>> "T.H. Ray" <thray123@aol.com> writes: >> >> > Yes, we know (by the Continuum Hypothesis, Cantor) >> that >> > betweeness is infinite--not just "huge." Suppose >> > we reject the CH? Every sequence is then finite >> > and differs from another sequence that infinitely >> > approaches, but does not reach it, by an >> infinitesimal >> > margin. (This is the insight, of course, that led >> > to the development of analysis--the study of >> continuous> > functions.) >> >> Sorry, but I have no idea what you are trying to >> express here. How is >> the Continuum Hypothesis relevant here?
[...]
> > Look at the context of the OP's original claim, along > with the rest of my explanation. > > We can do without the CH and have finite betweeness > (Weyl's term)or we can employ the CH and have infinite > betweeness. But we cannot have terms that are > simultaneously infinitely between, and finite, > between the same pair of integer terms (in the OP's > case, between 0 and 1). Our sequences > have to be either analytical with limits > (in which case 1.000... = 0.999...) or rational > numbers with terminating point. These are differentiable. > The OP allows no means to differentiate a finite sequence > from an infinite series. Both the analysis of continuous > functions, and Cantor's theory, do incorporate such means.
Again, I don't see how the hypothesis that
2^{aleph_0} = aleph_1
is relevant to any of these claims. Are you suggesting that if
2^{aleph_0} > aleph_1
then there would be finitely many reals between 0 and 1?
-- Jesse F. Hughes "Marriage.. ..is the union of two persons of different sex for life-long reciprocal possession of their sexual faculties" -- Immanuel Kant, who died an unmarried virgin
