INFINITY Revisited
Don Whitehurst
on the posting called "infinity":
<Hello everyone,
<I'm having an argument with a friend about the following problem:
<Suppose you have a giant vase and a bunch of ping pong balls with an
<integer written on each one, e.g. just like the lottery, so the balls
<are numbered 1, 2, 3, ... and so on. At one minute to noon you put
<balls 1 to 10 in the vase and take out number 1. At half a minute to
<noon you put balls 11 - 20 in the vase and take out number 2. At one
<quarter minute to noon you put balls 21 - 30 in the vase and take out
<number 3. Continue in this fashion. Obviously this is physically
<impossible, but you get the idea. Now the question is this: At noon,
<how many ping pong balls are in the vase?
<An 2nd experiment goes as follows:
<put no. 1 - 10 in, take no. 10 out, put no. 11-20 in, take no. 20
out, <put no. 21-30 in, take no. 30 out, etc.
<(of course at the same moments as above)
<My friend claims both experiments end up with an empty vase.
<I think however the 2nd experiment ends up with a vase with an
infinite
<number of balls:
<1-9, 11-19, 21-29 etc. are definitely in the vase.
<He says it all has to do with Cantor's set theory, cardinality etc...,
<but browsing the internet didn't really help me much.
<Any information or relevant links are very welcome,
<Thanks, Theo
I started a new thread because I could not access the old "infinity"
posting tonight.
I confess that I have not read anywhere near all of the voluminous
"Infinity" tree and if I am repeating what others have asserted, or
asked about, I apologize for wasting your time.
If I am following this correctly, the Cantorians are insisting and
teaching us that despite our intuition the outcome is that at noon the
vase is necessarily empty. While I agree with this reasoning, I think
when the problem is extended slightly, difficulties seem to me to
arise.
If the problem were changed as described below does the outcome change?
At 1 minute to noon, balls 1 - 9 are put in the vase and take out
number one.
At 1/2 minute to noon, balls 10 - 99 are put in the vase and take out
number two.
At 1/4 minute to noon, balls 100 - 999 are put in the vase and take out
number three.
Continue in this fashion.
How many balls are in the vase at noon?
Clearly the first group of 9 balls added to the vase will be removed
before noon, as will the second group of 90 balls, and eventually the
third group of 900 balls and so on. Before noon the number of balls in
the vase at any time seems to grow very rapidly and without bound. But
since nothing stops the slow but steady onslaught of the infinite
removal process, the magic of the infinite comes to our rescue and the
vase must still be empty at noon. Is it still true for this case that
the vase is empty at noon?
If at noon there are no balls remaining, does this have implications
about the naturals having a one to one correspondence with the real
decimalic numbers?
At 2 minutes to noon the vase is empty.
0 => 0
Consider A) at 1 minute to noon, balls 1 - 9 that are added to the vase
have the following printed on their respective surfaces:
1 => 0.1
2 => 0.2
: => :
9 => 0.9.
This corresponds to all of the single digit decimals between 0 and 1.
Consider also B) at 1/2 minute to noon, balls 10 - 99 that are added to
the vase have the following printed on their respective surfaces:
10 => 0.01
11 => 0.11
12 => 0.21
13 => 0.31
: => :
19 => 0.91
20 => 0.02
21 => 0.12
22 => 0.22
: => :
29 => 0.92
30 => 0.03
31 => 0.13
32 => 0.23
: => :
98 => 0.89
99 => 0.99
This along with (union) the printing on the balls from A) corresponds
to all of the two digit decimals between 0 and 1.
Consider also C) at 1/4 minute to noon, balls 100 - 999 have the
following printed on their respective surfaces:
100 => 0.001
101 => 0.101
102 => 0.201
103 => 0.301
: => :
108 => 0.801
109 => 0.901
110 => 0.011
111 => 0.111
112 => 0.211
: => :
189 => 0.981
190 => 0.091
191 => 0.191
: => :
198 => 0.891
199 => 0.991
200 => 0.002
201 => 0.102
202 => 0.202
203 => 0.302
: => :
989 => 0.989
990 => 0.099
991 => 0.199
: => :
998 => 0.899
999 => 0.999
This along with (union) the printing on the balls from A) and B)
corresponds to all of the three digit decimals between 0 and 1.
This procedure of adding and removing balls continues toward noon.
Any time before noon, there exists an "n" equal to the total number of
balls added up to that time, which is of course finite. For the
natural numbers corresponding to these balls, all naturals up to and
including that "n" are a finite subset of the infinite set of naturals.
The set of naturals associated with the n balls is not complete or
infinite until noon.
At noon the set of natural numbers associated with the balls is
infinite, and I believe all balls must have necessarily been removed.
By noon an infinite number of balls (with printing on the surface) has
been added to the vase and removed. This seems to me to imply that by
noon all balls with infinite decimalic representations such as 1/3 =
0.333..., Pi/10 = 0.31415..., and sqrt (2)/2 = 0.70710678..., as well
as many that can't be denumerated in real life, must also have been
placed in the vase and removed.
Consequently, if I am not mistaken, by the removal process all reals on
the interval 0 to 1 were set in a one to one correspondence with the
naturals or else the vase would not be empty. This occurred at noon
when 1) the naturals stopped being a finite subset and became the
infinite set of naturals and simultaneously when 2) the decimalic
representations printed on the balls changed from terminating rationals
into the repeating rationals and the irrationals. The terminating
rationals at each succesive interval formed numerous series that were
approahing an infinitely growing number of limit points.
Does this really establish what Cantor's diagonal proof disproved??
Dave Seaman
> If at noon there are no balls remaining, does this have implications
> about the naturals having a one to one correspondence with the real
> decimalic numbers?
> At 2 minutes to noon the vase is empty.
> 0 => 0
> Consider A) at 1 minute to noon, balls 1 - 9 that are added to the vase
> have the following printed on their respective surfaces:
> 1 => 0.1
> 2 => 0.2
>: => :
> 9 => 0.9.
> This corresponds to all of the single digit decimals between 0 and 1.
> Consider also B) at 1/2 minute to noon, balls 10 - 99 that are added to
> the vase have the following printed on their respective surfaces:
> 10 => 0.01
> 11 => 0.11
[ ... ]
> At noon the set of natural numbers associated with the balls is
> infinite, and I believe all balls must have necessarily been removed.
Correct.
> By noon an infinite number of balls (with printing on the surface) has
> been added to the vase and removed. This seems to me to imply that by
> noon all balls with infinite decimalic representations such as 1/3 =
> 0.333..., Pi/10 = 0.31415..., and sqrt (2)/2 = 0.70710678..., as well
> as many that can't be denumerated in real life, must also have been
> placed in the vase and removed.
No. According to the scheme you have described, only the balls with
terminating decimal expressions have been added and removed.
> Consequently, if I am not mistaken, by the removal process all reals on
> the interval 0 to 1 were set in a one to one correspondence with the
> naturals or else the vase would not be empty. This occurred at noon
> when 1) the naturals stopped being a finite subset and became the
> infinite set of naturals and simultaneously when 2) the decimalic
> representations printed on the balls changed from terminating rationals
> into the repeating rationals and the irrationals. The terminating
> rationals at each succesive interval formed numerous series that were
> approahing an infinitely growing number of limit points.
> Does this really establish what Cantor's diagonal proof disproved??
No, it merely establishes that the set of terminating decimal expressions
is countable.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
Don Whitehurst
I clearly agree that at any time before noon only the balls with
terminating decimal expressions have been added or removed and at any
time before noon only a finite number of balls have been added,
representing a finite subset of the infinite natural numbers. However,
by noon things have changed: the finite became infinite - the naturals
became an infinite set. What about the sequence of balls represented
by 0.3, 0.33, 0.333, ...? At noon don't there become an infinite
number of digits associated with the limit point of the sequence formed
from balls with only the numerals "3" printed on the surface? If not,
at noon how many digits are there printed on the surface of the ball
containing the largest number of numeral threes and which has only 3s
as numerals?
This thought experiment seems like a method prone to generate
misunderstanding.
What is the proper mathematical term for a decimalic digit "dj" for the
jth place to the right of the decimal point in a number represented in
decimalic form? For example: 0.333...dj. Knowing proper terms would
be useful when trying to precisely specify and discuss issues on these
matters. I hope the language I use in this post will not be
misunderstood.
The sequence of rational numbers 0.3, 0.33, 0.333, ... converges to a
real limit point (as dj the number of decimalic digits approaches
infinity) of 0.333...; where 0.333... is the "repeating" decimalic
representation of the rational number 1/3.
Does the list of natuarl numbers corresponding to the infinite set of
natural numbers enumerate all of the digits dj where j = 1 -> oo of
1/3 = 0.333... ?
0, 1, 2, 3, ...
0 . 3 3 3 ...
0,1,2,3, ..., n, ...
0.3 3 3 ..., dj, ...
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)?
Can there ever be any decimalic digits dj of a real number that such a
one to one correspondence (between the digits of the real number and
the infinite set of natural numbers) miss? Cantors diagonal proof
seems to depend upon this.
Does such a rule serve as an alternative definition of a real number
eliminating numbers such as 0.000...1 ( the termination of the sequence
of numbers 0.1, 0.01, 0.001, ... that apparently lacks a real limit
point) from consideration as real numbers?
<snip>
Don Whitehurst
Peter Webb
> This thought experiment seems like a method prone to generate
> misunderstanding.
>
> What is the proper mathematical term for a decimalic digit "dj" for the
> jth place to the right of the decimal point in a number represented in
> decimalic form? For example: 0.333...dj. Knowing proper terms would
> be useful when trying to precisely specify and discuss issues on these
> matters. I hope the language I use in this post will not be
> misunderstood.
>
Its normally written as the jth digit d_j but you are fine with dj.
> The sequence of rational numbers 0.3, 0.33, 0.333, ... converges to a
> real limit point (as dj the number of decimalic digits approaches
> infinity) of 0.333...; where 0.333... is the "repeating" decimalic
> representation of the rational number 1/3.
>
Yes
> Does the list of natuarl numbers corresponding to the infinite set of
> natural numbers enumerate all of the digits dj where j = 1 -> oo of
> 1/3 = 0.333... ?
>
No. 1/3 doesn't appear in the listas you have written it. It appears in
others, such as the standard mapping of p/q to Natural numbers.
> 0, 1, 2, 3, ...
> 0 . 3 3 3 ...
>
> 0,1,2,3, ..., n, ...
> 0.3 3 3 ..., dj, ...
>
> 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
> Can there ever be any decimalic digits dj of a real number that such a
> one to one correspondence (between the digits of the real number and
> the infinite set of natural numbers) miss? Cantors diagonal proof
> seems to depend upon this.
>
Sort of. We need to know that if you count up from 1 you eventually get to
all numbers. This is provable from the axioms of arithmetic.
> Does such a rule serve as an alternative definition of a real number
> eliminating numbers such as 0.000...1 ( the termination of the sequence
> of numbers 0.1, 0.01, 0.001, ... that apparently lacks a real limit
> point) from consideration as real numbers?
>
> <snip>
Yes, the rule says that the limit point of 0.1, 0.01, 0.001 is 0, and tough
titty if that is not part of the sequence. You can construct things that
look like 0.000..1, "infintismals", but they don't help with Cantor.
>
> Don Whitehurst
>
Dave Seaman
> Dave Seaman wrote:
>> > By noon an infinite number of balls (with printing on the surface) has
>> > been added to the vase and removed. This seems to me to imply that by
>> > noon all balls with infinite decimalic representations such as 1/3 =
>> > 0.333..., Pi/10 = 0.31415..., and sqrt (2)/2 = 0.70710678..., as well
>> > as many that can't be denumerated in real life, must also have been
>> > placed in the vase and removed.
>> No. According to the scheme you have described, only the balls with
>> terminating decimal expressions have been added and removed.
> I clearly agree that at any time before noon only the balls with
> terminating decimal expressions have been added or removed and at any
> time before noon only a finite number of balls have been added,
> representing a finite subset of the infinite natural numbers. However,
> by noon things have changed: the finite became infinite - the naturals
> became an infinite set. What about the sequence of balls represented
> by 0.3, 0.33, 0.333, ...? At noon don't there become an infinite
> number of digits associated with the limit point of the sequence formed
> from balls with only the numerals "3" printed on the surface? If not,
> at noon how many digits are there printed on the surface of the ball
> containing the largest number of numeral threes and which has only 3s
> as numerals?
What makes you think there is such a thing as a "largest number of
numeral threes and which has only 3s as numerals"? Do you also think
there is a largest integer?
Each ball that is marked with a terminating decimal string is added at a
specific time before noon. There is no time when any ball with a
nonterminating string is added.
> This thought experiment seems like a method prone to generate
> misunderstanding.
> What is the proper mathematical term for a decimalic digit "dj" for the
> jth place to the right of the decimal point in a number represented in
> decimalic form? For example: 0.333...dj. Knowing proper terms would
> be useful when trying to precisely specify and discuss issues on these
> matters. I hope the language I use in this post will not be
> misunderstood.
The language you are using is fine.
> The sequence of rational numbers 0.3, 0.33, 0.333, ... converges to a
> real limit point (as dj the number of decimalic digits approaches
> infinity) of 0.333...; where 0.333... is the "repeating" decimalic
> representation of the rational number 1/3.
> Does the list of natuarl numbers corresponding to the infinite set of
> natural numbers enumerate all of the digits dj where j = 1 -> oo of
> 1/3 = 0.333... ?
For each natural number n, the terminating decimal of the form
sum_{k=1}^n 3/10^k is on the label of a ball that is added before noon.
The nonterminating string sum_{k=1}^oo 3/10^k = 1/3 is not on the label
of any ball that is added to the vase according to your description.
> 0, 1, 2, 3, ...
> 0 . 3 3 3 ...
> 0,1,2,3, ..., n, ...
> 0.3 3 3 ..., dj, ...
> 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.
> Can there ever be any decimalic digits dj of a real number that such a
> one to one correspondence (between the digits of the real number and
> the infinite set of natural numbers) miss? Cantors diagonal proof
> seems to depend upon this.
No, there are no digit positions that are missed, and no, the Cantor
diagonal argument does not depend on any such thing.
How many 3-digit strings can you make from the ten decimal digits? 1000,
right? Isn't 1000 larger than 3?
The number of digits in a nonterminating decimal digit string is aleph_0
(that's the name we use for the infinity of the natural numbers). The
number of different digit strings of length aleph_0 is 10^aleph_0 = c,
the cardinality of the continuum. Just as 10^3 is bigger than 3,
10^aleph_0 is bigger than aleph_0. That's what the diagonal argument
shows.
> Does such a rule serve as an alternative definition of a real number
> eliminating numbers such as 0.000...1 ( the termination of the sequence
> of numbers 0.1, 0.01, 0.001, ... that apparently lacks a real limit
> point) from consideration as real numbers?
What rule?
A real number is not a digit string. A real number is a collection of
rational numbers that is nonempty, bounded above, downward closed, and
has no maximum element.
Timothy Little
> If at noon there are no balls remaining, does this have implications
> about the naturals having a one to one correspondence with the real
> decimalic numbers?
No.
> By noon an infinite number of balls (with printing on the surface) has
> been added to the vase and removed.
Correct: there are infinitely many balls with finite labels, and each
of those was added, and later removed before noon.
> This seems to me to imply that by noon all balls with infinite
> decimalic representations such as 1/3 = 0.333..., Pi/10 =
> 0.31415..., and sqrt (2)/2 = 0.70710678..., as well as many that
> can't be denumerated in real life, must also have been placed in the
> vase and removed.
At what time was the ball labelled with 1/3 placed in the vase? How
long did it stay there?
- Tim
Don Whitehurst
<snip>
DW wrote
> > Does the list of natuarl numbers corresponding to the infinite set of
> > natural numbers enumerate all of the digits dj where j = 1 -> oo of
> > 1/3 = 0.333... ?
> >
>
> No. 1/3 doesn't appear in the listas you have written it. It appears in
> others, such as the standard mapping of p/q to Natural numbers.
>
>
> >
> >
> > 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
>
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?
> <snip>
> >
> > Don Whitehurst
> >
Don Whitehurst
> Don Whitehurst wrote:
> > By noon an infinite number of balls (with printing on the surface) has
> > been added to the vase and removed.
>
> Correct: there are infinitely many balls with finite labels, and each
> of those was added, and later removed before noon.
>
>
think finally understood what you were saying and avoiding. While I
agree that every ball with a finite label will be eventually be removed
before noon, the number of balls in the vase at any time before noon
has increased and will continue to increase at each time interval (#
balls = 10^n - n). At any time before noon the number of balls is
larger than the previous time. At no time before noon are there zero
balls in the vase. Yet by noon the miracle has happened.
> > This seems to me to imply that by noon all balls with infinite
> > decimalic representations such as 1/3 = 0.333..., Pi/10 =
> > 0.31415..., and sqrt (2)/2 = 0.70710678..., as well as many that
> > can't be denumerated in real life, must also have been placed in the
> > vase and removed.
>
> At what time was the ball labelled with 1/3 placed in the vase? How
> long did it stay there?
>
I don't know when, but I am sure it was the same time the vase became
empty.
My other response to this would be at noon the time of of the
miraculous coming of the infinite. At noon (the time when our limits
are measured) the infinite set of naturals rose for the measuring, the
infinite decimalic balls arrived and disappeared instantly.
>
> - Don
ste...@nomail.com
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.
Stephen
Don Whitehurst
Of course there is no ball with the most (but finite) number of d_js =
3. Before noon the number of digits in any decimal representation on a
ball were obviously finite. However, by noon aleph_0 must have been
reached or else there would be very many balls left. When aleph_0 was
reached all digit strings were necessarily infinite. If it were
otherwise, how did the vase become empty?
I believe you recognized perfectly well the concept I was asking by
using sarcasm to emphasize the point; particularly given your follow-up
sarcastic question.
If indeed you were confused by or indeed strongly objected to my poor
choice of language, I apologize. I tried to ask what are the proper
mathematical terms associated with decimals to describe the concepts
such as a) meaning of digit in a decimal, b) digit place - which I have
been informed is (d_j), c) numerical value associated with a particular
digit place - where pi/10 has d_3 = 4, d) digit string, etc. Without
such precise terms, it is hard to discuss the concepts I would like to
learn about.
> Each ball that is marked with a terminating decimal string is added at a
> specific time before noon. There is no time when any ball with a
> nonterminating string is added.
I agree that there is no time before noon when any ball with a
nonterminating decimal string is added; but there is also no time
before noon when the next time interval will not increase the total
number of balls in the vase. Yet we both agree by noon the vase will
be empty.
Before noon both of the following statements apply to every interval
corresponding to the natural numbers and hence both are a property
associated with every interval of additon and removal of balls from the
vase before noon. Induction therefore applies to each before noon.
1) Every ball added to the vase before noon will be removed before
noon.
2) Before noon, whenever the nth ball is removed, there are 10^n -n
balls remaining in the vase.
Oops! You taught me that 10^aleph_0 - aleph_0 is not equal to zero. A
Paradox by noon - even if I assume that aleph_0 was reached by noon. Is
this due to completing the naturals which requires a last natural
number to exist?
>
> > This thought experiment seems like a method prone to generate
> > misunderstanding.
<snip>
> > Can there ever be any decimalic digits dj of a real number that such a
> > one to one correspondence (between the digits of the real number and
> > the infinite set of natural numbers) miss? Cantors diagonal proof
> > seems to depend upon this.
>
> No, there are no digit positions that are missed, and no, the Cantor
> diagonal argument does not depend on any such thing.
>
My point about dependence was that *if* there were any decimalic digits
d_j of some real number missed by a one to one corespondence with the
set of naturals, then the Cantor diagonal could not check the missed
digits.
>
> How many 3-digit strings can you make from the ten decimal digits? 1000,
> right? Isn't 1000 larger than 3?
>
> The number of digits in a nonterminating decimal digit string is aleph_0
> (that's the name we use for the infinity of the natural numbers). The
> number of different digit strings of length aleph_0 is 10^aleph_0 = c,
> the cardinality of the continuum. Just as 10^3 is bigger than 3,
> 10^aleph_0 is bigger than aleph_0. That's what the diagonal argument
> shows.
>
How was the continuum shown to be 10^aleph_0? If too difficult to
summarize here please direct me to a text that you would recommend.
<snip>
>
> A real number is not a digit string. A real number is a collection of
> rational numbers that is nonempty, bounded above, downward closed, and
> has no maximum element.
>
I do not understand this description of a real number. A further
exposition of the meaning of these terms would be appreciated.
Does the infinite digit string forming the number "0.01002000300004..."
exist as a real number?
This may also be expressed as the series 1/{[(10^1)*(10^1)] + 2/[(10^2)
*(10^(1+2))] + 3/[(10^3) *(10^(1+2+3))] + ... +
n/(10^n)*[10^sum_n=1^oo (n)]}
Is the limit of the corresponding infinite series as n => oo a real
number?
Is the same true of 0.0100002000000000300000000000000004... and of
0.010000000020000000000000000000000000003... ?
Such numbers seems to have some characteristics of an irrational number
and yet are almost a terminating rational number due to having an
ending digit string similar to what Peter Webb called the infintismals.
--
Don Whitehurst
Dave Seaman
> Dave Seaman wrote:
>> > I clearly agree that at any time before noon only the balls with
>> > terminating decimal expressions have been added or removed and at any
>> > time before noon only a finite number of balls have been added,
>> > representing a finite subset of the infinite natural numbers. However,
>> > by noon things have changed: the finite became infinite - the naturals
>> > became an infinite set. What about the sequence of balls represented
>> > by 0.3, 0.33, 0.333, ...? At noon don't there become an infinite
>> > number of digits associated with the limit point of the sequence formed
>> > from balls with only the numerals "3" printed on the surface? If not,
>> > at noon how many digits are there printed on the surface of the ball
>> > containing the largest number of numeral threes and which has only 3s
>> > as numerals?
>> What makes you think there is such a thing as a "largest number of
>> numeral threes and which has only 3s as numerals"? Do you also think
>> there is a largest integer?
> Of course there is no ball with the most (but finite) number of d_js =
> 3. Before noon the number of digits in any decimal representation on a
> ball were obviously finite. However, by noon aleph_0 must have been
> reached or else there would be very many balls left. When aleph_0 was
> reached all digit strings were necessarily infinite. If it were
> otherwise, how did the vase become empty?
At what time was a ball with the label aleph_0 added to the vase? Where
does it say this in the problem specification?
Why is it impossible to add all the finite-numbered balls without also
adding a ball marked aleph_0?
The vase becomes empty because each ball that is added is removed before
noon.
> I believe you recognized perfectly well the concept I was asking by
> using sarcasm to emphasize the point; particularly given your follow-up
> sarcastic question.
You asked many questions. I do not know which one you are referring to
here, particularly in light of your next paragraph, which completely
changes the subject.
> If indeed you were confused by or indeed strongly objected to my poor
> choice of language, I apologize. I tried to ask what are the proper
> mathematical terms associated with decimals to describe the concepts
> such as a) meaning of digit in a decimal, b) digit place - which I have
> been informed is (d_j), c) numerical value associated with a particular
> digit place - where pi/10 has d_3 = 4, d) digit string, etc. Without
> such precise terms, it is hard to discuss the concepts I would like to
> learn about.
And I told you the language you were using for representing the decimal
digits was fine. This has nothing to do with your previous question
about a largest finite string, which was nonsense.
>> Each ball that is marked with a terminating decimal string is added at a
>> specific time before noon. There is no time when any ball with a
>> nonterminating string is added.
> I agree that there is no time before noon when any ball with a
> nonterminating decimal string is added; but there is also no time
> before noon when the next time interval will not increase the total
> number of balls in the vase. Yet we both agree by noon the vase will
> be empty.
Yes.
> Before noon both of the following statements apply to every interval
> corresponding to the natural numbers and hence both are a property
> associated with every interval of additon and removal of balls from the
> vase before noon. Induction therefore applies to each before noon.
> 1) Every ball added to the vase before noon will be removed before
> noon.
> 2) Before noon, whenever the nth ball is removed, there are 10^n -n
> balls remaining in the vase.
> Oops! You taught me that 10^aleph_0 - aleph_0 is not equal to zero.
In the first place, subtraction of infinite cardinals is not generally
defined. In the second place, 10^aleph_0 has no relevance to the
balls-in-the-vase problem, since only aleph_0 balls go in and aleph_0
balls go out.
The reason the vase is empty has nothing to do with cardinalities. The
reason the vase is empty is that each ball is removed before noon.
>A
> Paradox by noon - even if I assume that aleph_0 was reached by noon. Is
> this due to completing the naturals which requires a last natural
> number to exist?
Where do you see a paradox? And why add assumptions that are not present
in the problem statement?
>> > This thought experiment seems like a method prone to generate
>> > misunderstanding.
No more so than the original problem.
>> > Can there ever be any decimalic digits dj of a real number that such a
>> > one to one correspondence (between the digits of the real number and
>> > the infinite set of natural numbers) miss? Cantors diagonal proof
>> > seems to depend upon this.
>> No, there are no digit positions that are missed, and no, the Cantor
>> diagonal argument does not depend on any such thing.
> My point about dependence was that *if* there were any decimalic digits
> d_j of some real number missed by a one to one corespondence with the
> set of naturals, then the Cantor diagonal could not check the missed
> digits.
No need to worry. No digits are missed.
>> How many 3-digit strings can you make from the ten decimal digits? 1000,
>> right? Isn't 1000 larger than 3?
>> The number of digits in a nonterminating decimal digit string is aleph_0
>> (that's the name we use for the infinity of the natural numbers). The
>> number of different digit strings of length aleph_0 is 10^aleph_0 = c,
>> the cardinality of the continuum. Just as 10^3 is bigger than 3,
>> 10^aleph_0 is bigger than aleph_0. That's what the diagonal argument
>> shows.
> How was the continuum shown to be 10^aleph_0? If too difficult to
> summarize here please direct me to a text that you would recommend.
First of all, it's easy to show that 10^aleph_0 = 2^aleph_0 by using the
laws of exponents. We surely have 2^aleph_0 <= 10^aleph_0 <=
(2^aleph_0)^aleph_0, but since 2^aleph_0 = 2^(aleph_0*aleph_0) =
(2^aleph_0)^aleph_0, it turns out that all three expressions are equal.
But 2^aleph_0 is the cardinality of P(N), the power set of the naturals,
which is the same as the set of all infinite binary strings.
Secondly, the cardinality of the unit interval [0,1] is easily seen to be
the same as the cardinality of R. It doesn't particularly matter whether
we use the closed interval [0,1], the open interval (0,1), or the
half-open interval [0,1). The cardinalities are the same.
Next, notice that each real number in [0,1) can be represented as an
infinite binary string. Some of the numbers have two representations,
such as 0.1000..._2 = 0.011111..._2, but for definiteness we'll take the
representation that doesn't terminate. This defines an injection from
[0,1) into P(N) and hence establishes that |[0,1)| <= |P(N)|.
We can also define an injection in the opposite direction by mapping each
S \subset N to the number f(S) = sum_{n in S} 2/3^n. The image of this
mapping is the Cantor set, which is a subset of [0,1], and this shows
that |P(N)| <= |[0,1]|.
To complete the proof, we invoke the Schroeder-Bernstein theorem, which
basically says that if |A| <= |B| and |B| <= |A|, then |A| = |B|.
>> A real number is not a digit string. A real number is a collection of
>> rational numbers that is nonempty, bounded above, downward closed, and
>> has no maximum element.
> I do not understand this description of a real number. A further
> exposition of the meaning of these terms would be appreciated.
Real numbers are usually defined using either Dedekind cuts or
equivalence classes of Cauchy sequences of rationals. The description I
gave was a simplified version of the Dedekind cut definition. The point
I was trying to make is that we do not define the reals by making rules
about which digit strings are allowed; the digit strings are merely an
afterthought. The properties of the reals come from the underlying
definition, and the properties of the decimal representations of the
reals are derived from that.
> Does the infinite digit string forming the number "0.01002000300004..."
> exist as a real number?
Any digit string that contains a decimal digit at digit position n for
each natural number n is a representation of a real number.
> This may also be expressed as the series 1/{[(10^1)*(10^1)] + 2/[(10^2)
> *(10^(1+2))] + 3/[(10^3) *(10^(1+2+3))] + ... +
> n/(10^n)*[10^sum_n=1^oo (n)]}
> Is the limit of the corresponding infinite series as n => oo a real
> number?
> Is the same true of 0.0100002000000000300000000000000004... and of
> 0.010000000020000000000000000000000000003... ?
Yes.
> Such numbers seems to have some characteristics of an irrational number
> and yet are almost a terminating rational number due to having an
> ending digit string similar to what Peter Webb called the infintismals.
There are no infinitesimals here. These are standard real numbers.
Besides, I don't understand where you are going with this. The standard
real numbers are already an uncountable set, and throwing in
infinitesimals will surely not make the set any smaller.
Don Whitehurst
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
ste...@nomail.com
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
Don Whitehurst
vase? Where
does it say this in the problem specification?"
The problem has time going to noon by intervals constantly reduced by
1/2. The time intervals are 1, 1/2, 1/4, ..., 1/2^n minutes to noon.
As n => oo, time approaches noon. Consequently, the time noon
represents actual infinite n.
> Why is it impossible to add all the finite-numbered balls without also
> adding a ball marked aleph_0?
>
Before noon each nth interval is represented by a natural number and
since there is no largest natural number at any time before noon (nth
interval) additonal balls always remain to be added and to be removed.
The paradox arises from the statements 1) and 2):
1) Every ball added to the vase at the nth interval before noon will be
removed before noon (at some later ( n+ r_kl )th interval).
2) Before noon, at the nth interval, whenever the nth ball is removed,
there are 10^n -n balls remaining in the vase.
> >> > This thought experiment seems like a method prone to generate
> >> > misunderstanding.
>
> No more so than the original problem.
>
I was referring to the original problem of which this is but a subset.
Thank you very much for this explanation of the continuum as
10^aleph_0, I follow the outline of the proof but not yet all of the
details. It appears simpler than I expected but I may return later
with further questions.
> >> A real number is not a digit string. A real number is a collection of
> >> rational numbers that is nonempty, bounded above, downward closed, and
> >> has no maximum element.
>
>
> > I do not understand this description of a real number. A further
> > exposition of the meaning of these terms would be appreciated.
>
> Real numbers are usually defined using either Dedekind cuts or
> equivalence classes of Cauchy sequences of rationals. The description I
> gave was a simplified version of the Dedekind cut definition. The point
> I was trying to make is that we do not define the reals by making rules
> about which digit strings are allowed; the digit strings are merely an
> afterthought. The properties of the reals come from the underlying
> definition, and the properties of the decimal representations of the
> reals are derived from that.
>
Thank you again for this exposition, I will ask further about the
Dedekind cuts after I have done some preliminary investigation.
> > Does the infinite digit string forming the number "0.01002000300004..."
> > exist as a real number?
>
> Any digit string that contains a decimal digit at digit position n for
> each natural number n is a representation of a real number.
>
So does this mean that all of the infintesimals 0.000...1 through
0.000...n the limit point of the sequences 0.1, 0.001, 0.0001, ...
through 0.n, 0.0n, 0.00n, ... respectively are also a real numbers all
taking the value zero? I thought these were not real numbers.
> > This may also be expressed as the series 1/{[(10^1)*(10^1)] + 2/[(10^2)
> > *(10^(1+2))] + 3/[(10^3) *(10^(1+2+3))] + ... +
> > n/(10^n)*[10^sum_n=1^oo (n)]}
>
> > Is the limit of the corresponding infinite series as n => oo a real
> > number?
>
> > Is the same true of 0.0100002000000000300000000000000004... and of
> > 0.010000000020000000000000000000000000003... ?
>
> Yes.
>
> > Such numbers seems to have some characteristics of an irrational number
> > and yet are almost a terminating rational number due to having an
> > ending digit string similar to what Peter Webb called the infintismals.
>
> There are no infinitesimals here. These are standard real numbers.
> Besides, I don't understand where you are going with this. The standard
> real numbers are already an uncountable set, and throwing in
> infinitesimals will surely not make the set any smaller.
>
>
I was trying to throw out the infinitesimals to make the reals. But I
was also trying to understand how the naturals could count (map) the
infinite representation of digits strings for numbers such as these.
NTON =
0.0100112000111222300001111222233334000001111122222333334444445000000111111222222333333444444555555600000001111111222222233333334444444555555566666666700000000111111112222222233333333444444445555555566666666777777778000000000111111111222222222333333333444444444555555555566666666677777777777888888888900000000001111111111222222222233333333334444444444555555555566666666667777777777888888888899999999991000000000000111111111112222222222233333333333444444444445555555555566666666666777777777778888888888899999999999101010101010101010101011000000000000111111111111222222222222333333333333444444444444555555555555666666666666677777777777788888888888899999999999910101010101010101010101011111111111111111111111112...
FACTND=
0.112112321312231213132123432143124231421341324123342134123241321431423124243124132341231421432134143214231342132412431234543215431254231542135413254123534215341253241532145314253124524315241352341523145214352134514325142351342513245124351234453214531245231452134513245123435214351243251432154315243125425314251342351423154215342135415324152341352413254125341235354213541235241352143514235124345213451234251342153415234125325413251432451324153215432145315423152431452314253125431245254312541325341253142514325134245312451324351243152415324135235412351423451234152315423145215432153421453214352135421345154321542315342153241524315234145321452314352143251425314235135421352413452134251325413245125431253412453124351235412345...
The infinite decimalic number above arises from taking factorials of
digits associated with each "n" arranged in descending order and
appending them to the decimal as in a sequence. This is shown in more
detail below. The number of added digits with each n is much larger
than 2^n.
0! = 1 => 1
1! = 1 => 1
2! = 2 => 21, 12
3! = 6 => 321, 312, 231, 213, 132, 123
4! = 24 => 4321, 4312, 4231, 4213, 4132, 4123, 3421, 3412, 3241, 3214,
3142, 3124, 2431, 2413, 2341, 2314, 2143, 2134, 1432, 1423, 1342, 1324,
1243, 1234
5! = 120 => 54321, 54312, 54231, 54213, 54132, 54123, 53421, 53412,
53241, 53214, 53142, 53124, 52431, 52413, 52341, 52314, 52143, 52134,
51432, 51423, 51342, 51324, 51243, 51234, 45321, 45312, 45231, 45213,
45132, 45123, 43521, 43512, 43251, 43215, 43152, 43125, 42531, 42513,
42351, 42315, 42153, 42135, 41532, 41523, 41352, 41325, 41253, 41235,
35421, 35412, 35241, 35214, 35142, 35124, 34521, 34512, 34251, 34215,
34152, 34125, 32541, 32514, 32451, 32415, 32154, 32145, 31542, 31524,
31452, 31425, 31254, 31245, 25431, 25413, 25341, 25314, 25143, 25134,
24531, 24513, 24351, 24315, 24153, 24135, 23541, 23514, 23451, 23415,
23154, 23145, 21543, 21534, 21453, 21435, 21354, 21345, 15432, 15423,
15342, 15324, 15243, 15234, 14532, 14523, 14352, 14325, 14253, 14235,
13542, 13524, 13452, 13425, 13254, 13245, 12543, 12534, 12453, 12435,
12354, 12345
The mapping by the naturals of the infinite representation of decimalic
numbers NTON and FACTND seems to require going "through" at least an
infinte number of digits each natural number n. These don't seem like
real numbers to me. Furthermore, if the naturals can map the infinite
digit strings associated with NTON and FACTND I really don't understand
how they cannot map the reals?
Don Whitehurst
Don Whitehurst
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
Don Whitehurst
> the set of naturals are infinite, where do you see a problem?
>
> >
> > Stephen
ste...@nomail.com
How did you possibly read that into what I said? pi
does not have a last digit. 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.
Stephen
Dave Seaman
> Dave asked "At what time was a ball with the label aleph_0 added to the
> vase? Where
> does it say this in the problem specification?"
> The problem has time going to noon by intervals constantly reduced by
> 1/2. The time intervals are 1, 1/2, 1/4, ..., 1/2^n minutes to noon.
> As n => oo, time approaches noon. Consequently, the time noon
> represents actual infinite n.
Correct.
>> Why is it impossible to add all the finite-numbered balls without also
>> adding a ball marked aleph_0?
> Before noon each nth interval is represented by a natural number and
> since there is no largest natural number at any time before noon (nth
> interval) additonal balls always remain to be added and to be removed.
This does not answer the question. Why is it impossible to add all the
finite-numbered balls without also adding a ball marked aleph_0?
>> Where do you see a paradox? And why add assumptions that are not present
>> in the problem statement?
> The paradox arises from the statements 1) and 2):
> 1) Every ball added to the vase at the nth interval before noon will be
> removed before noon (at some later ( n+ r_kl )th interval).
> 2) Before noon, at the nth interval, whenever the nth ball is removed,
> there are 10^n -n balls remaining in the vase.
That may seem paradoxical to you, but it is not a contradiction.
>> > Does the infinite digit string forming the number "0.01002000300004..."
>> > exist as a real number?
>> Any digit string that contains a decimal digit at digit position n for
>> each natural number n is a representation of a real number.
> So does this mean that all of the infintesimals 0.000...1 through
> 0.000...n the limit point of the sequences 0.1, 0.001, 0.0001, ...
> through 0.n, 0.0n, 0.00n, ... respectively are also a real numbers all
> taking the value zero? I thought these were not real numbers.
Those are not real numbers. They are character strings. You haven't
explained what they are supposed to mean.
>> There are no infinitesimals here. These are standard real numbers.
>> Besides, I don't understand where you are going with this. The standard
>> real numbers are already an uncountable set, and throwing in
>> infinitesimals will surely not make the set any smaller.
> I was trying to throw out the infinitesimals to make the reals. But I
> was also trying to understand how the naturals could count (map) the
> infinite representation of digits strings for numbers such as these.
In order to "throw out infinitesimals to make the reals", you need to
start with a set that contains some infinitesimals. You have not defined
such a set, nor have you explained why you think infinitesimals have
anything to do with the problem.
> NTON =
> 0.01001120001112223000011112222333340000011111222223333344444450000001111112222223333334444445555556000000011111112222222333333344444445555555666666667000000001111111122222222333333334444444455555555666666667777777780000000001111111112222222223333333334...
> FACTND=
> 0.11211232131223121313212343214312423142134132412334213412324132143142312424312413234123142143213414321423134213241243123454321543125423154213541325412353421534125324153214531425312452431524135234152314521435213451432514235134251324512435123445321453124...
There are no infinitesimals here.
> The infinite decimalic number above arises from taking factorials of
> digits associated with each "n" arranged in descending order and
> appending them to the decimal as in a sequence. This is shown in more
> detail below. The number of added digits with each n is much larger
> than 2^n.
The details are not important. These are ordinary real numbers. Each is
defined by an ordinary infinite series.
> The mapping by the naturals of the infinite representation of decimalic
> numbers NTON and FACTND seems to require going "through" at least an
> infinte number of digits each natural number n. These don't seem like
> real numbers to me. Furthermore, if the naturals can map the infinite
> digit strings associated with NTON and FACTND I really don't understand
> how they cannot map the reals?
The mapping that you described earlier does not include these numbers,
because your mapping covers only the reals that have terminating decimal
representations. A number such as 1/3 = 0.3333... does not appear among
the values represented by your mapping.
Don Whitehurst
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. 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.
ste...@nomail.com
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
Jesse F. Hughes
> pi cannot be the last element because there is no last element.
Right.
> The set of naturals are infinite and so there is no last
> natural number to map to pi.
But this reasoning is just wrong. The fact that the set of naturals
is infinite does not imply there is no greatest element. After all,
omega + 1 is also infinite, but it has a greatest element (omega).
--
"Eventually the truth will come out, and you know what I'll do then?
Probably go to the beach. I'll also hang out in some bars. Yup, I'll
definitely hang out in some bars, preferably near a beach."
-- JSH on the rewards of winning a mathematical revolution
Don Whitehurst
to the internet.
Dave Seaman wrote:
> >> Why is it impossible to add all the finite-numbered balls without also
> >> adding a ball marked aleph_0?
>
>
> > Before noon each nth interval is represented by a natural number and
> > since there is no largest natural number at any time before noon (nth
> > interval) additonal balls always remain to be added and to be removed.
>
> This does not answer the question. Why is it impossible to add all the
> finite-numbered balls without also adding a ball marked aleph_0?
>
My response answered the first part of the why all the balls could not
be added to the vase before noon which I thought should be adequate for
the answer; but apparently not.
Since there is no last (n+1)th interval that is 1/2 of the time
remaining from 1/(2^n) minutes to noon just as there is no last
natural, there is no last time of addition for the balls. Only by
noon, the at actual infinity was it possible to add the all the balls
which must include those with an infinite number of digits as well.
I treated the time noon as being infinity and used aleph_0 only because
you said that aleph_0 was the name of infinity for the naturals. I
don't think I said there was a ball marked aleph_0 added. But it may be
a proper inference from what I did say.
At noon I think there are an infinite number of balls added all having
decimals each with no last digit printed on them (perhaps this is only
aleph_0 balls but I think it may be 10^ aleph_0 balls but I am just
learning these concepts). It seems to me that any decimalic number
with no last digit must be either a repeating rational or an irrational
number. If there are decimalic numbers with no last digit that are not
a repeating rational or irrational number, please provide a me with an
example of such a number.
> >> Where do you see a paradox? And why add assumptions that are not present
> >> in the problem statement?
>
>
> > The paradox arises from the statements 1) and 2):
> > 1) Every ball added to the vase at the nth interval before noon will be
> > removed before noon (at some later ( n+ r_kl )th interval).
> > 2) Before noon, at the nth interval, whenever the nth ball is removed,
> > there are 10^n -n balls remaining in the vase.
>
> That may seem paradoxical to you, but it is not a contradiction.
>
I tried to bring 1) and 2) closer by making the statements more
specific - apparently too close. Let me try rephrasing these and add
another.
1) Every ball added to the vase before noon will also be removed before
noon.
2) After the first balls are added there are always balls remaining in
the vase.
3) At no time before noon is the vase empty.
With the proper understanding regarding statement 1); namely, that only
a portion of balls remaining are removed at anytime before noon and
that statements 2) and 3) follow as a consequence of the fact that the
addition and removal process has no ending before noon - you are
correct that these statements, which on face seem paradoxical, are not
contradictory.
> >> > Does the infinite digit string forming the number "0.01002000300004..."
> >> > exist as a real number?
>
> >> Any digit string that contains a decimal digit at digit position n for
> >> each natural number n is a representation of a real number.
>
>
> > So does this mean that all of the infintesimals 0.000...1 through
> > 0.000...n the limit point of the sequences 0.1, 0.001, 0.0001, ...
> > through 0.n, 0.0n, 0.00n, ... respectively are also a real numbers all
> > taking the value zero? I thought these were not real numbers.
>
> Those are not real numbers. They are character strings.
I didn't think they were real numbers but I thought they met your
definition (which I believe may be a simplified description relative to
the Dedekind cuts and Gauchy sequences about which you have been
teaching me).
What is the difference between a digit string and a character sting?
How do the infintesimals 0.00...01 not meet the meaning of a "decimal
digit at digit position n for each position n"? Is it because there is
no last n in the naturals?
> You haven't
> explained what they are supposed to mean.
>
I don't understand this question. I am not sure what most irrational
numbers mean or even how they are obtained. These infintesimals were
the limit points of the sequences such as 0.1, 0.01, 0.001, ..., ...
and 0.2, 0.02, 0.002, ..., ... and includes 0.23, 0.023, 0.0023, ...,
... and in general 0.n, 0.0n, 0.00n, ..., ... for all n. I am not
trying to develop infintesimal math, I assume from your and Peter
Webb's responses that such numbers have been studied. Nevertheless, it
is clear as I presented them each successive number in the sequence is
1/10th the value of the previous number. As an aside, this may be
analogous to the time being reduced by 1/2 for each successive
interval. These sequences arose from the way I numbered balls in order
to include values less than 0.1. The limit points 0.00...01 through
0.000...0n (all equal to zero) arise as this sequence approaches
infinty in a similar manner as the limit point 0.333... arises from the
sequence 0.3, 0.33, 0.333, ....
<snip>
>
> > NTON =
> > 0.01001120001112223000011112222333340000011111222223333344444450000001111112222223333334444445555556000000011111112222222333333344444445555555666666667000000001111111122222222333333334444444455555555666666667777777780000000001111111112222222223333333334...
>
> > FACTND=
> > 0.11211232131223121313212343214312423142134132412334213412324132143142312424312413234123142143213414321423134213241243123454321543125423154213541325412353421534125324153214531425312452431524135234152314521435213451432514235134251324512435123445321453124...
>
> There are no infinitesimals here.
>
> > The infinite decimalic number above arises from taking factorials of
> > digits associated with each "n" arranged in descending order and
> > appending them to the decimal as in a sequence. This is shown in more
> > detail below. The number of added digits with each n is much larger
> > than 2^n.
>
> The details are not important. These are ordinary real numbers. Each is
> defined by an ordinary infinite series.
>
>
> > The mapping by the naturals of the infinite representation of decimalic
> > numbers NTON and FACTND seems to require going "through" at least an
> > infinte number of digits each natural number n. These don't seem like
> > real numbers to me. Furthermore, if the naturals can map the infinite
> > digit strings associated with NTON and FACTND I really don't understand
> > how they cannot map the reals?
>
> The mapping that you described earlier does not include these numbers,
> because your mapping covers only the reals that have terminating decimal
> representations. A number such as 1/3 = 0.3333... does not appear among
> the values represented by your mapping.
>
If by the mapping you are referring to the problem with the balls being
added before and the question being asked by noon, I disagree.
If you are referring to the mapping with columns that I presented in an
earlier post and for which I was recently being questioned by Stephen,
I agree that the representation in columnar form was poor particularly
since it did not even show pi.
Let me try to improve the representation (using rows instead of
columns) to better show what I intended with the mappings.
E) 0, 0.3, 0.33, ..., 0.33..(3_n), ..., 0.33...(3_n)... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
D) 0, 1, 2, ..., n, ..., ... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
C) 0, 0.3, 0.33, ..., 0.33...(3_n), ..., 0.33...(3_n)... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
\ \_____ |
\ \ |
B) 0, 3, 3, ..., d_n(n <- d_dec1/3), ..., ... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
A) 0, 1, 2, ..., n, ..., ... as n => oo
Where d_n(n <- d_dec1/3) is the nth digit of this digit string that
arises from the nth digit to the right of the decimal point of the
decimalic expression for 1/3 which is 0.33....
Where (3_n) represents the nth digit of 0.333....
Where 0.33...(3_n) is the finite decimal approximation to 0.333... with
n digits to the right of the decimal point.
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.
I don't see how the mapping shown above (which includes mapping the
infinite number of naturals to the decimalic numbers based on 0.333...
) does not include 0.333... which has no last "3" as n => oo.
Don Whitehurst
Don Whitehurst
You correctly pointed out that my columnar representation of the
mapping that I tried to describe in words was poor, in fact it did not
even include pi. I will try to form a better representation of the
mapping to demonstrate more precisely what I meant to communicate.
E) 3, 3.1, 3.14, ..., 0.3.14...(d_n), ..., 3.14...(d_n)... as n=>oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
/
D) 0, 1, 2, ..., n, ..., ... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
C) 3, 3.1, 3.14, ..., 3.14...(d_n), ..., 0.33...(d_n)... as n => oo
^ ^ ^ ^ ^ ^ ^
| | | | | | |
\
B) 3, 1, ..., d_n(n <- d_dec pi), ..., ... as n => oo
^ ^ ^ ^ ^ ^
| | | | | |
A) 1, 2, ..., n, ..., ... as n => oo
Where d_n(n <- d_dec pi) is the nth digit of this digit string that
arises from the nth digit of the decimalic expression for pi which is
3.14....
Where (d_n) represents the nth digit of pi = 3.14... and where (d_n)
has the same numerical value as the the nth digit of pi. For example,
(d_6) is the sixth digit of pi = 3.14159265... or "9".
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
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.
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.
Don
Dave Seaman
> I apologize for not responding sooner, my computer lost its connection
> to the internet.
> Dave Seaman wrote:
>> >> Why is it impossible to add all the finite-numbered balls without also
>> >> adding a ball marked aleph_0?
>> > Before noon each nth interval is represented by a natural number and
>> > since there is no largest natural number at any time before noon (nth
>> > interval) additonal balls always remain to be added and to be removed.
>> This does not answer the question. Why is it impossible to add all the
>> finite-numbered balls without also adding a ball marked aleph_0?
> My response answered the first part of the why all the balls could not
> be added to the vase before noon which I thought should be adequate for
> the answer; but apparently not.
It was not a two-part question.
> Since there is no last (n+1)th interval that is 1/2 of the time
> remaining from 1/(2^n) minutes to noon just as there is no last
> natural, there is no last time of addition for the balls. Only by
> noon, the at actual infinity was it possible to add the all the balls
> which must include those with an infinite number of digits as well.
There is no ball that is added at noon. Noon is simply the earliest time
at which we can say all the balls have been added.
> I treated the time noon as being infinity and used aleph_0 only because
> you said that aleph_0 was the name of infinity for the naturals. I
> don't think I said there was a ball marked aleph_0 added. But it may be
> a proper inference from what I did say.
Can we agree, then, that there is no ball marked oo or aleph_0? Each
ball is marked with a natural number in the original version, or with a
terminating decimal in your version. Each natural number maps to a
terminating decimal.
> At noon I think there are an infinite number of balls added all having
> decimals each with no last digit printed on them (perhaps this is only
> aleph_0 balls but I think it may be 10^ aleph_0 balls but I am just
> learning these concepts). It seems to me that any decimalic number
> with no last digit must be either a repeating rational or an irrational
> number. If there are decimalic numbers with no last digit that are not
> a repeating rational or irrational number, please provide a me with an
> example of such a number.
There is no ball added at noon. There is no ball added at any time that
has a nonterminating decimal on it. When would such a ball be added?
>> >> Where do you see a paradox? And why add assumptions that are not present
>> >> in the problem statement?
>> > The paradox arises from the statements 1) and 2):
>> > 1) Every ball added to the vase at the nth interval before noon will be
>> > removed before noon (at some later ( n+ r_kl )th interval).
>> > 2) Before noon, at the nth interval, whenever the nth ball is removed,
>> > there are 10^n -n balls remaining in the vase.
>> That may seem paradoxical to you, but it is not a contradiction.
> With the proper understanding regarding statement 1); namely, that only
> a portion of balls remaining are removed at anytime before noon and
> that statements 2) and 3) follow as a consequence of the fact that the
> addition and removal process has no ending before noon - you are
> correct that these statements, which on face seem paradoxical, are not
> contradictory.
>> >> > Does the infinite digit string forming the number "0.01002000300004..."
>> >> > exist as a real number?
>> >> Any digit string that contains a decimal digit at digit position n for
>> >> each natural number n is a representation of a real number.
Let me put it a different way. Each decimal digit string uniquely
specifies a Cauchy sequence of rational numbers, and each such Cauchy
sequence determines a unique real number. Does that answer your
question?
> What is the difference between a digit string and a character sting?
> How do the infintesimals 0.00...01 not meet the meaning of a "decimal
> digit at digit position n for each position n"? Is it because there is
> no last n in the naturals?
A decimal digit string has, for each natural number n, a digit at
position n following the decimal point. There are no infinite digit
positions. The Cauchy sequence corresponding to a digit string is
affected only by the digits appearing at finite positions in the string.
>> You haven't
>> explained what they are supposed to mean.
In other words, what Cauchy sequence does 0.00...01 correspond to? Am I
understanding your meaning correctly? Do you intend to say that there
are infinitely many 0's before the 1? Then my question stands. What
real number is this?
> I don't understand this question. I am not sure what most irrational
> numbers mean or even how they are obtained. These infintesimals were
> the limit points of the sequences such as 0.1, 0.01, 0.001, ..., ...
> and 0.2, 0.02, 0.002, ..., ... and includes 0.23, 0.023, 0.0023, ...,
You keep talking about infinitesimals. The definitions I have mentioned
(Dedekind cuts or Cauchy sequences) are from standard analysis and have
nothing to do with infinitesimals. A nonterminating digit string
represents a perfectly standard real number, with no infinitesimals
involved.
Furthermore, I have explained before that adding infinitesimals to the
discussion will not enable you to claim the reals are countable. The
reals are already uncountable, even without considering infinitesimals.
> ... and in general 0.n, 0.0n, 0.00n, ..., ... for all n. I am not
> trying to develop infintesimal math, I assume from your and Peter
> Webb's responses that such numbers have been studied. Nevertheless, it
> is clear as I presented them each successive number in the sequence is
> 1/10th the value of the previous number. As an aside, this may be
> analogous to the time being reduced by 1/2 for each successive
> interval. These sequences arose from the way I numbered balls in order
> to include values less than 0.1. The limit points 0.00...01 through
> 0.000...0n (all equal to zero) arise as this sequence approaches
> infinty in a similar manner as the limit point 0.333... arises from the
> sequence 0.3, 0.33, 0.333, ....
Taking 1/10 of a number with a terminating decimal representation gives
you another number with a terminating decimal representation. This is
the basis of a proof by induction that for every n, the n'th ball is
marked with a terminating decimal string. You never do get to the point
where even a single ball has a nonterminating decimal associated with it.
>> > The mapping by the naturals of the infinite representation of decimalic
>> > numbers NTON and FACTND seems to require going "through" at least an
>> > infinte number of digits each natural number n. These don't seem like
>> > real numbers to me. Furthermore, if the naturals can map the infinite
>> > digit strings associated with NTON and FACTND I really don't understand
>> > how they cannot map the reals?
A nonterminating decimal represents a perfectly standard real number, but
it does not represent a label on any of the balls that is added to the
vase before noon according to your scheme. Your labels account for only
a countable subset of the real numbers.
>> The mapping that you described earlier does not include these numbers,
>> because your mapping covers only the reals that have terminating decimal
>> representations. A number such as 1/3 = 0.3333... does not appear among
>> the values represented by your mapping.
> If by the mapping you are referring to the problem with the balls being
> added before and the question being asked by noon, I disagree.
> If you are referring to the mapping with columns that I presented in an
> earlier post and for which I was recently being questioned by Stephen,
> I agree that the representation in columnar form was poor particularly
> since it did not even show pi.
The problem was not with the way you arranged your columns. Any mapping
of a countable set to the reals is going to omit almost all of the reals.
The only question is which ones.
Even if you change your mapping in such a way that some nonterminating
decimals are included in the sequence (which you have not done yet), the
Cantor diagoal proof still shows that you cannot possibly include all of
them.
> Let me try to improve the representation (using rows instead of
> columns) to better show what I intended with the mappings.
Pointless. You still fail to understand what uncountability of the reals
means.
> I don't see how the mapping shown above (which includes mapping the
> infinite number of naturals to the decimalic numbers based on 0.333...
> ) does not include 0.333... which has no last "3" as n => oo.
Your mapping includes the number a_n = 1/3 * (1-1/10^n) for each n, but
a_n differs from 1/3 by 1/(3*10^n) for each n.
You seem to think that all sets are closed. That is, you think if x is a
limit point of a set A, then x is necessarily a member of A. Sets that
have this property are called closed sets.
There are lots of sets that are not closed. For example, the open
interval (0,1) = { x in R : 0 < x < 1 } is not a closed set. The numbers
0 and 1 are both limit points of the set, but they are not members.
Similarly, if f: N -> R is an arbitrary mapping, it is a mistake to claim
without proof that f(N), the image of f, must be closed. That is exactly
what you have been claiming. Merely because 1/3 is a limit point of
f(N), it does not follow that 1/3 is a member of f(N).
ste...@nomail.com
<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
imagin...@despammed.com
Don Whitehurst wrote:
> What is the difference between a digit string and a character sting?
> How do the infintesimals 0.00...01 not meet the meaning of a "decimal
> digit at digit position n for each position n"? Is it because there is
> no last n in the naturals?
Basically yes (you've answered your own question). The idea of a
nonterminating decimal fraction is that it, uh, has no end (which is
what 'nonterminating' means, of course). For some reason, lots of
people have terrible difficulty with this - they have never (obviously)
in real life encountered something which extended linearly to left and
to right, without having an end at the left and an end at the right. So
when we say, consider the naturals as the unending sequence
0, 1, 2, 3, 4, ...
where the dots represent the fact that this extends endlessly to the
right, all goes well for about one or two sentences, then they start
talking about the rightmost element. Well, with luck, you can see that
this is nonsense.
Incidentally, your first question above isn't _quite_ right. Dave
Seaman said "those are character strings", but of course so are the
valid representations of real numbers. He might more helpfully have
said "Those are merely character strings (with no interpretation as
real numbers)". The fact that one can string symbols together is no
guarantee that they mean anything, as witness large quantities of word
salad one sees in sci.math threads.
Brian Chandler
http://imaginatorium.org
Don Whitehurst
>
> There is no ball that is added at noon. Noon is simply the earliest time
> at which we can say all the balls have been added.
>
Are you then saying that by noon an infinite number of balls have been
added?
If so, by noon an infinite number of balls have been added: where they
were added many at a time during each interval and where one ball is
immediately removed for each of the infinite nth time intervals
corresponding to the infinite number of naturals.
Clearly, this should be impossible since there is no last natural.
Similarly there is no time before noon when all of the balls could have
been added.
Nevertheless, if one assumes that by noon an infinite number of balls
have been added (where the balls were added many at a time during each
interval and where one ball was immediately removed for each of the
infinite nth time intervals (before noon) corresponding to the infinite
number of naturals associated one each with the removal of the balls),
I believe it follows that balls with an infinite number of digits must
also have been added. Since the only way I can conceptually understand
the completion of the naturals (adding all the balls) is to reach
infinity, it would seem to me under such circumstance when infinity was
reached there must also have been added an infinite number of balls
each having an infinite number of digits. The one at a time removal
of balls "n" corresponds to the balls being added at the nth interval
having "n" digits. If all the balls are added, an infinite number of
them corresponding to all the naturals (n reahed infinity) must have
been added and some of the balls added must have had an infinite number
of digits since there is one digit for each nth interval at which the
nth ball is removed.
> > I treated the time noon as being infinity and used aleph_0 only because
> > you said that aleph_0 was the name of infinity for the naturals. I
> > don't think I said there was a ball marked aleph_0 added. But it may be
> > a proper inference from what I did say.
>
> Can we agree, then, that there is no ball marked oo or aleph_0? Each
> ball is marked with a natural number in the original version, or with a
> terminating decimal in your version. Each natural number maps to a
> terminating decimal.
>
I am not sure how we got to noon (infinity) without adding such a ball.
But let's assume you are correct. By noon, all balls have been added
(an infinite number of intervals has occurred n->oo and an infinite
number of balls corresponding to the infinite number of naturals has
been removed). Since all the balls were added they must include some
with an infinite number of digits. For each interval n, all the ball
added at interval n have n digits. Just as there is no last natural,
there must be balls with no last digit. If all the balls were added
these balls must also have been added. I believe these are the
nonterminating decimals.
> >> >> > Does the infinite digit string forming the number "0.01002000300004..."
> >> >> > exist as a real number?
>
> >> >> Any digit string that contains a decimal digit at digit position n for
> >> >> each natural number n is a representation of a real number.
>
> Let me put it a different way. Each decimal digit string uniquely
> specifies a Cauchy sequence of rational numbers, and each such Cauchy
> sequence determines a unique real number. Does that answer your
> question?
>
It probably would if I understood what constitutes a Cauchy sequence of
natural numbers. Are the sequences that I wrote (eg 0.1, 0.01, ... )
Cauchy sequences all of which determine the unique real number zero?
Zero seems to be the proper real limit of the sequence.
> > What is the difference between a digit string and a character sting?
> > How do the infintesimals 0.00...01 not meet the meaning of a "decimal
> > digit at digit position n for each position n"? Is it because there is
> > no last n in the naturals?
>
> A decimal digit string has, for each natural number n, a digit at
> position n following the decimal point. There are no infinite digit
> positions. The Cauchy sequence corresponding to a digit string is
> affected only by the digits appearing at finite positions in the string.
>
That sounds reasonable and I think I comprehend what you are saying.
Nevertheless, I am not sure how it applies to FACTND or NTON, which
both have some of the infinite digit position characteristics that I
thought you described above. You indicated previously that FACTND and
NTON each describe a real number.
> >> You haven't
> >> explained what they are supposed to mean.
> In other words, what Cauchy sequence does 0.00...01 correspond to?
The sequence (0.1, 0.01, ... ); which may or may not be a Cauchy
sequence.
Am I
> understanding your meaning correctly? Do you intend to say that there
> are infinitely many 0's before the 1?
Good question. This question shows me that the representation I wrote
for this number does not match my interpretation of a decimal with an
infinite number of digits. Perhaps 0.000...(0_n-1)1... is better where
the interior ... indicates that n-1 digits (all zeroes) are present
followed by a 1 and the trailing ... indicates that n => oo.
> Then my question stands. What
> real number is this?
>
The real number from the sequence (0.1, 0.01, ... ) as the sequence
approaches infinity.
> You keep talking about infinitesimals. The definitions I have mentioned
> (Dedekind cuts or Cauchy sequences) are from standard analysis and have
> nothing to do with infinitesimals. A nonterminating digit string
> represents a perfectly standard real number, with no infinitesimals
> involved.
>
> Furthermore, I have explained before that adding infinitesimals to the
> discussion will not enable you to claim the reals are countable. The
> reals are already uncountable, even without considering infinitesimals.
>
I indicated previously that I entered this discussion to extend my
understanding with respect decimalic representations that I had
started with Timothy Little, Josh Purinton, and Robert Kolker where I
was trying to discern differences if any at limit points as n => oo.
It strikes me that mathematicians discern characteristic differences
between the two such types of numbers; although I am unaware of any
formal accepted representation for them. Without such a difference
being present, I do believe that the reals would be countable. Since
they are not countable, I am trying to understand what makes this
characteristic difference.
<snip>
... in a similar manner as the limit point 0.333... arises from the
> > sequence 0.3, 0.33, 0.333, ....
>
> Taking 1/10 of a number with a terminating decimal representation gives
> you another number with a terminating decimal representation. This is
> the basis of a proof by induction that for every n, the n'th ball is
> marked with a terminating decimal string. You never do get to the point
> where even a single ball has a nonterminating decimal associated with it.
>
Similarly, each successive decimalic number of the sequence has one
extra digit. This could be the basis of a proof by induction that just
as there are an infinite number of naturals, the number of digits to
the right of the decimal point must also be infinite (the number of
digits to the right of the decimal point in the number with no threes
(0.) is zero, the number of digits to the right of the decimal point in
the next number (0.3) is 1, the number of digits to the right of the
decimal point in the next number (0.33) is 2, the number of digits to
the right of the decimal point of the next number (0.333) is 3, ...).
There is no last digit associated with this sequence. My main question
in this posting deals with how this differs from a nonterminating
decimal. I have always regarded the nonterminating decimalic
representations as indicating that there is no last digit, and hence
there are an infinite number of digits for such numbers as 0.333...,
and 3.1415.... In other words, 0.333... is the decimalic
representation from the infinite process obtained by using long
division for 1/3 where there is no last digit.
>
> A nonterminating decimal represents a perfectly standard real number, but
> it does not represent a label on any of the balls that is added to the
> vase before noon according to your scheme. Your labels account for only
> a countable subset of the real numbers.
>
> >> The mapping that you described earlier does not include these numbers,
> >> because your mapping covers only the reals that have terminating decimal
> >> representations. A number such as 1/3 = 0.3333... does not appear among
> >> the values represented by your mapping.
>
>
> > If by the mapping you are referring to the problem with the balls being
> > added before and the question being asked by noon, I disagree.
>
> > If you are referring to the mapping with columns that I presented in an
> > earlier post and for which I was recently being questioned by Stephen,
> > I agree that the representation in columnar form was poor particularly
> > since it did not even show pi.
>
> The problem was not with the way you arranged your columns. Any mapping
> of a countable set to the reals is going to omit almost all of the reals.
> The only question is which ones.
>
The reason I switched from columns to rows was to try to show a
representation that produced a similarity between a) the trailing ...,
... after the list of naturals that indicated 1) there were more
numbers and 2) there was no last natural number and b) the decimalic
representaion of ..., 0.33...(3_n)... that indicated 1) there are more
terminating decimals approacing 0.333, and 2) the decimalic digits of
1/3 are being added one by one for each natural and that there is no
last digit of the decimalic representation of 0.333.... Again you
showed me my representation was not as good as it should have been.
The 0.33...n would probably have been better as 0.33...(3_n) and that
0.33... as n => oo should may have best been written as 0.33...(3_n)...
as n => oo.
> Even if you change your mapping in such a way that some nonterminating
> decimals are included in the sequence (which you have not done yet), the
> Cantor diagoal proof still shows that you cannot possibly include all of
> them.
>
While I think I understand the Cantor proof , this problem with the
balls makes me question one aspect of it. But I am sure you have
readily dealt with such concerns in the past.
> > Let me try to improve the representation (using rows instead of
> > columns) to better show what I intended with the mappings.
>
> Pointless. You still fail to understand what uncountability of the reals
> means.
>
> > I don't see how the mapping shown above (which includes mapping the
> > infinite number of naturals to the decimalic numbers based on 0.333...
> > ) does not include 0.333... which has no last "3" as n => oo.
>
> Your mapping includes the number a_n = 1/3 * (1-1/10^n) for each n, but
> a_n differs from 1/3 by 1/(3*10^n) for each n.
>
Nevertheless there are an infinite number of naturals n and there must
also be an infinite number of a_ns (one a_n corresponding to each
natural n), and since each a_n has n digits, there must be an infinite
number of digits associated with the a_ns.
> You seem to think that all sets are closed. That is, you think if x is a
> limit point of a set A, then x is necessarily a member of A. Sets that
> have this property are called closed sets.
>
> There are lots of sets that are not closed. For example, the open
> interval (0,1) = { x in R : 0 < x < 1 } is not a closed set. The numbers
> 0 and 1 are both limit points of the set, but they are not members.
>
> Similarly, if f: N -> R is an arbitrary mapping, it is a mistake to claim
> without proof that f(N), the image of f, must be closed. That is exactly
> what you have been claiming. Merely because 1/3 is a limit point of
> f(N), it does not follow that 1/3 is a member of f(N).
>
While you may be mostly correct, I think my misunderstanding instead
arises from what a limit point is and what the nonterminating decimalic
representation of numbers means.
Don Whitehurst
Dave Seaman
> Dave Seaman wrote:
>> > I treated the time noon as being infinity and used aleph_0 only because
>> > you said that aleph_0 was the name of infinity for the naturals. I
>> > don't think I said there was a ball marked aleph_0 added. But it may be
>> > a proper inference from what I did say.
Aleph_0 is identical to the set of all natural numbers. Since no set is
a member of itself, it logically follows that aleph_0 cannot *be* a
natural number, else it would have to be a member of itself.
>> Can we agree, then, that there is no ball marked oo or aleph_0? Each
>> ball is marked with a natural number in the original version, or with a
>> terminating decimal in your version. Each natural number maps to a
>> terminating decimal.
> I am not sure how we got to noon (infinity) without adding such a ball.
There is no ball added at noon. Each ball whose number is less than
aleph_0 is added before noon, and no other balls are added.
> But let's assume you are correct. By noon, all balls have been added
> (an infinite number of intervals has occurred n->oo and an infinite
> number of balls corresponding to the infinite number of naturals has
> been removed). Since all the balls were added they must include some
> with an infinite number of digits.
How many balls are there having only a finite number of digits? (Pick
one):
(a) There are infinitely many balls, each having only finitely many
digits.
(b) There are only finitely many balls, each having only finitely many
digits.
If you choose (a), then you have contradicted yourself, because you have
just agreed that it is possible to add infinitely many balls without any
of them having an infinite number of digits.
If you choose (b), then you have a problem. If there are only finitely
many balls having only finitely many digits, then there must be a largest
one, say M. How many digits are there in M+1?
So which is it? You need to answer this question in order for this
discussion to make any progress.
>> Let me put it a different way. Each decimal digit string uniquely
>> specifies a Cauchy sequence of rational numbers, and each such Cauchy
>> sequence determines a unique real number. Does that answer your
>> question?
> It probably would if I understood what constitutes a Cauchy sequence of
> natural numbers. Are the sequences that I wrote (eg 0.1, 0.01, ... )
> Cauchy sequences all of which determine the unique real number zero?
> Zero seems to be the proper real limit of the sequence.
Yes. It turns out that the Cauchy sequences are precisely those that
converge to some real number, although that is not exactly the
definition. It's actually a consequence of the way the reals are
defined.
>> > What is the difference between a digit string and a character sting?
Character strings can contain characters that are not digits.
>> > How do the infintesimals 0.00...01 not meet the meaning of a "decimal
>> > digit at digit position n for each position n"? Is it because there is
>> > no last n in the naturals?
Infinitesimals are not represented as "super-long" decimal digit strings
like that, mainly because such strings are not powerful enough to
represent all the infinitesimals. Entirely different methods are used.
But that need not concern us here. I simply do not accept that strings
like 0.00...01 have any recognizable meaning, either as reals or as
infinitesimals.
The answer to your question is yes. The string you wrote down clearly
has a last digit and therefore it is "terminating", although it is
infinitely long. The only digit strings that make sense as real numbers
are the ones whose digit positions are all finite (though there may be
infinitely many of them).
>> A decimal digit string has, for each natural number n, a digit at
>> position n following the decimal point. There are no infinite digit
>> positions. The Cauchy sequence corresponding to a digit string is
>> affected only by the digits appearing at finite positions in the string.
> That sounds reasonable and I think I comprehend what you are saying.
> Nevertheless, I am not sure how it applies to FACTND or NTON, which
> both have some of the infinite digit position characteristics that I
> thought you described above. You indicated previously that FACTND and
> NTON each describe a real number.
Neither of those has a "last" digit, which makes them unlike the
"superlong" string you wrote above.
>> >> You haven't
>> >> explained what they are supposed to mean.
>> In other words, what Cauchy sequence does 0.00...01 correspond to?
> The sequence (0.1, 0.01, ... ); which may or may not be a Cauchy
> sequence.
That is a Cauchy sequence, and the real number that it represents is
simply 0. You can write it 0.000... if you like, but the string has no
last digit and it doesn't contain any nonzero digits.
> I indicated previously that I entered this discussion to extend my
> understanding with respect decimalic representations that I had
> started with Timothy Little, Josh Purinton, and Robert Kolker where I
> was trying to discern differences if any at limit points as n => oo.
> It strikes me that mathematicians discern characteristic differences
> between the two such types of numbers; although I am unaware of any
> formal accepted representation for them. Without such a difference
> being present, I do believe that the reals would be countable. Since
> they are not countable, I am trying to understand what makes this
> characteristic difference.
The real numbers are uncountable, but the rationals are a countable set
with the property that each real number can be represented arbitrarily
closely by rationals. We say that the rationals are "dense" in the
reals, and therefore the reals are an example of a topological space that
has a countable dense subset. Such a space is called "separable".
(Don't ask me why that particular word was chosen to describe the
property of having a countable dense subset; it just was.)
If you still insist on thinking that every set is topologically closed
(i.e., contains all its limit points), then it's quite understandable
that you must be having a hard time distinguishing the rationals from the
reals. After all, the reals are the closure of the rationals in R.
Therefore, it's precisely the fact that the rationals are not a closed
set in R that is a stumbling block.
[ ... snip ... ]
>> > I don't see how the mapping shown above (which includes mapping the
>> > infinite number of naturals to the decimalic numbers based on 0.333...
>> > ) does not include 0.333... which has no last "3" as n => oo.
Being a limit of a sequence is not the same as being a member of that
sequence.
>> There are lots of sets that are not closed. For example, the open
>> interval (0,1) = { x in R : 0 < x < 1 } is not a closed set. The numbers
>> 0 and 1 are both limit points of the set, but they are not members.
> While you may be mostly correct, I think my misunderstanding instead
> arises from what a limit point is and what the nonterminating decimalic
> representation of numbers means.
A nonterminating decimal digit string is naturally associated with a
sequence of rationals. The number represented by the whole string is the
limit of that sequence, but is not a member.
Don't forget to pick one of the choices (a) or (b) from above when you
reply.
Don Whitehurst
<snip>
DW
> > 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.
>
I agree with the first statement; but the mapping is with the naturals
for which there is no last natural (an infinite number of them). Since
this mapping goes to successive numbers approximating pi each with one
additional digit drawn from pi, then the no last natural goes to a
decimalic representation with no last digit.
> > 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.
>
Yes, I assumed base 10 - hence the word decimalic in the same sentence.
Sorry if my language seemed to be generalizing beyond this implied
limitation about base 10 and drawing upon the concept of remainder from
long hand division (base 10). Certainly the use of (the whole) was a
poor word choice if not downright incorrect.
> > 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.
>
Column B as presented cannot have pi (Row four, Column B should be
3.141). In the interest of continuing this discussion, I will assume
this typographical error is not present.
Let me begin by saying I never meant to say pi was necessarily in
column B.
The no last natural number corresponds to the decimalic representation
with no last digit where each of those digits are drawn from the
nonterminating decimalic representation of pi which also has no last
digit. I believe the nonterminating decimalic representation of pi
(3.14...) is pi and this representation of pi displays no differences,
that I can discern, from (3.14...(d_n<-pi)...) the decimalic
representation with no last digit where every successive digit in the
representation is drawn from the nonterminating decimalic
representation of pi.
A B
1, <--> 3,
2, <--> 3.1,
3, <--> 3.14,
..., <--> ...,
n, <--> 3.14...(d_n<-pi),
..., <--> ...,
... <--> 3.14...(d_n<-pi)... ?=? 3.14... ?=? pi
no no
last <--> last
natural digit
Here column A represents both the natural numbers and the number of
digits of the decimalic representations associated with column B. In
neither mapping is there a last position. No last natural and no last
digit in the decimalic representation. Both columns A & B have an
infinite number of elements but neither has a terminal element;
nevertheless, a decimalic representation for the no last digit
(infinite number of digits) does exist.
The fact that "... " is the representation for both the concept of no
last natural and the infinite number of naturals but that no numeric
representation exists, does not preclude a decimalic representation (
3.14...(d_n<-pi)... ) from exisiting. Particularly where the decimalic
representation ( 3.14...(d_n<-pi)... ) also corresponds to the concept
of no last digit associated with the no last natural, and corresponds
to an infinite number of digits (there is always one more)
corresponding to an infinite number of naturals.
To me it looks exactly like the decimalic representaion of pi, but I
gather you believe it actually differs from pi.
If the decimalic representation 3.14...(d_n<-pi)... in which there is
no last digit is not equal to pi and you can demonstrate or prove this
point to me, we will have no disagreement. How is it possible for the
decimalic representaion with no last digit formed from all of the
consecutive decimalic digits of pi to not be a part of column B?
> 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.
>
When the mapping is infinite, as are the naturals (no endpoint), I
cannot understand how there can possibly not be an infinite number of
digits (no ending digit) in column B.
Don Whitehurst
Virgil
"Don Whitehurst" <whit...@umn.edu> wrote:
> Dave Seaman wrote:
>
> >
> > There is no ball that is added at noon. Noon is simply the
> > earliest time at which we can say all the balls have been added.
> >
>
> Are you then saying that by noon an infinite number of balls have
> been added?
>
> If so, by noon an infinite number of balls have been added: where
> they were added many at a time during each interval and where one
> ball is immediately removed for each of the infinite nth time
> intervals corresponding to the infinite number of naturals.
>
> Clearly, this should be impossible since there is no last natural.
> Similarly there is no time before noon when all of the balls could
> have been added.
>
> Nevertheless, if one assumes that by noon an infinite number of balls
> have been added (where the balls were added many at a time during
> each interval and where one ball was immediately removed for each of
> the infinite nth time intervals (before noon) corresponding to the
> infinite number of naturals associated one each with the removal of
> the balls), I believe it follows that balls with an infinite number
> of digits must also have been added.
Since no natural number has more than finitely many digits in any n-ary
representation, where do those infinitely many digited balls come from?
They cannot come from any natural number!
Virgil
"Don Whitehurst" <whit...@umn.edu> wrote:
> A B
> 1, <--> 3,
> 2, <--> 3.1,
> 3, <--> 3.14,
> ..., <--> ...,
> n, <--> 3.14...(d_n<-pi),
> ..., <--> ...,
> no no
> last <--> last
> natural digit
As there is no number representing a last number in column A, there is
no number representing pi by no last digit in column B.
>
> Here column A represents both the natural numbers and the number of
> digits of the decimalic representations associated with column B. In
> neither mapping is there a last position. No last natural and no last
> digit in the decimalic representation. Both columns A & B have an
> infinite number of elements but neither has a terminal element;
> nevertheless, a decimalic representation for the no last digit
> (infinite number of digits) does exist.
But not IN the sequence of column B, only as the limiting value of such
a sequence.
> When the mapping is infinite, as are the naturals (no endpoint), I
> cannot understand how there can possibly not be an infinite number of
> digits (no ending digit) in column B.
Which (necessarily finite) natural number in column A corresponds to (is
in the same row as) pi-to-infinitely-many-digits in column B?
And if so, what is in the next row of the table, below that?
imagin...@despammed.com
> <snip>
> DW
> > > 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.
> >
Please read your own statements _very_ carefully!
>
> I agree with the first statement; but the mapping is with the naturals
> for which there is no last natural*** (an infinite number of them).
*** Right: "there is no last natural". The "last natural" does not
exist.
Since
> this mapping goes to successive numbers approximating pi each with one
> additional digit drawn from pi, then the no last natural goes *** to a
> decimalic representation with no last digit.
*** Huh? This statement doesn't quite fit normal grammar, but what does
it mean to say "the nonexistent something does something"? Is this part
of the mapping? The last natural does not exist, but it is still mapped
to something?
Anyway, just an aside. Pleease stop using the nonexistent word
"decimalic". The adjective is "decimal". Using standard words for
things will help you get the attention of people enthusiastic to help.
Brian Chandler
http://imaginatorium.org
Randy Poe
This is all correct.
> nevertheless, a decimalic representation for the no last digit
> (infinite number of digits) does exist.
I can't understand this. What you possibly mean by
the DECIMAL representation of the phrase "no last digit"?
> If the decimalic representation 3.14...(d_n<-pi)... in which there is
> no last digit is not equal to pi and you can demonstrate or prove this
> point to me, we will have no disagreement. How is it possible for the
> decimalic representaion with no last digit formed from all of the
> consecutive decimalic digits of pi to not be a part of column B?
All of the things in column B have the property that:
digits 1-n are the same as digits 1-n of pi
digits n+1, n+2, ... are 0.
Obviously pi does not have the property that there are
an infinite number of zeros beginning at some position n
(if it did, it would be rational). Yet everything in column
B has that property. Hence pi is not in column B.
- Randy
Minus XVII
for establishing pi; only evaluating it.
since exp(pi*i) = -1,
it can be represented in base-e, so long
as one goes to the complex-valued logarithms;
is that correct?
one can just as easily say that
the surface-area of the sphere (or circumference) is unit,
then the diameter is "pith."
> All of the things in column B have the property that:
> digits 1-n are the same as digits 1-n of pi
> digits n+1, n+2, ... are 0.
>
> Obviously pi does not have the property that there are
> an infinite number of zeros beginning at some position n
> (if it did, it would be rational). Yet everything in column
> B has that property. Hence pi is not in column B.
--Hemp for Haemarrhoids (Bogart that Poultice, Friend) !!
http://members.tripod.com/~ame rican_almanac
Don Whitehurst
> In article <1126068402.2...@g47g2000cwa.googlegroups.com>,
> "Don Whitehurst" <whit...@umn.edu> wrote:
>
>
> > A B
> > 1, <--> 3,
> > 2, <--> 3.1,
> > 3, <--> 3.14,
> > ..., <--> ...,
> > n, <--> 3.14...(d_n<-pi),
> > ..., <--> ...,
>
> > no no
> > last <--> last
> > natural digit
>
> As there is no number representing a last number in column A, there is
> no number representing pi by no last digit in column B.
> >
You changed the content of my columns by deleting the last row
containing mathematical representations. This may have been due to
thinking you were correcting a mistake in what I wrote but you should
take credit for the change, not give the impression that it matches
what I originally wrote. Perhaps you were mislead by the words at the
end so I will try to make them more clear.
A B
1, <--> 3,
2, <--> 3.1,
3, <--> 3.14,
..., <--> ...,
n, <--> 3.14...(d_n<-pi),
..., <--> ...,
... <--> 3.14...(d_n<-pi)...
Where the last " ... "in Column A indicates that no last natural
exists,
and where the trailing " ... " in the decimal representation "
3.14...(d_n<-pi)... " indicates that no last digit exists.
Note also that the " ..., " indicates an additional unspecified number
of elements of each sequence (before and after n in Column A and before
and after 3.14...(d_n<-pi) in Column B).
Similarly, the interior " ... " present in both "3.14...(d_n<-pi) "
and in " 3.14...(d_n<-pi)... " indicates an additional unspecified
number of digits within the decimal and the decimal representation
following 3.14 (the first three digit of pi).
I am not claiming this decimal representation " 3.14...(d_n<-pi)... "
is pi, I am merely claiming that it is a perfectly valid decimal
representation for the sequence in Column B as n approaches infinity.
It is certainly as valid as the " ... " being understood to be the
continuation of the sequence in Column A as opposed to the achievement
of actual infinity ( n becoming infinite ). If you like, I could
change this decimal repesentation somewhat to more clearly
differentiate that it possibly has a nonstandard interpretation.
> > Here column A represents both the natural numbers and the number of
> > digits of the decimalic representations associated with column B. In
> > neither mapping is there a last position. No last natural and no last
> > digit in the decimalic representation. Both columns A & B have an
> > infinite number of elements but neither has a terminal element;
> > nevertheless, a decimalic representation for the no last digit
> > (infinite number of digits) does exist.
>
> But not IN the sequence of column B, only as the limiting value of such
> a sequence.
>
The decimal representation 3.14...(d_n<-pi)::: is in fact as completely
immersed IN the sequence of Column B as the final ... is IN the
sequence of Column A.
> > When the mapping is infinite, as are the naturals (no endpoint), I
> > cannot understand how there can possibly not be an infinite number of
> > digits (no ending digit) in column B.
>
Both columns have an infinite number of elements. There is no last
natural number in Column A just as there is a decimal representation
in Column B with no last digit.
> Which (necessarily finite) natural number in column A corresponds to (is
> in the same row as) pi-to-infinitely-many-digits in column B?
>
> And if so, what is in the next row of the table, below that?
Both the representations " ... " for Column A and "
3.14...(d_n<-pi)::: " for Column B each indicate in thier own way that
their respective columns each have elements extending successively
without end.
Don
Don Whitehurst
> Don Whitehurst wrote:
> > >ste...@nomail.com wrote:
> > >> Don Whitehurst <whit...@umn.edu> wrote:
>
> > <snip>
> > DW
> > > > 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.
> > >
>
> Please read your own statements _very_ carefully!
> >
A B
1, <--> 3,
2, <--> 3.1,
3, <--> 3.14,
..., <--> ...,
n, <--> 3.14...(d_n<-pi),
..., <--> ...,
... <--> 3.14...(d_n<-pi):::
Where the last " ... "in Column A indicates that no last natural
exists,
and where the trailing " ::: " in the decimal representation "
3.14...(d_n<-pi)::: " indicates that no last digit exists.
Note also that the " ..., " indicates an additional unspecified number
of elements of each sequence (before and after n in Column A and before
and after 3.14...(d_n<-pi) in Column B).
Similarly, the interior " ... " present in both "3.14...(d_n<-pi) "
and in " 3.14...(d_n<-pi):::" indicates an additional unspecified
number of digits within the decimal and the decimal representation
following 3.14 (the first three digit of pi).
The fact that 3.14...(d_n<-pi) is the finite decimal approximation to
pi with n digits exactly matching the first n digits of pi, does not
preclude the decimal representation 3.14...( d_n<-pi)::: from being a
valid representation of this sequence having decimals each having one
more successive decimal digit (taken from the respective digit of pi)
added to the right of the previous decimal number. In other words,
like the naturals having no last number (through the representation "
... ") this sequence has no decimal with the most decimal digits
(through the decimal representation " 3.14...( d_n<-pi)::: ") - there
is always a decimal number in this sequence with one more decimal
digit. Just as the number of naturals in the sequence of naturals
approaches infinity and for which there are considered infinitely many
so does the number of decimal digits associated with the infinite
number of decimal numbers.
> > I agree with the first statement; but the mapping is with the naturals
> > for which there is no last natural*** (an infinite number of them).
>
> *** Right: "there is no last natural". The "last natural" does not
> exist.
>
> Since
> > this mapping goes to successive numbers approximating pi each with one
> > additional digit drawn from pi, then the no last natural goes *** to a
> > decimalic representation with no last digit.
>
> *** Huh? This statement doesn't quite fit normal grammar, but what does
> it mean to say "the nonexistent something does something"? Is this part
> of the mapping? The last natural does not exist, but it is still mapped
> to something?
The "no last natural" means that the naturals do not terminate; there
is always one more succesive natural number. Similarly, the no last
decimal digit means the number of decimal digits does not terminate.
>
> Anyway, just an aside. Pleease stop using the nonexistent word
> "decimalic". The adjective is "decimal". Using standard words for
> things will help you get the attention of people enthusiastic to help.
>
Thank you Brian for pointing out my ignorance that decimal is also an
adjective. It seems obvious now that I think about the many phrases or
compound words, such as decimal notation, decimal place, and decimal
point, where decimal is used as an adjective. I am surprised that I
had not been corrected earlier in my life.
Does the word "decimal " have two meanings; namely, either a) the
entire decimal fraction or the decimal number (such as a particular
finite and/or nonterminating decimal) or b) a particular digit in a
particular decimal place to the right of the decimal point or decimal
marking?
Does this also mean that I have been using the word "digit"
incorrectly? Is the word digit(s) restricted to the symbol(s) to
denote a number or does digit(s) include their combinatory use to form
a number. For example, there are a finite number (10) of unique
symbols (0,1,2, ..., 9) that form the decimal digits; but that decimal
numbers may have any number of digits. How many digits to the right of
the decimal point does the the decimal number 0.333... possess in
proper and precise mathematical jargon? Is it only one (the digit 3)
or is it a number approaching infinity. Is it correct to say that the
integer "119" has a) three digits, b) only two digits in three place
value positions, or c) both are correct usages of the word digit?
Would it be more correct or precise for me to be using "no last decimal
place" instead of "no last digit"?
Don Whitehurst
Don Whitehurst
Just as " ... " at the trailing end of the sequence of natural numbers
represents the concept that there is no last natural number, the
"3.14...(d_n<-pi)::: " is a decimal representation of the concept that
both a) there is no last decimal number, and that there is no last
number of decimal digits. Since there is always one more successor of
each, both grow without bound and there are an infinite number of each.
I used the ::: in 3.14...(d_n<-pi)::: to have a different
representation than 3.14...(d_n<-pi)... so as to make my distinction in
meaning clearer.
> > If the decimalic representation 3.14...(d_n<-pi)... in which there is
> > no last digit is not equal to pi and you can demonstrate or prove this
> > point to me, we will have no disagreement. How is it possible for the
> > decimalic representaion with no last digit formed from all of the
> > consecutive decimalic digits of pi to not be a part of column B?
>
> All of the things in column B have the property that:
> digits 1-n are the same as digits 1-n of pi
> digits n+1, n+2, ... are 0.
>
> Obviously pi does not have the property that there are
> an infinite number of zeros beginning at some position n
> (if it did, it would be rational). Yet everything in column
> B has that property. Hence pi is not in column B.
>
Do you not believe that a one to one correspondence exists between the
natural numbers and any single real nonterminating decimal?
Doesn't Cantor's diagonal proof that the set of real numbers is
nondenumerably infinite require such a one to one correspondence?
Don
Don Whitehurst
> On 6 Sep 2005 16:25:21 -0700, Don Whitehurst wrote:
> > Dave Seaman wrote:
>
>
> >> > I treated the time noon as being infinity and used aleph_0 only because
> >> > you said that aleph_0 was the name of infinity for the naturals. I
> >> > don't think I said there was a ball marked aleph_0 added. But it may be
> >> > a proper inference from what I did say.
>
> Aleph_0 is identical to the set of all natural numbers. Since no set is
> a member of itself, it logically follows that aleph_0 cannot *be* a
> natural number, else it would have to be a member of itself.
>
I have little understanding of this Aleph_0 concept. More explanation
might be helpful.
> >> Can we agree, then, that there is no ball marked oo or aleph_0? Each
> >> ball is marked with a natural number in the original version, or with a
> >> terminating decimal in your version. Each natural number maps to a
> >> terminating decimal.
>
>
> > I am not sure how we got to noon (infinity) without adding such a ball.
>
Each ball is marked with a decimal. Each natural number maps to 1) a
group of balls each marked with a decimal having the same number of
digits as the natural numbers (these are the balls added at the nth
interval before noon), and to 2) a ball with a decimal marking that is
removed from the vase at the nth interval. I am not sure that every
ball marked with a decimal has a terminating decimal. If by noon all
balls have been added, that means there must have been an infinite
number of groups of balls added, each group with all the decimals in
that group having one more digit than the all the decimals in the
previous group.
If 1) the naturals make a one to one correpondence with the decimal
digits of any particular group of nonterminating decimals, and if 2)
there are an infinite number of naturals due to the concept of no last
natural, doesn't it follow that there must be a corresponding
requirement of no last decimal digit in the groups of decimals added.
After all as Brian Chandler pointedly wrote, "The idea of a
nonterminating decimal fraction is that it, uh, has no end (which is
what 'nonterminating' means, of course)."
> There is no ball added at noon. Each ball whose number is less than
> aleph_0 is added before noon, and no other balls are added.
>
All the balls cannot be added at any time interval before noon. The
naturals (corresponding to each of the diminishing time intervals
before noon) are only approaching infinity before noon. Since 1) the
no last natural requires one more natural to follow and since 2) there
are a group of balls added at each nth interval equal to 1/2^nth of an
hour before noon, all the balls cannot be added at any time interval
before noon since the time intervals before noon associated with adding
the groups of balls approach but never reach zero.
> > But let's assume you are correct. By noon, all balls have been added
> > (an infinite number of intervals has occurred n->oo and an infinite
> > number of balls corresponding to the infinite number of naturals has
> > been removed). Since all the balls were added they must include some
> > with an infinite number of digits.
>
> How many balls are there having only a finite number of digits? (Pick
> one):
>
> (a) There are infinitely many balls, each having only finitely many
> digits.
> (b) There are only finitely many balls, each having only finitely many
> digits.
>
My first inclination, based on meager knowledge of the properties of
the naturals, is to agree with the premise (a). But I interpret the
phrase "infinitely many" to arise from the property of the naturals
that for any natural n there is always a successor natural n+1 (in this
case the group of balls associated with n all have n decimal digits,
while the succesor group of balls all have n+1 decimal digits) and no
last natural n exists which means there is no group of ballls with
decimal number that have the most dugits. Consequently, I believe
there are infinitely many groups of balls (each natural associated
with a group of balls) upon which decimals are written, one decimal
number for each ball of the specific group and each decimal having the
same number of digits as there are groups that are added. But since
the number of groups of balls are infinitely many only in the sense
that there is no termination of the sequence of groups - the number of
groups approaches infinity but here is no largest number of groups.
There are not an infinite number of balls added at any time before
noon, so there is not an infinite number of balls in the sense of
capablility of adding balls or of terminating or ending the
denumeration of the naturals with a last natural.
> If you choose (a), then you have contradicted yourself, because you have
> just agreed that it is possible to add infinitely many balls without any
> of them having an infinite number of digits.
>
Based upon my interpretation above, it is not possible to "add"
infinitely many balls. The adding can never end. Infinity must be
achieved (or become actualized) for the adding to end. It never does
for the naturals; nor at any time before noon.
> If you choose (b), then you have a problem. If there are only finitely
> many balls having only finitely many digits, then there must be a largest
> one, say M. How many digits are there in M+1?
>
If I chose (a) with what I believe is your interpretation of meaning,
then I agree that I have contradicted myself and so should give up my
requirement that there must be an infinite number of digits. But if I
choose (a) and give up my requirement regarding the highest number of
digits associated these "all the balls having been added by noon", I
also have a problem. I believe one property of every natural number in
a positional numeration system, (a place value system or a noninfinite
natural base system) is that there are very many other numbers (in fact
infinitely many) with more digits than the number n. In this case,
there is always many groups of balls have more decimal digits written
on the surface than the ball n. If there are only groups of balls
each with only with finitely many digits, there must be one group with
the most digits, say M. How many digits are there in M+1?
> So which is it? You need to answer this question in order for this
> discussion to make any progress.
>
Perhaps a third alternative (c) is more appropriate.
(c) All three statements 1), 2), and 3) each meet certain aspects of
the problem posed by Theo Jacobs and which I have extended from groups
of ten balls added at each 1/2^nth interval before noon to the adding
groups of balls (totaling 9*10^n) each with n decimal digits at each
1/2^nth interval before noon.
1) There are only finitely many balls each having a finite number of
decimal digits (associated with a particular group of balls added)
written on the surface of the ball that can be read or actually counted
at any particular time before noon. This corresponds to case (b).
2) This reading of the decimals written on the balls or the
denumeration process may be extended toward infinity. This extension
toward infinity of the denumeration produces the realization that there
must be no last group of balls added and consequently that the adding
of balls continues without end before noon. Each group of balls has
the same number of digits in the decimal number written on any ball
added with that group as equals the total number of separate groups of
balls added up to that time interval before noon. This denumeration
process extends toward infinity and is without end, and so there may be
properly regarded to be infinitely many balls. This infinitely namy
balls corresponding to infinitely many groups of balls added each with
a finite number of groups for any actual time. At any actual time
before noon there are only balls with a finite number of digits since
the number of balls added with finite digits does not end before noon.
This corresponds to case (a). The infinitely many balls, each having
only finitely many digits. But these infinitely many balls are only
being added or conceptually or potentially being added before noon, not
actually added.
3) Every natural number has a successor number. There is no last
natural number. Every natural number, n, expressed in a positional
place system has the property that there exists very many other
(infinitely many) natural numbers having one or more digit places than
the natural number n. There is no natural number expressed in a
positional place system having the most digit places. These statements
readily translate to the balls. Every nth group of balls, where each
ball in the group has a decimal written on the surface with n decimal
digits, has a successor n+1th group of balls, where each ball in the
successor group has a decimal written on the surface with n+1 decimal
digits. There is no last group of balls added, and there is no group
of balls having the most digits. The number of groups of balls
increases without bound, the number of digits on those groups of balls
also increases without bound. (c) The number of groups of balls added
at any time approaches infinity, and the number of digits associated
with the groups of balls added at any time also approaches infinity.
The same meaning of infinitely many groups of balls applies to the
number of digits (infinitely many digits) associated with the groups as
well as the successor relationship associated with the number of digits
with each group that is added.
4) If it were possible to complete the denumeration process as
represented by noon in this problem from Theo Jacobs, then the infinite
number of naturals must have been actualized or completely specified.
The neverending process of adding balls as n approaches infinity is
being evaluated at infinity as if the process of adding balls could
reach an end - which by assumption, inference and induction should not
be possible.
While it was not possible to add the last group of balls at any
specified time before noon or to add all of the balls at any time
before noon, one can examine the situation by noon. This phrase "by
noon" appears to represent that the infinite was actualized, the
approach to infinity became at infinity. By noon, all the balls would
seem to have been added - all time intervals for addition of the balls
before noon were somehow used up in the infinitely closer approach to
infinity.
If by noon "all" of the groups of balls were added, then an infinite
number of groups of balls must have been added before noon. This
infinite number must include the last natural that is known not to
exist; otherwise all of the balls were not added.
Although I do not know how to prove it, I suspect that last natural
that seems outside of the naturals by definition, must be infinite in
extent (have an infinite number of digits). Since each group of balls
added has the same number of digits as the corresponding natural that
by noon has miracuously reached infinity, the number of digits
associated with each successive groups of balls also grew without bound
and reached infinity by noon.
If one insists that all of the groups of balls were actually added
before noon, none of which had an infinite number associated with it
because the successive process of adding one more group continued for
shorter time intervals without end before noon, then the number of
decimal digits written on the balls also grew without bound making an
infinite number of them forming what are all of the nonterminating
decimals. Just as there is no last or largest natural there is no
last or longest digit string on the balls. By noon, the digit strings
are nonterminating in extent - the succession process is done. How
exactly does nonterminating in extent digit string differ from a
nonterminating decimal?
Is my reasoning correct regarding the different representations of the
naturals and the corresponding expression in various bases. It strikes
me that the naturals expressed using a positional place system all have
the property that given any particular natural, there are very many
other naturals (infinitely many) with at least one more digit, and with
at least two more digits, etc. This property of the naturals relating
to the number of digits in a particular representation seems to be
associated with any integer base 2 or greater but less than infinity.
If this reasoning is correct, it seems to suggest there exists a
property of the naturals based on positional place system
representations that may not exist in naturals formed from sets (or
possibly an infinite base system with an infinite number of unique
symbols). If so, how and why is this possible?
<snip>
> The real numbers are uncountable, but the rationals are a countable set
> with the property that each real number can be represented arbitrarily
> closely by rationals. We say that the rationals are "dense" in the
> reals, and therefore the reals are an example of a topological space that
> has a countable dense subset. Such a space is called "separable".
> (Don't ask me why that particular word was chosen to describe the
> property of having a countable dense subset; it just was.)
>
> If you still insist on thinking that every set is topologically closed
> (i.e., contains all its limit points), then it's quite understandable
> that you must be having a hard time distinguishing the rationals from the
> reals. After all, the reals are the closure of the rationals in R.
> Therefore, it's precisely the fact that the rationals are not a closed
> set in R that is a stumbling block.
>
You may be correct that I may may have such a stumbling block despite
the fact that I don't completely understand all the concepts you
discuss. Nenertheless, I suspect my problem arises from my
interpretation of what constitutes a nonterminating decimal. I have
viewed them as arising from a successor operation applied infinitely
many times as opposed to a limit taking process. Both concepts seem to
be represented perfectly adequately by the decimal representations of
the nonterminating decimals such as 0.333.. or 3.1415..., yet there
seems to be no acceptance, recognition, or acknowledgement that such
representations meet both concepts.
>
> A nonterminating decimal digit string is naturally associated with a
> sequence of rationals. The number represented by the whole string is the
> limit of that sequence, but is not a member.
>
Are you saying that my concept of the meaning for a nonterminating
decimal representaion of 1/3 from the long hand division is incorrect.
Namely, 0.333... (or if you prefer 0.333::: to represent my concept
for the meaning of a nonterminating decimal which may differ from the
standard concept 1/3 = 0.333...) is the decimal representation of the
series of long hand division steps of 1/3 where each succesive digit to
the decimal associated with the additional remainder is successively
added as the rightmost decimal digit. The ::: (if you prefer) in
0.333::: represents the approach to completion of the process of
division but not its actual completion which never occurs. For me all
real nonterminating decimal numbers have had this meaning, and this
concept seems to differ somewhat from the requirement that the decimal
0.333... = 1/3 is the limit of the sequence. The 0.333::: represents
successor decimal numbers as the number of digits approaches infinity
of the nonterminating elements of the infinite sequence derived from
dividing 1 by 3, just as ... represents succesor naturals as n
approaches infinity in the 1, 2, 3, ..., n, ..., ... infinite sequence
associated with the naturals.
Are you then saying that the mapping of the infinite number of naturals
in a one to one correspondence with the decimals of any single
nonterminating decimal does not include all of the digit decimals of
the nonterminating decimal?
If so, how then would the Cantor proof of the nondenumerability of the
reals work if the generated diagonal 0.(z_1)(z_2)(z_3)... does not
extend to all of the decimal digits of a nonterminating decimal so as
to be compared and shown to be different than all the members of the
assumed complete list of nonterminating decimals =/=
0.(a_1)(a_2)(a_3)..., =/= 0.(b_1)(b_2)(b_3)..., =/=
0.(c_1)(c_2)(c_3)..., =/= ..., ....?
Don
Virgil
"Don Whitehurst" <whit...@umn.edu> wrote:
> n, <--> 3.14...(d_n<-pi),
>
>
How is it that the ultimate and antipenultimate rows shown in your
complete diagram are identical?
Does this indicate that the proces become cyclic, or that there is a
last row?
> Where the last " ... "in Column A indicates that no last natural
> exists,
But apparently, according to your diagam, a last natural is supposed to
exist.
> and where the trailing " ... " in the decimal representation "
> 3.14...(d_n<-pi)... " indicates that no last digit exists.
"3.14..." is understandable, but what does "(d_n<-pi)" signify?
>
> Note also that the " ..., " indicates an additional unspecified number
> of elements of each sequence (before and after n in Column A and before
> and after 3.14...(d_n<-pi) in Column B).
What does the lack of any ellipsis after your last row signify? That
there is a last row?
Randy Poe
Don Whitehurst wrote:
> Randy Poe wrote:
> > All of the things in column B have the property that:
> > digits 1-n are the same as digits 1-n of pi
> > digits n+1, n+2, ... are 0.
> >
> > Obviously pi does not have the property that there are
> > an infinite number of zeros beginning at some position n
> > (if it did, it would be rational). Yet everything in column
> > B has that property. Hence pi is not in column B.
>
> Do you not believe that a one to one correspondence exists between the
> natural numbers and any single real nonterminating decimal?
Yes, I believe that a nonterminating decimal has one digit
for every natural number. So what? How does that prove that
any particular nonterminating decimal is in your list?
If you want to claim that proves pi appears on the list,
the same argument says that sqrt(2), e, and log(17) are
in your list. And of course they are not.
A sequence of terminating decimals, the n-th value terminating
at position n, will not contain a non-terminating decimal.
- Randy
Jonathan Hoyle
>> \with infinite decimalic representations such as 1/3 =
>> 0.333..., Pi/10 = 0.31415..., and sqrt (2)/2 =
>> 0.70710678..., as well as many that can't be
>> denumerated in real life, must also have been
>> placed in the vase and removed.
Not true. to see this, Note that there is no time prior to noon where
such a ball is ever added. We know that Ball 0.5 was added at 1/2
minute before noon, and Ball 0.0001 was added 1/16 minute before noon,
but when was Ball 0.333... added? According to your scheme, never.
You did not say, "Also at noon, please add all the following balls...".
You specified what to do at each time 1/2^n minutes before noon, for
each finite integer n, but without explicit instructions, no actions
were described for noon itself.
Jonathan Hoyle
>> 1) Every ball added to the vase at the nth interval before noon
>> will be removed before noon (at some later ( n+ r_kl )th
>> interval).
>> removed, there are 10^n -n balls remaining in the vase.
This is not a paradox. You are confusing the limit of the sizes with
the size of the limit. Let Sn be the set of balls at the nth interval.
lim |Sn| is infinite whereas |lim Sn| is 0. There is no paradox here.
That you have a size discontinuity at noon is hardly a problem.
Counter-intuitive maybe, but not paradoxical.
Try mapping f(x) = e^(1/x) sometime. Specifically, compare lim x->0+
f(x) with lim x->0- f(x). One is infinite, the other is 0. This is
math, things like happen. Learn to like it. :-)
Jonathan Hoyle
Dave Seaman
> Dave Seaman wrote:
>> On 6 Sep 2005 16:25:21 -0700, Don Whitehurst wrote:
>> > Dave Seaman wrote:
>> >> > I treated the time noon as being infinity and used aleph_0 only because
>> >> > you said that aleph_0 was the name of infinity for the naturals. I
>> >> > don't think I said there was a ball marked aleph_0 added. But it may be
>> >> > a proper inference from what I did say.
>> Aleph_0 is identical to the set of all natural numbers. Since no set is
>> a member of itself, it logically follows that aleph_0 cannot *be* a
>> natural number, else it would have to be a member of itself.
> I have little understanding of this Aleph_0 concept. More explanation
> might be helpful.
Aleph_0 is the smallest infinite cardinal. It is the same set as w, the
smallest infinite ordinal. The natural numbers are identical to the
finite ordinals, which are exactly the members of w.
There is indeed a set that contains all the natural numbers and also
contains w. This is the set w+1 = { 0, 1, 2, 3, ..., w }. It's a
different set than w, because w is a member of w+1, but w is not a member
of w. No set is a member of itself.
>> >> Can we agree, then, that there is no ball marked oo or aleph_0? Each
>> >> ball is marked with a natural number in the original version, or with a
>> >> terminating decimal in your version. Each natural number maps to a
>> >> terminating decimal.
>> > I am not sure how we got to noon (infinity) without adding such a ball.
We added balls corresponding to all the members of w. We did not cover
all the members of w+1, which is a different and larger set. Why is that
concept difficult?
> Each ball is marked with a decimal. Each natural number maps to 1) a
> group of balls each marked with a decimal having the same number of
> digits as the natural numbers (these are the balls added at the nth
> interval before noon), and to 2) a ball with a decimal marking that is
> removed from the vase at the nth interval. I am not sure that every
> ball marked with a decimal has a terminating decimal. If by noon all
> balls have been added, that means there must have been an infinite
> number of groups of balls added, each group with all the decimals in
> that group having one more digit than the all the decimals in the
> previous group.
There are infinitely many terminating decimals, and each of them is added
and removed before noon. There is no ball marked with a nonterminating
decimal according to the scheme you have described. The balls added at
step n have at most n decimal digits, which is a finite number for each
n.
> If 1) the naturals make a one to one correpondence with the decimal
> digits of any particular group of nonterminating decimals, and if 2)
> there are an infinite number of naturals due to the concept of no last
> natural, doesn't it follow that there must be a corresponding
> requirement of no last decimal digit in the groups of decimals added.
> After all as Brian Chandler pointedly wrote, "The idea of a
> nonterminating decimal fraction is that it, uh, has no end (which is
> what 'nonterminating' means, of course)."
No, it doesn't follow. Why would that follow?
>> There is no ball added at noon. Each ball whose number is less than
>> aleph_0 is added before noon, and no other balls are added.
> All the balls cannot be added at any time interval before noon.
Quantifier dyslexia. I am saying that for each n, there is a time before
noon when ball n is added.
(1) (An)(Et) (t < 0 and ball n is added at or before time t).
You are trying to reverse the order of the quantifiers, obtaining
(2) (Et)(An) (t < 0 and ball n is added at or before time t).
You are quite correct in stating that (2) is false, but this does not
imply that (1) is false. Those statements are not equivalent. The
quantifiers do not commute.
For each t < 0, the number of balls added at or before time t is finite.
However, the number of balls added before time t=0 is infinite.
So you agree that it is possible to speak of the set of all natural
numbers (w) without including w itself as a member? This contradicts
what you have been claiming up to now. For example, it implies that we
can talk about the set of all terminating decimal strings without
including any nonterminating decimals.
>> If you choose (a), then you have contradicted yourself, because you have
>> just agreed that it is possible to add infinitely many balls without any
>> of them having an infinite number of digits.
> Based upon my interpretation above, it is not possible to "add"
> infinitely many balls.
I don't care whether you think it is physically possible to add
infinitely many balls. The problems statement specifies that balls are
added at time t=1/n for each n, and that implies that infinitely many
balls are added before t=0. If you reject that premise, then you and I
are discussing different problems and any further discussion is
pointless. You must accept the premises stated in the problem.
> The adding can never end. Infinity must be
> achieved (or become actualized) for the adding to end. It never does
> for the naturals; nor at any time before noon.
Why do you keep adding the irrelevant phrase "before noon"? All of the
balls are added by time t=0 (noon), and therefore the process has ended
at noon, even though there was no last ball to be added. Noon is the
least upper bound of the set of times when balls are added, but it is not
a member of that set of times.
>> If you choose (b), then you have a problem. If there are only finitely
>> many balls having only finitely many digits, then there must be a largest
>> one, say M. How many digits are there in M+1?
> If I chose (a) with what I believe is your interpretation of meaning,
> then I agree that I have contradicted myself and so should give up my
> requirement that there must be an infinite number of digits. But if I
> choose (a) and give up my requirement regarding the highest number of
> digits associated these "all the balls having been added by noon", I
> also have a problem. I believe one property of every natural number in
> a positional numeration system, (a place value system or a noninfinite
> natural base system) is that there are very many other numbers (in fact
> infinitely many) with more digits than the number n. In this case,
> there is always many groups of balls have more decimal digits written
> on the surface than the ball n. If there are only groups of balls
> each with only with finitely many digits, there must be one group with
> the most digits, say M. How many digits are there in M+1?
You were doing fine up until that last sentence. What makes you think
there has to be a group with the most digits?
The difference between this statement of yours and the reasoning I used
regarding (b) earlier, is that I was talking about a finite set, but you
are talking about an infinite set. A finite totally ordered set
necessarily has a largest member, but an infinite set need not.
>> So which is it? You need to answer this question in order for this
>> discussion to make any progress.
> Perhaps a third alternative (c) is more appropriate.
> (c) All three statements 1), 2), and 3) each meet certain aspects of
> the problem posed by Theo Jacobs and which I have extended from groups
> of ten balls added at each 1/2^nth interval before noon to the adding
> groups of balls (totaling 9*10^n) each with n decimal digits at each
> 1/2^nth interval before noon.
Are you referring to your numbered statements that follow? I count 4,
not 3.
> 1) There are only finitely many balls each having a finite number of
> decimal digits (associated with a particular group of balls added)
> written on the surface of the ball that can be read or actually counted
> at any particular time before noon. This corresponds to case (b).
No, it doesn't correspond to case (b), because case (b) asserts (wrongly)
that there are only finitely many balls added _in toto_. Your statement
1) is consistent with case (a).
> 2) This reading of the decimals written on the balls or the
> denumeration process may be extended toward infinity. This extension
> toward infinity of the denumeration produces the realization that there
> must be no last group of balls added and consequently that the adding
> of balls continues without end before noon. Each group of balls has
> the same number of digits in the decimal number written on any ball
> added with that group as equals the total number of separate groups of
> balls added up to that time interval before noon. This denumeration
> process extends toward infinity and is without end, and so there may be
> properly regarded to be infinitely many balls. This infinitely namy
> balls corresponding to infinitely many groups of balls added each with
> a finite number of groups for any actual time. At any actual time
> before noon there are only balls with a finite number of digits since
> the number of balls added with finite digits does not end before noon.
> This corresponds to case (a). The infinitely many balls, each having
> only finitely many digits. But these infinitely many balls are only
> being added or conceptually or potentially being added before noon, not
> actually added.
The problem statement says they are actually added. Aside from that,
your statement (2) is consistent with case (a).
> 3) Every natural number has a successor number. There is no last
> natural number. Every natural number, n, expressed in a positional
> place system has the property that there exists very many other
> (infinitely many) natural numbers having one or more digit places than
> the natural number n. There is no natural number expressed in a
> positional place system having the most digit places. These statements
> readily translate to the balls. Every nth group of balls, where each
> ball in the group has a decimal written on the surface with n decimal
> digits, has a successor n+1th group of balls, where each ball in the
> successor group has a decimal written on the surface with n+1 decimal
> digits. There is no last group of balls added, and there is no group
> of balls having the most digits. The number of groups of balls
> increases without bound, the number of digits on those groups of balls
> also increases without bound. (c) The number of groups of balls added
> at any time approaches infinity, and the number of digits associated
> with the groups of balls added at any time also approaches infinity.
> The same meaning of infinitely many groups of balls applies to the
> number of digits (infinitely many digits) associated with the groups as
> well as the successor relationship associated with the number of digits
> with each group that is added.
Again, all consistent with case (a).
> 4) If it were possible to complete the denumeration process as
> represented by noon in this problem from Theo Jacobs, then the infinite
> number of naturals must have been actualized or completely specified.
> The neverending process of adding balls as n approaches infinity is
> being evaluated at infinity as if the process of adding balls could
> reach an end - which by assumption, inference and induction should not
> be possible.
The problem statement specifies that the process of adding balls is in
fact completed by noon. Infinitely many balls are added, each bearing a
finite number, as specified by case (a).
> While it was not possible to add the last group of balls at any
> specified time before noon or to add all of the balls at any time
> before noon, one can examine the situation by noon. This phrase "by
> noon" appears to represent that the infinite was actualized, the
> approach to infinity became at infinity. By noon, all the balls would
> seem to have been added - all time intervals for addition of the balls
> before noon were somehow used up in the infinitely closer approach to
> infinity.
Correct.
> If by noon "all" of the groups of balls were added, then an infinite
> number of groups of balls must have been added before noon. This
> infinite number must include the last natural that is known not to
> exist; otherwise all of the balls were not added.
We are talking about the set w, not w+1. The set w includes all of the
natural numbers but does not have a last member. The set w+1 includes
all of the naturals and also has a last member (namely w), but w is not a
member of w.
> Although I do not know how to prove it, I suspect that last natural
> that seems outside of the naturals by definition, must be infinite in
> extent (have an infinite number of digits).
The set w is not "the last natural". It is the first non-natural. That
is, it's the smallest ordinal that is *not* a natural number.
>Since each group of balls
> added has the same number of digits as the corresponding natural that
> by noon has miracuously reached infinity, the number of digits
> associated with each successive groups of balls also grew without bound
> and reached infinity by noon.
No, there is no group of balls added at any time that has infinitely many
digits. Infinitely many groups are added, each marked by a finite number
of digits as in case (a), which you have agreed to.
> If one insists that all of the groups of balls were actually added
> before noon, none of which had an infinite number associated with it
> because the successive process of adding one more group continued for
> shorter time intervals without end before noon, then the number of
> decimal digits written on the balls also grew without bound making an
> infinite number of them forming what are all of the nonterminating
> decimals. Just as there is no last or largest natural there is no
> last or longest digit string on the balls. By noon, the digit strings
> are nonterminating in extent - the succession process is done. How
> exactly does nonterminating in extent digit string differ from a
> nonterminating decimal?
If there were a group having infinitely many digits, then surely that
group would be the last one. But you just admitted that there is no last
group. Therefore, it logically follows that no group can have infinitely
many digits.
> Is my reasoning correct regarding the different representations of the
> naturals and the corresponding expression in various bases. It strikes
> me that the naturals expressed using a positional place system all have
> the property that given any particular natural, there are very many
> other naturals (infinitely many) with at least one more digit, and with
> at least two more digits, etc. This property of the naturals relating
> to the number of digits in a particular representation seems to be
> associated with any integer base 2 or greater but less than infinity.
Correct.
> If this reasoning is correct, it seems to suggest there exists a
> property of the naturals based on positional place system
> representations that may not exist in naturals formed from sets (or
> possibly an infinite base system with an infinite number of unique
> symbols). If so, how and why is this possible?
Given any natural number n, there are infinitely many naturals that are
greater than n. I don't see what is gained by talking about positional
place systems or digit strings.
>> The real numbers are uncountable, but the rationals are a countable set
>> with the property that each real number can be represented arbitrarily
>> closely by rationals. We say that the rationals are "dense" in the
>> reals, and therefore the reals are an example of a topological space that
>> has a countable dense subset. Such a space is called "separable".
>> (Don't ask me why that particular word was chosen to describe the
>> property of having a countable dense subset; it just was.)
>> If you still insist on thinking that every set is topologically closed
>> (i.e., contains all its limit points), then it's quite understandable
>> that you must be having a hard time distinguishing the rationals from the
>> reals. After all, the reals are the closure of the rationals in R.
>> Therefore, it's precisely the fact that the rationals are not a closed
>> set in R that is a stumbling block.
> You may be correct that I may may have such a stumbling block despite
> the fact that I don't completely understand all the concepts you
> discuss. Nenertheless, I suspect my problem arises from my
> interpretation of what constitutes a nonterminating decimal. I have
> viewed them as arising from a successor operation applied infinitely
> many times as opposed to a limit taking process. Both concepts seem to
> be represented perfectly adequately by the decimal representations of
> the nonterminating decimals such as 0.333.. or 3.1415..., yet there
> seems to be no acceptance, recognition, or acknowledgement that such
> representations meet both concepts.
It reduces to your erroneous claim that the set w must be a member of
itself. No set is a member of itself. A set that contains all the
finite ordinals need not have an infinite ordinal (w) as a member.
>> A nonterminating decimal digit string is naturally associated with a
>> sequence of rationals. The number represented by the whole string is the
>> limit of that sequence, but is not a member.
> Are you saying that my concept of the meaning for a nonterminating
> decimal representaion of 1/3 from the long hand division is incorrect.
> Namely, 0.333... (or if you prefer 0.333::: to represent my concept
> for the meaning of a nonterminating decimal which may differ from the
> standard concept 1/3 = 0.333...) is the decimal representation of the
> series of long hand division steps of 1/3 where each succesive digit to
> the decimal associated with the additional remainder is successively
> added as the rightmost decimal digit. The ::: (if you prefer) in
> 0.333::: represents the approach to completion of the process of
> division but not its actual completion which never occurs.
Here is where we differ. The notation 0.333... represents a real number,
and real numbers do not "approach" anything. The value of 0.333... is
*exactly* 1/3, because 1/3 happens to be the limit of the sequence of
partial sums associated with that decimal, but the limit is not a
*member* of that sequence of partial sums.
>For me all
> real nonterminating decimal numbers have had this meaning, and this
> concept seems to differ somewhat from the requirement that the decimal
> 0.333... = 1/3 is the limit of the sequence. The 0.333::: represents
> successor decimal numbers as the number of digits approaches infinity
> of the nonterminating elements of the infinite sequence derived from
> dividing 1 by 3, just as ... represents succesor naturals as n
> approaches infinity in the 1, 2, 3, ..., n, ..., ... infinite sequence
> associated with the naturals.
Perhaps you want to say that your 0.333::: represents the actual sequence
of partial sums, rather than the limit of that sequence. Then 0.333:::
and 0.333... are not the same thing, and are not even the same kind of
thing. One is a sequence, while the other is a real number.
> Are you then saying that the mapping of the infinite number of naturals
> in a one to one correspondence with the decimals of any single
> nonterminating decimal does not include all of the digit decimals of
> the nonterminating decimal?
Yes, I am saying that w is not a member of w, because each member of w is
finite, but w itself is infinite.
> If so, how then would the Cantor proof of the nondenumerability of the
> reals work if the generated diagonal 0.(z_1)(z_2)(z_3)... does not
> extend to all of the decimal digits of a nonterminating decimal so as
> to be compared and shown to be different than all the members of the
> assumed complete list of nonterminating decimals =/=
> 0.(a_1)(a_2)(a_3)..., =/= 0.(b_1)(b_2)(b_3)..., =/=
> 0.(c_1)(c_2)(c_3)..., =/= ..., ....?
The Cantor proof only requires that for each i in w and for each j in w,
there is a j-th digit in the i-th number of the list. There is no
requirement that there be an w-th number or that any number have an w-th
digit. The requirement extends only to members of w, and w is not a
member of w.
Vivian Gage
> following problem:
>
> <Suppose you have a giant vase and a bunch of ping
> pong balls with an
> <integer written on each one, e.g. just like the
> lottery, so the balls
> <are numbered 1, 2, 3, ... and so on. At one minute
> to noon you put
> <balls 1 to 10 in the vase and take out number 1. At
> half a minute to
> <noon you put balls 11 - 20 in the vase and take out
> number 2. At one
> <quarter minute to noon you put balls 21 - 30 in the
> vase and take out
> <number 3. Continue in this fashion. Obviously this
> is physically
> <impossible, but you get the idea.
>
> <My friend claims both experiments end up with an
> empty vase.
only way I can see this happening is after so many years, someone came along and kicked the bucket