Can we raise aleph cardinals to other aleph cardinals?


What would aleph0 ^ aleph0 be?
What would aleph1 ^ aleph1 be?

What would cardinality of P^k(N) be?

[That is taking power sets k times on naturals. If k=1, then it is
P(N) = 2^aleph0

c^alpeh_i is easier since c = 2^aleph_0, therefore
c^alpeh_i = 2^aleph_i*aleph0 = 2^aleph_i

Can anyone give me a set with cardinality aleph_1?

No derivations required.
.

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