Eliezer,

Very minor quibble/question. I assume you mean 2^Aleph_0 rather than Aleph_1. Unless one is doing something with the cardinals/ordinals themselves, it is almost always the numbers Aleph_0, 2^Aleph_0, 2^2^Aleph_0... that come up rather than Aleph_n. You may therefore like the convenient Beth numbers instead, where:

Beth_0 = Aleph_0 Beth_n+1 = 2^Beth_n