T.H. Ray
Posts:
1,107
Registered:
12/13/04
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Re: Proof 0.999... is not equal to one.
Posted:
May 31, 2007 1:38 PM
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> "T.H. Ray" <thray123@aol.com> writes:
>
> >> "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
> ant, who died an unmarried virgin
No, of course not. Again, you need to read everything
in context of the claims. There is much more to
Cantor's theory than this simple proposition. The OP's
claim depends on a geometrical interpretation of the
difference between 0.999... and unity. If the CH
is true, no such interpretation is possible. If the
CH is not used, the OP's terms are not differentiable.
The CH was used in my explanation to illustrate the
rules of geometrical constraint, betweeness as Weyl
referred to it. Cantor's theory teaches us that there
is no abstract betweenness, and real analysis teaches
us that limit functions produce real results. Then the
OP's proposition has no harbor--we are either talking
about measured real results, or not. Any proof of his
claim would have to incorporate measured real results,
and when attempted would necessarily show the
equivalence of his terms; i.e. things that are not
differentiable are identical.
Tom
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