User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-25418284-20140509201345/@comment-1605058-20140509205528

Think about this way: let's start with something small, like Aleph_0. From this cardinal, we can easily go to the next one - Aleph_1. We can keep taking successors, getting Aleph_n for all finite n. But we reached all of these ordinals with no effort! We just kept applying simple "next largest" rule. So we can think of these as "small" cardinals. Now, to break out of it, to get above Aleph_n, we might want to sum up all these ordinals. There is Aleph_0 of these ordinals, and summing small number of small things should result in small thing. So Aleph_w, which we get, can be thought of as a "small" cardinal too. We can go further - Aleph_w+1, Aleph_w2, Aleph_w^2, Aleph_e_0, Aleph_CK, even Aleph_Aleph_1, Aleph_Aleph_2, Aleph_Aleph_Aleph_1 and so on, if we define "small" cardinals using this rule, that we can take successors and "small" sums, we get vast amount of numbers. But look at the definition of weakly inaccessible: it's a limit cardinal, so we can't take it as a successor, and it's also regular, which means this: if there was a "small" sum resulting in inaccessible I, then by regularity this sum would have I summands. So I is small only because I is small, which is a circular definition. This also means we can't reach it from below (using smaller cardinals). So we shouldn't think of this cardinal as small, as we can't construct it, really. So this cardinal has to be large.