Cardinalities, aleph, and beth

In <http://www.ii.com/math/ch/>, Appendix 2, you quote Gamow as saying "there are aleph_2 different curves."  You correctly point out that he's saying aleph_2 when he really means 2^2^aleph_0, but you missed a different mistake: the number of curves is actually only 2^aleph_0, i.e., equal to the number of points, not greater.  A correct statement would be "there are 2^2^aleph_0 different functions from R to R."  (Most of those functions are discontinuous, and hence their graphs are not curves.)

Incidentally, this first confusion -- using aleph_1 to mean the continuum and aleph_2 to mean its power set -- would be reduced, I believe, if the beth notation were more widely known and used.  I was never formally taught this notation, but my understanding is that beth_0 = aleph_0, beth_(k+1) is 2^beth_k, and for a limit ordinal l, beth_l = sup { beth_k | k < l }.  Then CH can be stated "beth_1 = aleph_1", and GCH can be stated "beth_k = aleph_k for all ordinals k."

Karl Heuer