# Aleph null !

How might one show that (aleph_null)! = aleph_1?

HallsofIvy
Homework Helper
One might start by defining the "factorial function" for aleph_null!

Ok, so the person who proposed this problem to me gave me a way to understand (aleph_null)!.

So, consider two sets, A and B. Then |A|*|B|=|A x B|, where AxB is the cartesian product of A and B.

Thus, consider N_m={1,2,3,...m}, and |N_m|=m.

Then (aleph_null)! = |N_1 x N_2 x N_3 x ...... |.

So how can I find a bijection from N_1 x N_2 x N_3 x ...... to, say, P(N), the power set of the naturals?

CRGreathouse
This shows that the set has cardinality $\beth_1$, not $\aleph_1$ unless you have the CH.