Re: New countable infiniity logic
whit0911_at_umn.edu
Date: 10/27/04
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Date: 27 Oct 2004 09:06:31 -0700
joshp@bayes.joshpurinton.com (Josh Purinton) wrote in message news:<xQved.2354$3Q4.907@news.flashnewsgroups.com>...
> In article <b453b903.0410230702.18d6c6ed@posting.google.com>,
> <whit0911@umn.edu> wrote:
> >joshp@bayes.joshpurinton.com (Josh Purinton) wrote in message
> >news:<Ezged.2037$3Q4.1998@news.flashnewsgroups.com>...
> > Thank you for pointing out that I did not recreate |-|erc's list
> > exactly; as there should not have been any zero digits to the right of
> > the natural numbers at each equivalent stage; they should have all
> > been truncated.
>
> Then the new list doesn't contain "0.100...", right? (Actually, "0.1"
> is the same as "0.100...", but you seem to be distinguishing between
> them.)
>
> > In the corrected case listed, is it not true that as X (where X
> > corresponds to the number(s) whose digits consist of only 3s)
> > approahes infinity, the limit of 1/3-M(X) = 0?
>
> Roughly speaking, yes. But there is a difference between "approaching"
> and actual equality.
>
> For example, let f be the function
> x --> 1 - sin(x)/x
>
> Then f(x) approaches 0 as x approaches 0, but f(x) is never actually
> equal to 0.
>
> Similarly, for all X, M(X) is never actually equal to 1/3.
Mike Oliver's and Josh Purinton's responses provide me with a better
understanding of why mathematicians may not view this function as
mapping the reals, namely it is felt the numbers it outputs approach
but never reach *any* actual rational repeating or irrational
decimalic numbers, making the difference between the positive real in
the range 0 -1 with such an infinite expansion and the generated
number not equal to zero. This seems very sound reasoning if the two
numbers are not as equally infintesimally close as to be the same as
per the number(s?) that might be described as 0.49999... = 0.5 (as
distinct from 0.49999 =/= 0.5 where the finite form truncates at some
finite number of decimalic digits in this case 6). I think, if am
mistaken about this function, that I may also have had (perhaps still
do) some confusion as to what is a number, its representation, and its
value. But if I do, I cannot currently articulate what I mean or
understand by that last statement.
Your discussion of picking epsilons to describe the difference is
compelling, but I believe this function may also generate the
infintesimals that you describe as being the difference between 1
-0.999... = 0
I am not sure what "separable" means exactly, but I think it may have
to do with the dense approach but not reaching by the function which
produces a series of numbers such as 0.3, 0.33, 0.333, 0.333, ... not
being the same as or ever achieving 0.333... = 1/3 or 0.01417... =
(sqrt 2)/100, etc. I don't see why you do not believe this function
outputs all such actual rational repeating and irrational decimalic
numbers as well as those that approach. It seems to me you do not
accept that the infinite set of naturals (or the set of naturals of
infinte extent) must produce a list infinitely long in extent for this
function. I believe and assert that the only way to produce a list
infinitely long in extent from this function is if the rational
repeating and irrational decimalic numbers are included. I would
appreciate it if you could show me where I am wrong on this account.
The function is described by:
"Convert the natural number directly to its natural decimalic
expansion by placing a decimal point in front of the number; unless it
ends with a zero, in which case all the consecutive ending zero digits
are moved from the end of the number to become placholders behind (to
the right of) the decimal point and in front of the remaining part of
the number. "
1 becomes 0.1
3 becomes 0.3
9 becomes 0.9
10 becomes 0.01
11 becomes 0.11
12 becomes 0.12
19 becomes 0.19
30 becomes 0.03
31 becomes 0.31
89 becomes 0.89
90 becomes 0.09
91 becomes 0.91
99 becomes 0.99
100 becomes 0.001
101 becomes 0.101
109 becomes 0.109
110 becomes 0.011
111 becomes 0.111
123 becomes 0.123
300 becomes 0.003
314 becomes 0.314
899 becomes 0.899
900 becomes 0.009
901 becomes 0.901
999 becomes 0.999
:
n, where n grows without bound.
Notice that with each power of ten of the naturals, an extra digit "d"
is added to the finite decimalic numbers (those which truncate or do
not have an infinite number of digits).
Rearranging the order of this list (which list represents a set of
infinite extent) produces a series of decimalic numbers including but
not limited to:
0.1, 0.01, 0.001, ... whose series includes forming the number I call
the lowest infintesimal;
0.1, 0.12, 0.123, ... whose series includes forming the Champernowne's
irrational number ;
0.3, 0.31, 0.314, ... whose series includes forming the number Pi/10;
0.3, 0.33, 0.333, ... whose series includes forming the number 1/3;
0.9, 0.09, 0.009, ... whose series includes forming the ninth lowest
infintesimal; and
0.9, 0.99, 0.999, ... whose series includes forming the number
0.999... which almost everyone accepts as equal to 1.
In set notation, these individual finite decimalic numbers are
elements that each appear only once, nevertheless all such numbers are
used infinitely many times to form all of the series representing the
repeating rational and irrational decimalic expansions to complete the
reals and further include what I call the infintesimals (where the
number of zeroes before the finite decimal becomes infinite in
extent).
I guess where I am still confused is why a function which should
generate a list that is infinite in extent - since it is based on the
naturals (which are themselves infinite in extent) - is nevertheless
somehow not allowed to generate an infinite number of digits. If the
naturals are infinite in extent, they must generate a list (from the
specified function) that is also infinite in extent. Note: because
the only way the list for this specified function (where a new digit
is added for each succesive power of ten numbers inserted into the
function) can be infinite in extent is if the number of digits
themselves are infinite in extent. After all the function keeps
adding one more digit every time an additional power of ten from the
naturals are utilized. The number of digits therefore have the same
property as the number of naturals; namely there is no largest number
associated with them.
In the same way that the naturals are a list of infinite extent
expressed as 1, 2, 3, ... (where "..." means the naturals continue to
add one higher number forever), the decimalic expansion list of
infinite extent from this function will generate not only an infinite
number of decimalic finites, but also the entire infinite number of
repeating rationals and irrationals (reals) from which 0.2222...,
0.13291329..., 0.123456789101112..., etc. (where the "..." means
the decimalic expansion adds the next digit for the intended number
forever) are examples. Is it not obvious that the only way such
infinite decimalic expanded numbers are able not to be included in
this function is if the naturals themselves are not infinite in
extent? While I can no more specify the value of x in X=f(x) that
will produce any particular decimal of infinite extent from this
specified function (anymore than I can specify the n that is any
finite natural number less than the largest N), this fact has no
bearing upon the matter of whether such numbers are not part (or if
you prefer - will not become a part) of the list produced by the
specified function; anymore than whether the numbers not specifically
listed on the list of naturals are not part (or if you prefer - will
not become a part) of the set of naturals numbers of infinite extent.
After all, the only way that this function would not produce an
infinite number of digits is if there was a largest finite number N in
the naturals and if that were the case, the function would have an
endpoint and the number of digits "d" would truncate. But since this
is not the case (there is no largest N), d must be infinite in extent
and numbers such as:
0.111...
0.12345678910111213...
0.31415...
0.33333...
0.00271828...
along with all the other nonfinite decimalic numbers (repeating
rationals and the irrational decimalic numbers between 0 - 1) must be
found on the list.
One can no more expect a person to be able to name the particular
natural number that generates any particular one of these numbers any
more than one can expect a person to name the particular largest
number or the next one hundred numbers thereafter. This unnameability
arises from the infinite extent of both sets. Just as it is obvious
that one does not have to name or specify every number on the set of
naturals of infinite extent to know what elements are included, if one
accepts that the number of digits for this function is infinite in
extent (is there any other choice?) then not only do the finite
decimalic numbers between 0 -1 approach inifintesimally close to the
repeating rational and irrational decimalic numbers but the "actual"
rational and irrational decimalic numbers are included as expressed by
the use of the "..." in 0.333... The number which expresses the
infinite extent of the number of digits "d" with the series 0.3, 0.33,
0.333, etc. all off which finite forms are found on the list is
0.333... = 1/3. That number (as one example) must be present on the
list because no other number from this particular series (0.3, 0.33,
0.333 etc.) expresses anything other than the finiteness of the number
of digits "d" and they are known to be of infinite extent for this
function.
Did I miss anything?
Don Whitehurst
- Next message: Keith A. Lewis: "Re: Explaining e?"
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- In reply to: Josh Purinton: "Re: New countable infiniity logic"
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