hagman
Posts:
1,923
Registered:
1/29/05


Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 5:31 AM


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). 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.
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}. 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)?
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) ?
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.
Thus sum(S) = sum(S), sum(10*S)=10 *sum(S) and finally sum(S u 10*S) = 9*sum(S). Hence sum(S) = sum(C)/9 = 1.
hagman

