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