
Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 4:35 PM


On Jun 1, 2:31 am, hagman <goo...@voneitzen.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 nonzero 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

