# [Haskell-cafe] Re: Theory about uncurried functions

Hans Aberg haberg at math.su.se
Thu Mar 5 08:58:08 EST 2009

```On 5 Mar 2009, at 13:29, Daniel Fischer wrote:

> In standard NBG set theory, it is easy to prove that card(P(N)) ==
> card(R).

No, it is an axiom: Cohen showed in 1963 (mentioned in Mendelson,
"Introduction to Mathematical Logic") that the continuum hypothesis
(CH) is independent of NBG+(AC)+(Axiom of Restriction), where AC is
the axiom of choice. Thus you can assume CH or its negation (which is
intuitively somewhat strange). AC is independent of NGB, so you can
assume it or its negation (also intuitively strange), though GHC
(generalized CH, for any cardinality) + NBG implies AC (result by
Sierpinski 1947 and Specker 1954). GHC says that for any set x, there
are no cardinalities between card x and card 2^x (the power-set
cardinality). Since card ω < card R by Cantors diagonal method, and
card R <= card 2^ω since R can be constructed out of binary sequences
(and since the interval [0, 1] and R can be shown having the same
cardinalities), GHC implies card R = card 2^ω. (Here, ω is a lower
case omega, denoting the first infinite ordinal.)

Hans Aberg

```