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


Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 3:56 AM


On 1 Jun., 05:34, chaja...@mail.com wrote: > > Near the bottom of page 1 you write: > > 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 have never defined this operator '+' for intervals. > > The standard use (derived from the '+' for numbers) is to take the set > > of all possible sums x+y where x is in S and y is in T. > > However, this would result in S + T = (1,1]. > > The operator is not applied to the intervals. It is applied to two > sets whose elements are all real numbers contained within these > intervals.
OK, then you have not defined the operator '+' for sets. And you should have seen that I identified the intervals with the set of real numbers contained in these intervals. Do you se any ontological difference between an interval of real numbers ond the set of real numbers contained in that interval?
> > To save my own sanity, I scrolled ahead until I found this example > > of downright nonsense: > > > However, defining it as a limit is all but directly saying that it > > may > > not be exactly equal to one. > > I'm surprised you see this as nonsense. A limit is a bound, that which > can be infinitely approached. A limit does not indicate that this > limit can be reached, only that any real number between this limit and > our entity will be exceeded. > > It's the classic story of crossing half a distance, then half the > remaining distance, on and on. The limit is 1. But you can never reach > the other side, even should you attempt to do so infinitely. I'm > always boggled why people say that a journey taken in ninetenths the > remaining distance would in fact reach 1...
Classic story as in Archilles and the Tortoise, that is.
> > [0, 1], [0.9, 1], [0.99, 1], ..., [0.999..., 1] > > > But what about the closed set [1]? How could we have felt sure > > 0.999... > > would be in this set as well? The nested set theorem only guaranteed > > you'd get to a single value. > > > Indeed, you'd get only a single value AND you'd get 0.999... AND you'd > > get 1. This implies 0.999...=1. > > You can only be sure that 0.999... will be in the final nested set if > you define 0.999... as a limit, which again, I do not do.
Do you define 0.999... at all?
> Recall, that > 0.999... is strictly equal to 1 is taught to students everyday outside > of a concept of limits. This brings up a natural reservation that they > are being taught a untruth. Within the definition of limit, I do not > argue it yields 1.
So if you want to give any interpretation where 0.999...=1 does not hold, you should give a definition of the meaning of 0.999...
> > > Thus one can hope to define sum(S) and in fact whatever you claim > > 0.999... to be is what you define as sum in this case. > > > If r is a positive rational number, one can define the set T > > containing > > all elements of S multiplied by r, i.e. the numbers r*9/10^n. > > > It should definitely be true that, whenever either sum(S) or sum(T) > > is defined so is the other and the equality > > sum(T) = r * sum(S) > > holds, shouldn't it? > > Agreed. > > > > > If s is a rational number not contained in S, we can define the set > > U = S u {s} containing all elements of S and also s. > > It should definitely be true that, whenever either sum(S) or sum(U) > > is defined so is the other and the equality > > sum(U) = s + sum(S) > > holds, shouldn't it? > > Agreed. > > > With your set S defined as above, I choose r=10 to form T and I choose > > s=9 > > to form U. > > If you look at it closely, you will note that T=U holds. > > It should definitely be true that the equality > > sum(U) = sum(T) > > holds in this case, shouldn't it? > > I disagree. Your claim is that 0.999... * 10 = 9.999...
No, my first claim is T = U My second claim is T=U => sum(T)=sum(U)
I assume you disagree with the first claim. This implies that you disagree with the following argument at some place. Please indicate where:
1) Two sets are equal if and only if they contain exactly the same elements.
2) x in T if and only if x is a rational number such that x/10 is in S.
3) x in T if and only if x/10 is a rational number of the form 9/10^n for some integer n >= 1.
4) x in T if and only if x is a rational number of the form 90/10^n for some integer n >= 1.
5) x in T if and only if x is a rational number of the form 9/10^n for some integer n >= 0.
6) x in U if and only if either x=9 or x is in S.
7) x in U if and only if either x=9/10^0 or x is a rational number of the form 9/10^n for some integer n>=1.
8) x in U if and only if x is a rational number of the form 9/10^n for some integer n>=0.
9) x in T if and only if x in U.
10) T=U.
11) sum(T)=sum(U) if either sum is defined.
Note that 11) does not claim that the sums are real numbers, only that they are equal.
> > S is an infinite set. Should you lay two copies of S upon itself they > are onetoone mappable  you cannot point to an element in one copy > that does not directly correspond to an element in the other copy. > Should you point to all elements within one copy, no element in the > other does not correspond in value to something you have pointed out > in the first. They are infinitely pairable. > > T is also onetoone mappable with S. If you lay S upon T, there is no > element that in one set does not directly correspond to an element of > the other. Should you point to all elements at once within one, there > is no element within the other that does not correspond to all that > you pointed out in the other. Indeed the very elements of T are > generated one per each element in S. Should you point out all of S, no > T exists that is not ten times something you have pointed out. > > U is not infinitely mappable in a onetoone corresponance with either > S or T.
This is even stronger nonsense than saying U and T are different. Now you claim these sets have different cardinality. Have you ever noted that n>n^2 is a onetoone correspondence between the set N of natural numbers and a proper subset of N, namely the set of perfect squares? The correspondence you describe between T and S is of the same kind as S is a proper subset of T.
>Should you remove s from U, it becomes S and is onetoone > mappable with either S or T. In this respect, U has more in it than S > or T. We could remove elements in pairs, one from U and one from S or > T, and have one left over, not having omitted any other element from S > or T that we attempted to map with U.
> > Note that both intervals [0, 1) and [0, 1) are infinite, and mappable > in onetoone correspondance of real number. [0, 1) and [0, 1] are not > mappable in onetoone correspondance in infinite completion. If you > attempted to map all values from one onto values in the other of equal > value, the 1 would be left unpaired.
Do you consider only correspondences of equal value?? With that restriction, you could not have a onetoone correspondence between S and T either!
> > For Sum(U) to equal Sum(T), we would need to be able to remove equal > elements in a onetoone mapping, thus yielding 0. Should we remove s, > or 9, from U, the result (or S) is directly mappable in a one to one > correspondace with T as expressed earlier. The correspondance is not > in value. It is of an element of S with ten times this element in T. > But one exists for each.
I may discuss that once we are clearer about your opinion about whether or not U=T.
> > If we remove 9 from both U and T, the resulting sets are still not one > toone mappable. There will always remain an unpairable element in U > with the rest directly mappable. Should we remove a count of elements > approaching infinite from both sets in pairs of equal value, still U > lest these elements will not be directly mappable to T lest these > elements. > > To be able to assert that U is equal to T, we would need to be able to > map all values in onetoone correspondance. Traversing their elements > from left to right, this appears possible, but we know that U will > always have some aspect that is left unpairable. This makes U larger, > for all else lest some nature can be removed in pairs. > > This concept is difficult to relate as our ideas of infinity have not > been honed and trained to the extent that our arithmetic has been.
Pluralis maiestatis?
> > 9 + 0.999... = 9.999... > 9.999... > 10*0.999... > > charlie > > > > > But then > > r * sum(S) = sum(T) = sum(U) = s + sum(S) > > i.e. > > 10 * sum(S) = 9 + sum(S). > > > In my math, this can be transformed to > > sum(S) = 1. > > Your mileage my vary, but then you have not given > > adequate definitions of '+', '*', '='. > > > hagman

