Re: Is 0.123... a member of the set {0, 0.1, 0.12, 0.123, ...}?
From: Don Whitehurst (whit0911_at_umn.edu)
Date: 02/06/05
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Date: 6 Feb 2005 11:44:50 -0800
Timothy Little wrote:
> Don Whitehurst wrote:
> > <snip>
> If we let a* = that irrational number 0.123...;
>
> More precisely, for any epsilon > 0 there exists a in A* such that
> |a - a*| < epsilon.
>
> So a* is a limit point of A* (in fact, it is the only limit point of
> A*). This does not mean that a* is in A*, since some sets of reals
> are not closed.
>
I appreciate your clarity on establishing the irrational number and
showing that a* is a limit point of A*. I am not sure how you
determined it is only the only limit point of A* or even what that
means.
>
> > The irrational number ( IR = 0.123... ) has the identical
> > representation as the element ( ELE* = 0.123... ) of Set B*.
>
> In my notation, B* = { a* }, and so B* is a subset of the closure of
> A*.
>
In the notation that I used, where Bn = { 0.123...n}, and as n=> oo, B*
= { 0.123...n... } = { 0.123... }, I let the element of Set B* be the
representation of the infinite approach of 0.123... to the irrational
you defined as a*.
>
> > The element of Set B* ( ELE* = 0.123... ), is the representation
of
> > infinite approach of 0.123... to the irrational number IR =
> > 0.123... since each and every element of Set Bn as n => oo is a
> > terminating rational decimal.
>
> You can define B* to be the set containing the limit of the sequence,
> sure.
>
As I indicated above, I did not define B* as the set containing the
limit of the sequence, rather I used the representation of ELE* =
0.123... to indicate that as n =>oo, each successive element in A* has
a corresponding element in B*. I had been taught there was a subtle
but important difference between approaching infinity and the concept
of actual infinity (if it exists).
>
> > Is the representation of the element of Set B* = ELE* = 0.123...
> > correct?
>
> Looks fine to me.
>
If you would prefer, I could use underlined numerals (if I knew how to
do that here) or some other convention (exclamation points) to
delineate the fundamental representaional difference between a* and
ELE*.
For example, a* = 0.123..., ELE* = 0.!123...!
>
> > Is the element of Set B* a number?
>
> Sure.
>
I suspect that if I explained myself clearly, you might now change your
answer.
When an infinite set contains an infinite sequence of points (quasi
linear) the convergence to a limit point that is a number makes sense
as you showed with A* and the limit point need not be a member of the
infinite set. But are you also then saying that when an infinite
grouping of sets each having singular points forms a progression, if
ther exists a 'limit set', the 'limit set' of this progression of sets
is a number that is not a member of the progression?
>
> > When two numbers have the same representation and the difference
> > between the two numbers is zero, are they the same number?
>
> Which two numbers?
The two numbers I meant were a* and ELE* .
> For any member a of A*, |a* - a| is nonzero. So
> no member of A* is a*. You constructed a sequence which converges to
> a*, but again no member of the sequence is a*, and a sequence is not
a
> number. The limit of the sequence is a number, and is a*.
>
I believe the progression of unitary point sets as n approaches
infinity that are represented by set B* and element ELE* either 1) do
not create a valid number as represented by ELE* (despite its
representation being identical to the irrational a*) or 2) since the
representation results from a progression the 'limit set' that looks
identical to a* is not part of A*.
>
> - Tim
I appreciate your response, it has allowed me to recognize that the
embodiment of decimalic numbers in my mind has always been based on
sequences. For example, I now realize that 1/3 = 0.333... has always
been interpreted by me to be the number resulting from the inifinte
application by inference of the long hand division of 1 by 3 where the
remainder is always 1. Similarly Pi, sqrt2, and all other irrationals
all seemed to arise from sequences and for irrationals there was no way
to predict the next digit of the decimal without calculating to that
number of digit placeholders; since no single formula using integers
and simple arithmetic functions could generate these numbers. I
recognize that mathematicians distiguish between the sequences and the
limits, but I still remain perplexed by what constitutes a real number.
Consider the set Amn = { 0, 0.1, 0.12, 0.123, ..., 0.123...m,
0.123...mn } where m = n-1
Consider also the set Amm = { 0, 0.1, 0.12, 0.123, ..., 0.123...m,
0.123...mm }
and the set Amp = {0, 0.1, 0.12, 0.123, ..., 0.123...m, 0.123...mp}
where p = n+1.
Thus for example at m = 5, n = 6, p = 7
Set Amn = { 0, 0.1, 0.12, ..., 0.12345, 0.123456 }
Set Amm = { 0, 0.1, 0.12, ..., 0.12345, 0.123455 }
Set Amp = { 0, 0.1, 0.12, ..., 0.12345, 0.123457 }
Consider Set Bmn = { 0.123...mn }
Consider Set Bmm = { 0.123...mm}
Consider Set Bmp = { 0.123...mp }
As n => oo one obtains:
Set Amn* = { 0, 0.1, 0.12, ..., 0.12...mn...} = { 0, 0.1, 0.12, ...,
0.12...n...} = A*
Set Amm* = { 0, 0.1, 0.12, ..., 0.12...mm...}
Set Amp* = { 0, 0.1, 0.12, ..., 0.12...mp..}
and
Set Bmn* = { 0.123...mn... }
Set Bmm* = { 0.123...mm... }
Set Bmp* = { 0.123...mp... }
Do all three infinite A sets (Amn*, Amm*, and Amp*) have real limit
points. If so, are they all the irrational number a* = 0.123...?
I believe Bmn* has the same decimalic representation as a* but again in
the way I defined it, Bmn* is the point representing the progression of
single point sets as n => oo. Consequently, I doubt whether the
element of Bmn* 0.123...mn... is strictly a number.
Both Bmm* and Bmp*, are sets each representing the infinite progression
of sets, as n=> oo, that have as a property - a numerical value that is
either one digit lower or higher (respectively) than the last decimal
place (in the finite last element) of the ordered set A* as n => oo.
Despite, being well defined for all finite values throughout the
infinite sets Amm* and Amp*, neither the infinite limit point
representation Bmm*, and Bmp* seems to be a real number.
Is this why mathematicians use limits to define points (numbers) and
avoid sequences?
Thanks,
Don
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