By N, dkarapetyan presumably means the set of natural numbers, or aleph_0. So it isn't aleph_1. That said, it is also true that aleph_1 x aleph_1 = aleph_1. More generally, any aleph_alpha x aleph_alpha = aleph_alpha, for any ordinal alpha; and if we assume the axiom of choice, then any infinite cardinal A can be written as some aleph_alpha, and hence satisfies A x A = A.
However, ultimately none of this is relevant; number of steps is properly measured with ordinals, not cardinals.
However, ultimately none of this is relevant; number of steps is properly measured with ordinals, not cardinals.