Re: INFINITY Revisited
- From: "Don Whitehurst" <whit0911@xxxxxxx>
- Date: 2 Sep 2005 17:42:10 -0700
stephen@xxxxxxxxxx wrote:
> Don Whitehurst <whit0911@xxxxxxx> wrote:
> > stephen@xxxxxxxxxx wrote:
> >> Don Whitehurst <whit0911@xxxxxxx> wrote:
> >> > stephen@xxxxxxxxxx wrote:
> >> >> Don Whitehurst <whit0911@xxxxxxx> wrote:
> >> >> > stephen@xxxxxxxxxx wrote:
> >> >> >> pi cannot be the last element because there is no last element.
> >> >> >> The set of naturals are infinite and so there is no last
> >> >> >> natural number to map to pi.
> >> >> >>
> >> >>
> >> >>
> >> >> > This is the same issue that I began to address with Timothy Little
> >> >> > about six months ago before I became too busy to gain adequate
> >> >> > understanding.
> >> >>
> >> >> > To me it seems like a perfect match for mapping. The digit string
> >> >> > corresponding to pi is infinite and has no last digit, the set of
> >> >> > natural numbers is infinite and has no last digit, the approximations
> >> >> > to pi are finite, pi is finite and has an infinite digit string with
> >> >> > with no last digit.
> >> >>
> >> >> > A B C D E
> >> >> > 1 -> 3 -> 3. -> 1 -> 3
> >> >> > 2 -> 1 -> 3.1 -> 2 -> 3.1
> >> >> > 3 -> 4 -> 3.14 -> 3 -> 3.14
> >> >> > 4 -> 1 -> 3.141 -> 4 -> 3.141
> >> >> > 5 -> 5 -> 3.1415 -> 5 -> 3.1415
> >> >> > . . . . .
> >> >> > : -> : -> : -> : -> :
> >> >>
> >> >>
> >> >> > Do you agree that the infinite naturals (column A) map in a one to one
> >> >> > correspondence with the infinite list of digits (column B) having the
> >> >> > same representation as the corresponding successive digits of pi?
> >> >>
> >> >> > Do you agree that the infinite list of digits (column B) map in a one
> >> >> > to one correspondence with the real numbers (coulmn C) {there are an
> >> >> > infinite number of such reals} associated with the infinite string of
> >> >> > numbers that start with "3." and place one additional corresponding
> >> >> > digit from pi to the right of the previous number?
> >> >>
> >> >> > Do you agree that the real numbers from column C map in a one to one
> >> >> > correspondence with the infinite naturals in column D?
> >> >>
> >> >> > Do you agree that the infinite naturals (column D) map in a one to one
> >> >> > correspondence with the infinite list of real numbers in column E ?
> >> >>
> >> >> > If the infinite naturals in columns A & D (A = D) map the infinite
> >> >> > digit string B having the same digits as pi, how can the infinite
> >> >> > naturals 1, 2, 3, ... not map the infinite list of real numbers
> >> >> > presented in columns E and C (E = C) and represented by 3, 3.1, 3.14,
> >> >> > ..., 3.1415...?
> >> >>
> >> >>
> >> >> Neither 1,2,3, .... or 3, 3.1, 3.14, ... have a last element.
> >> >> pi is not an element of the sequence 3, 3.1, 3.14, 3.141, ....
> >> >> Sure there exists a one to one correspondence
> >> >> between the two, but neither has a last element. oo is
> >> >> not a natural number, and pi is not an element of the sequence
> >> >> 3, 3.1, 3.14, ...
> >> >>
> >> >> Why do you think there should be a last element to an unending
> >> >> sequence?
> >>
> >> > Are you suggesting that pi has a last digit? My infinite column of
> >> > numbers is represented as 3., 3.1, 3.14,..., 3.1415... where the there
> >> > is no last digit of pi. I know you know that pi is infinite and that
> >> > the naturals are infinite, where do you see a problem?
> >>
>
>
>
> >> How did you possibly read that into what I said? pi
> >> does not have a last digit.
>
> > When you wrote "pi is not an element of the sequence 3, 3.1, 3.14,
> > 3.141, .... Sure there exists a one to one correspondence between the
> > two, but neither has a last element.", I thought the neither that you
> > wrote was comparing pi to the sequence. I now see you used the word
> > element which I mistakenly read as digit. I occasionally invert
> > concepts such as right vs left.
>
> >> The sequence
> >> 3, 3.1, 3.14, 3.141, ...
> >> does not have a last element. pi is not an element
> >> of that sequence. The sequence
> >> 1, 2, 3, ...
> >> does not have a last element. oo is not an element
> >> of that sequence. You cannot map an element that
> >> is not in the the latter sequence to an element
> >> that is not in the former sequence when constructing
> >> a one to one correspondence between the two sequences.
> >>
>
>
>
> >> You seem to think that infinite sequences have a last
> >> element that equals the limit of the sequence. They
> >> do not in general.
> >>
>
>
> > I think pi = 3.14... exactly (its decimalic representation) and that
> > there is no last digit of pi.
>
> But that has nothing to do with the fact that the sequence
> 3, 3.1, 3.14, 3.1415, ..
> does not contain pi.
>
> > So that when the one to one
> > correspondence is written it extends from the the terminating 3., 3.1,
> > to the 3.14... = pi which makes definitivly clear that this sequence 1)
> > extends to all numbers formed from the digits associated with pi, and
> > 2) the sequence is infinite since it is formed from a number having no
> > last digit.
>
> > What other mathematical representation of this sequence shows these two
> > features thereby making clear that numbers such as 3.14152 are not
> > included in this sequence.
>
> > Since the decimalic form of pi forms the sequence and has no last
> > digit, I think that representation is correct.
>
> I am not sure what you are talking about now. The question
> you asked, and which I tried to answer, was:
>
> > Here is where I get lost. Above in essence you said that the infinite
> > set naturals can "trivially" be placed in a one to one correspondence
> > with all of the digits of pi; and yet you now seem to be suggesting
> > there are not enough natural numbers in the infinite set of natural
> > numbers for a mapping between N the approximations of pi and pi, unless
> > pi is placed as an indivdual element corresponding
> > to some finite natural (in other words pi cannot be the last element).
> > Why not if the set of naturals is infinite?
>
> pi cannot be the last element because there is no last element.
> Consider the following correspondences:
>
> A B A C
> 1 --> 3 1 --> 3
> 2 --> 3.1 2 --> 1
> 3 --> 3.14 3 --> 4
> 4 --> 3.145 4 --> 1
> ... ...
>
> You were asking why we could not put pi at the end of the B
> sequence. Well, if we did, what would be in the A and C columns?
>
> A B A C
> ? --> pi ? --> ?
>
> There is no last element of the sequence A, or the sequence C,
> and there is also no last element of the sequence B.
>
> You seem to be making this a lot harder than it is and
> confusing different concepts.
>
You correctly pointed out that my columnar representation of the
mapping that I tried to describe in words was poor, in fact it did not
even include pi. I will try to form a better representation of the
mapping to demonstrate more precisely what I meant to communicate.
E) 3, 3.1, 3.14, ..., 0.3.14...(d_n), ..., 3.14...(d_n)... as n=>oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
/
D) 0, 1, 2, ..., n, ..., ... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
C) 3, 3.1, 3.14, ..., 3.14...(d_n), ..., 0.33...(d_n)... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
\
B) 3, 1, ..., d_n(n <- d_dec pi), ..., ... as n => oo
^ ^ ^ ^ ^ ^
| | | | | |
A) 1, 2, ..., n, ..., ... as n => oo
Where d_n(n <- d_dec pi) is the nth digit of this digit string that
arises from the nth digit of the decimalic expression for pi which is
3.14....
Where (d_n) represents the nth digit of pi = 3.14... and where (d_n)
has the same numerical value as the the nth digit of pi. For example,
(d_6) is the sixth digit of pi = 3.14159265... or "9".
Where 3.14...(d_n) is the finite decimal approximation to the number
with with n digits exactly matching the first n digits of pi. For
example, 3.14...5 is the the decimal approximation to pi with five
digits; namely, 3.1415
Where 0.33...(3-n)... = 0.333... = 1/3. I believe the number 0.333...
where there is no last digit of the decimalic expression is the number
1/3 which coincides with the observation that unity (the whole) cannot
be evenly subdivided into three parts; there is always a remainder.
I don't see how the mapping shown above (which includes mapping the
infinite number of naturals to the decimalic numbers based on 3.14... )
does not include 3.14... which has no last (d_n) as n => oo.
Don
.
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