Re: INFINITY Revisited

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?

Stephen
.

Relevant Pages

  • Re: An uncountable countable set
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  • Re: Well Ordering the Reals
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  • Re: Galileos Paradox and the Project of the Reals
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  • Re: Well Ordering the Reals
    ... >>> rightmost zero in an unending sequence of ever more rightward ... >> naturals, you might as well call it something, I suppose. ... > In TO's system of "whole numbers", there is a most significant digit and ... >> infinite unending string of bits, even if most are generally ignored. ...
    (sci.math)
  • Re: Calculus XOR Probability
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