[Haskell-cafe] Re: Theory about uncurried functions

[Daniel Fischer]

Am Donnerstag, 5. März 2009 15:12 schrieb Daniel Fischer:

> Yes, but the continuum hypothesis is 2^Aleph_0 == Aleph_1, which is quite
> something different from 2^Aleph_0 == card(R).
>
> You can show the latter easily with the Cantor-Bernstein theorem,
> independent of CH or AC.

Just to flesh this up a bit:

let f : P(N) -> R be given by f(M) = sum [2*3^(-k) | k <- M ]
f is easily seen to be injective.

define g : (0,1) -> P(N) by 
let x = sum [a_k*2^(-k) | k in N (\{0}), a_k in {0,1}, infinitely many a_k = 
1]
and then g(x) = {k | a_k = 1}

again clearly g is an injection.
Now the Cantor-Bernstein theorem asserts there is a bijection between the two 
sets.