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Re: Proof 0.999... is not equal to one.
Posted:
Jun 1, 2007 2:47 PM
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On Jun 1, 12:56 am, hagman <goo...@von-eitzen.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 nine-tenths 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 one-to-one 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 one-to-one 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 one-to-one 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 one-to-one 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 one-to-one
> > 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 one-to-one correspondance of real number. [0, 1) and [0, 1] are not
> > mappable in one-to-one 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 one-to-one 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 one-to-one 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-
> > to-one 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 one-to-one 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 -
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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 re-reply later. I
agreed with all of your points up to U = T because U is not mappable
to T in a one-to-one 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 one-to-one 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 one-to-one mapping.
I will respond in full after class.
--charlie
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