How many integer sequences?

[Warut Roonguthai]

On Fri, 22 Oct 1999, Jud McCranie wrote:

> How many integer sequences are there in terms of Alephs?  Aleph-1?

I'm also interested to know the solution to this question.  Instead of
just waiting for replies from experts, I'd like to present my own answer
and let them correct me if I'm wrong.

The set you mentioned is the collection of all subsets of Z.  So its
cardinality is 2^Aleph_0 = c (cardinality of continuum) > Aleph_0.  Your
question about the order of this transfinite cardinal is equivalent to
asking for resolving the continuum hypothesis which is undeciable under
the ZFC models.