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:
>> >> >> 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.
>>

<snip>

> Where 3.14...(d_n) is the finite decimal approximation to the number
> with with n digits exactly matching the first n digits of pi. For
> example, 3.14...5 is the the decimal approximation to pi with five
> digits; namely, 3.1415

Yes, and there does not exist a finite k such that the
approximation to pi with k digits equals pi. That is
why pi does not show up in your mapping.

> Where 0.33...(3-n)... = 0.333... = 1/3. I believe the number 0.333...
> where there is no last digit of the decimalic expression is the number
> 1/3 which coincides with the observation that unity (the whole) cannot
> be evenly subdivided into three parts; there is always a remainder.

What? 1 can be divided into three parts. 1 divided by 3 is 1/3.
Just because the decimal representation is non-terminating does
not mean that 1 cannot be divided by three. If we used base 12,
1/3 would equal .4.

> I don't see how the mapping shown above (which includes mapping the
> infinite number of naturals to the decimalic numbers based on 3.14... )
> does not include 3.14... which has no last (d_n) as n => oo.

Because there is no last position in the mapping. I'll
repeat what I posted before. Consider the following
correspondence:

A B
1 --> 3
2 --> 3.1
3 --> 3.14
4 --> 3.145
.. ...
.. ...
? --> pi

If pi is in column B, what is in column A at
the same location? Remember, there is no last position,
and oo is not a natural number, and A only contains
natural numbers.

A set or sequence does not have to include its limit.
It is as simple as that. There is no reason to think
that the limit must be part of the set or sequence.

Stephen
.

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