Daniel Fischer daniel.is.fischer at web.de
Thu Mar 5 09:12:11 EST 2009

```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

```