Re: INFINITY Revisited

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?

>
> Stephen

.

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