omega, \omega ^{\omega}\]$

There are "concrete" examples of these orders: $\[0,\frac{1}{2},\frac{3}{4},\dots ,1\]$ has order type $\omega +1$ .

Cardinal Numbers

Two collections have the same quantity of stuff in them if you can place them into one-to-one correspondence, pair them off (equinumerous). The order type $\omega$ is less than the order type $\omega +1$ as ordinals, but they obviously have the same quantity of elements: we can pair them off by sticking the 1 on the front instead of at the end: $\[0,1,2,\dots \]$ $\[\omega ,0,1,2,\dots \]$

That probably makes it look like there is only one infinite cardinality, but in fact for every infinite cardinality or sequence of cardinalities, there is a greater one: For any set, the set of all of its subsets is bigger.

We have cardinal numbers $\aleph _{0},\aleph _{1},\dots ,\aleph_{\omega},\dots$

-- ProfessorShaughanLavine? - 11 Feb 2005