Re: INFINITY Revisited

stephen@xxxxxxxxxx wrote:
> Don Whitehurst <whit0911@xxxxxxx> wrote:
> > Peter Webb wrote:
> > <snip>
> >> > Don wrote:
> >> > Can the set of natural numbers be put in a one to one correspondence
> >> > with all of the digits of any specific decimalic number (including any
> >> > nonterminating rational or irrational number)?
> >> >
> >>
> >> Yes, trivially.
> >>
> >> 1 -> 3
> >> 2 -> 1
> >> 3 -> 4
> >> 4 -> 1
> >> 5 -> 5
> >>
> >> If you want a mapping between N and approximations to pi, pick
> >> 1 -> 3
> >> 2 -> 3.1
> >> 3 -> 3.14
> >>
> >> If you want pi on the list,
> >>
> >> 1 -> pi
> >> 2 -> 3
> >> 3 -> 3.1
> >> 4 -> 3.14
> >>
> > 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.
> 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...?



> Don Whitehurst

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... >> in my map to P, requiring something to map to N does NOT mean ... > naturals, Pwill contain all possible subsets ... Let's say they are uncountably long, without end, truly infinite. ... In reality, the power set is indeed bigger, ...
    (sci.math)
  • Re: An uncountable countable set
    ... Indexing the digit number n is equivalent to covering the ... string up to digit number n. ... Even a finite string of an infinite set of finite ... naturals does exist, and by Cantor, the cardinality is aleph-0. ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... > Okay, I don;t think I understood what you were saying. ... > 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: Well Ordering the Reals
    ... > in my map to P, requiring something to map to N does NOT mean ... > an "infinite natural", whatever that looks like, is mapped. ... naturals, Pwill contain all possible subsets ... only finite naturals. ...
    (sci.math)
  • Re: The Modified Halting Problem, Take ??? .
    ... Turing tapes" will not be a set or even a proper class in NAFL, ... each tape itself is an infinite entity. ... They're usually described as potentially infinite or finitely unbounded. ... and there is no last digit of Pi ever computed which one would think ...
    (sci.logic)
Loading