\aleph_0 is the first countable infinity because the alephs (like the beth numbers, and the levels of the cumulative hierarchy) are indexed by ordinals, which s

\aleph_0 is the first countable infinity because the alephs (like the beth numbers, and the levels of the cumulative hierarchy) are indexed by ordinals, which start from zero, i.e. the empty set in the usual constructions of von Neumann and Zermelo. However, whether we start from 0 or 1 is ultimately arbitrary and more a question of symbol choice than semantics, since the sequence of natural numbers beginning with 1 is isomorphic to the sequence of natural numbers beginning with 0. Set-theoretically it makes a little more sense to start with 0, but when considering arithmetic alone, it really makes no difference.