Re: INFINITY Revisited

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. 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.

.

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