
Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 2:47 PM


On Jun 1, 12:56 am, hagman <goo...@voneitzen.de> wrote: > 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 Hide quoted text  > >  Show quoted text  Hide quoted text  > >  Show quoted text  Hide quoted text  > >  Show quoted text  Hide quoted text  > >  Show quoted text  Hide quoted text  > >  Show quoted text  Hide quoted text  > >  Show quoted text 
I did reply to this in full and posted it, but it seems to have gotten lost in transmission. I have class soon so will rereply later. I agreed with all of your points up to U = T because U is not mappable to T in a onetoone correpondace. T is mappable to S in a one to one corresponce as T is simple all elements of S times 10. U is every element in S + {s}, so U is no longer onetoone mappable to S or T.
If you had a set of all integers >= 0 and a set of a all integers >= 0 squared, I would fully 100% agree there is a onetoone mapping.
I will respond in full after class.
charlie

