Re: INFINITY Revisited

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.

Stephen
.

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