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


Re: Proof 0.999... is not equal to one.
Posted:
Jun 2, 2007 3:20 AM


On 1 Jun., 22:35, chaja...@mail.com wrote: > 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.
X is a set (of real numbers). Then with X I denote the set { x  x in X }. Finally \ denotes set difference, i.e. if A and B are sets then A\B = { x  x in A and not x in B }
> > > 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. >
OK, reread the post with the definition of set difference I gave above.
> > > > 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.
Then give *any* definition of sum(A) for as many subsets of the real numbers as possible. Here's a suggestion of things that should hold (or specify which you prefer to remove) 1) sum({x}) = x for any real x. 2) A and B disjoint, sum(A),sum(B) defined => sum(A u B) is defined and is sum(A)+sum(B) 3) A symmetric with respect to 0 (i.e x in A => x in A) => sum(A) is defined and is 0 4) sum(A),sum(B) defined, C a subset of the reals, f:C>A, g:C>B bijective mappings and for each c in C we have c=f(c)+g(c) => sum(C) is defined and is sum(A)+sum(B) 5) A={a_1,a_2,...} with a_1 < a_2 < ..., B={b_1,b_2,...} with b_1 < b_2 < ..., C=(a_1+b_1, a_2+b_2, ...}, sum(A), sum(B) defined => sum(C) is defined and is sum(A)+sum(B) 5) A subset B, sum(A),sum(B) defined => sum(B\A) is defined and is sum(B)sum(A) 6) sum(A) defined => sum(A\(A)) is defined and is sum(A)
> > > > > 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.
If you don't require them to be disjoint yuo should at least allow them to be disjoint. Note that I started with the condition "If A and B are disjoint". Since you disagree, do you think there are disjoint sets such that sum(A) and sum(B) are disjoint but sum(A u B) is either not defined or is different from 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. > > I'm still not sure what \ means. >
Granted.
> > > > 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. >
Sigh

