> "T.H. Ray" <email@example.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? > > -- > "But remember, as long as one human being follows the > rules of > mathematics, then mathematics as a human discipline > survives. > Right now I'm that one human being, so mathematics > survives." > -- James S. Harris
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.
The point is, geometrical constraint can be either analytically measured in limit points of a continuous function, or transcendentally counted and measured, but not both (as the OP would have it).