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Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 4:35 PM
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On Jun 1, 2:31 am, hagman <goo...@von-eitzen.de> wrote:
> On 1 Jun., 08:55, chaja...@mail.com wrote:
>
>
>
>
>
> > > If you insist, here is the first obvious mistake in the paper:
>
> > > -------------------------------------------
> > > Let S be the set of all real numbers in the interval [0,1]
> > > Let T be the set of all real numbers in the interval (-1,0]
>
> > > If you applied an operator '+' to these two sets that sums the elements
> > > of both sets, you would get: S + T = 1
> > > -------------------------------------------
>
> > > You are trying to sum an uncountable set of numbers. Please define what
> > > this means. Also consider: if you pair the numbers differently, can you
> > > make the sum come out to something else?
>
> > > FInally, I didn't notice anything looking like a proof in your article,
> > > though I admit I haven't read all of it.
>
> > > --
> > > Eric Schmidt
>
> > > --
> > > Posted via a free Usenet account fromhttp://www.teranews.com
>
> > Hi Eric,
>
> > what I am attempting to point out in the S + T result is that any
> > element of the infinite set T corresponds to exactly one element in
> > the infinite set S of equal value and opposite sign, except for the
> > positive 1 in set S. Should you hold the infinity of values in your
> > awareness, all cancel except for one that cannot.
>
> > You would not be able to have a final result upon summing the entire
> > infinity of elements in S union T other than 1.
>
> > --charlie
>
> Oh, so you want to define sum(X) for certain subsets X of the reals?
> This should be easy if X is finite: just sum the elements (fortunately
> addition is associative and commutative).
> It should also be easy to define sum(X) if sum(Y) is defined
> where Y = X\(-X): just set sum(X)=sum(Y).
I don't understand what the backslash means.
> One should investigate any further definitions as to whether all that
> stuff yields a consistent definition of sum.
>
> Your example amounts to obtaining sum( (-1,1] ).
> With X = (-1,1], I would calculate Y = X\(-X) = {1} and obtain
> sum(X)=sum(Y)=1.
I would agree sum(X) = 1, I don't understand the construction of Y to
say.
>
> Nothing has been said yet about sum(S) if S\(-S) is infinite, esp.
> this has no relevance yet if S is the set of all rationals of the form
> 9/10^n.
>
> However, we can form the set union of -S and 10 times S, i.e.
> let C = (-S) u (10*S).
> It turns out that C\(-C) is {9}.
I don't believe this would be accurate. If C were defined as (-S) u (S
+ {9}) I would agree.
> We are forced to conclude sum(C)=9.
>
> If t is a non-zero number and sum(S) is defined, can sum(t*S) be
> anything
> but defined and equal to t*sum(S)?
I wouldn't think so.
>
> If A and B are disjoint and both sum(A) and sum(B) are defined,
> can sum(A u B) be anything but defined and equal to sum(A)+sum(B) ?
>
I wouldn't think so, and I would tend to use a multiset when doing
this kind of thing, so I wouldn't require they be disjoint.
> In fact, the last two issues are what motivates the way to
> calculate sum(X) from sum( X\(-X) ) used above and introduced by you.
I'm still not sure what \ means.
>
> Thus sum(-S) = -sum(S), sum(10*S)=10 *sum(S)
Agreed
> and finally
> sum(-S u 10*S) = 9*sum(S).
It should
> Hence sum(S) = sum(C)/9 = 1.
I disagree sum(C) = 9. I find it strictly less.
>
> hagman- Hide quoted text -
>
> - Show quoted text -
--charlie
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