Am Donnerstag, 5. März 2009 14:58 schrieb Hans Aberg: > 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.
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. > 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 _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe