Note Regarding Aleph-one equalling Aleph-C:
The validity of this equality is dependant upon the version of Set Theory being used and thus should not be stated as an unconditional truth in the article.
You inserted a reference to aleph_c, but from the context it is clear that you meant simply c, the cardinality of the continuum. The article already said, before you edited it, that equality of aleph_one and c does not follow from the Zermelo-Fraenkel axioms, even with the axiom of choice; there was no need for you to add that. You also inserted an error: that aleph-one is by definition the next cardinal after aleph-null. That follows from ZF with the axiom of choice, but need not hold in ZF without the axiom of choice. Michael Hardy 20:28, 29 Jul 2003 (UTC)
Though I accept most of these corrections, I believe your definitions of Aleph-1 and Aleph-2 (on Aleph) to be flawed. My professors and references have led me to believe that Aleph-1 is by definition the cardinality of the smallest set that cannot be placed in a one-to-one correspondense with the integers, and that Aleph-2 is by definition the next smallest. The question of whether Aleph-1 = c (ie 2^Aleph_0) is that of the continuum hypothesis and is not part of the definition of Aleph-1.
GulDan 16:33, 30 Jul 2003 (UTC)
The definition of aleph_1 that I gave is completely standard and is the one given by Cantor himself: aleph_1 is the cardinality of the set of all countable ordinals. Most mathematics professors do not know this area, and will admit as much if you get barely beyond the elementary matters you say you've asked them about. I suggest that you did not run past them the definition I gave and that they have probably never heard it or though about it. I also suggest you look in a textbook on the subject rather than some appendix in a calculus book or a theory-of-computability book or something like that. Paul Halmos' book Naive Set Theory will suffice, as will E. Kamke's Theory of Sets. Michael Hardy 21:14, 30 Jul 2003 (UTC)
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