 Thus, just as there is an infinite hierarchy of infinite ordinal numbers, which Cantor represented with lowercase Greek letters, there is also an infinite hierarchy of infinite cardinal numbers, which Cantor represented with Hebrew letters, and in particular aleph, “\(\aleph\)”. The finite cardinals are 0, 1, 2, 3, ….  The first infinite cardinal, that of the natural numbers (and all countable sets), is \(\aleph_0\). Cantor’s “well-ordering principle”, stating that every set can be put into some well-ordered form (and equivalent to the Axiom of Choice), implies that every cardinal number can be represented by an ordinal number, and the definition of ordinal numbers ensures that for any non-empty set of ordinal numbers there is always a first. So the cardinals must in fact be well-ordered. So Cantor used ordinal numbers to designate each cardinal’s position in their ordering. \(\aleph_1\) is the first cardinal beyond \(\aleph_0\), \(\aleph_2\) is the next beyond that, and after \(\aleph_3\), \(\aleph_4\), and so on, we eventually reach \(\aleph_\omega\), \(\aleph_{\omega+1}\), \(\aleph_{\omega+2}\), and so on with one cardinal for every ordinal.