Re: New countable infiniity logic
whit0911_at_umn.edu
Date: 11/10/04
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Date: 10 Nov 2004 11:56:43 -0800
Barb Knox <see@sig.below> wrote in message news:<cm7bca$umk$1@lust.ihug.co.nz>...
> In article <b453b903.0410270806.5d16d12c@posting.google.com>,
> whit0911@umn.edu wrote:
>
<snip>
>
> Avoid infintesimals, with which one can not do sensible arithmetic or
> algebra. The epsilon definition of limit is a successful means of putting
> the whole notion of infinite/infintesimal onto a sound mathematical footing.
This seems like good advice, and I will try to refrain from the use of
infintesimals.
>
> >I am not sure what "separable" means exactly, but I think it may have
> >to do with the dense
>
> Yes.
>
> >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.
>
> Because each sequence 0.3, 0.33, ... is a FINITE sequence of digits.
Am I wrong in believing that the sequence (0.3, 0.33, ...) is an
infinite sequence where each specified element in this representation
of the sequence has a finite number of digits?
> That
> is, the Nth number in this list has exactly N 3's (followed by an infinite
> sequence of 0's if you care to look at it that way). Whereas 0.33...
> clearly has an infinite sequence of NON-ZERO digits.
If (0.3, 0.33, ...) is an infinite sequence where each specified
element in this representation of the sequence has a finite number of
digits, isn't (0.3, 0.33, ..., 0.3333...) a valid representation of
the infinite nature of this sequence showing that as the number of
elements in the infinite sequence approaches infinity, the finite
number of digits associated with each element also approaches
infinity.
> Note that the sequence with N 3's equals (1/3) - (1/3)*10^(-N). That is,
> the sequence's decimal value is exactly (1/3)*10^(-N) away from 1/3. So,
> for EACH N, there is an epsilon e (anything less than (1/3)*10^(-N) will do)
> such that the value is NOT within epsilon of 1/3. Note there is a DIFFERENT
> epsilon for each N.
>
As N approaches infinity, doesn't the difference disappear, making the
"infinite sequence or list of numbers" or thier representation of
those numbers the same as 1/3 since the number of digits also grows
without bound?
<snip>
>
> No no no no no. Consider the naturals again: one INFINITE SET, but
> containing only FINITE ELEMENTS.
>
> >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.
>
> Correct. But each one is still FINITE, just as each natural is finite.
>
> >In the same way that the naturals are a list of infinite extent
>
> Correct.
>
If there are an infinite number of naturals - or if the number of
naturals approaches or extends toward infinity, then for this function
there are an infinite number of digits - or the number of digits
approaches or extends toward infinity.
Unlike natural numbers, the decimalic numbers have representations for
numbers having these properties with respect to digits; namely the
repeating decimalic rationals, and the identifiable irrationals. For
example 0.555... = 5/9, or 0.027182818128... = e/100.
<snip>
>
> Yes, since NO individual sequence (like no individual natural N) is infinite.
>
Are not the individual sequences infinite in extent (0.3, 0.33, 0.333,
...)? Don't the sequences go on forever just as the infinite set of
naturals do not truncate and the number of elements in the sequence
(1, 2, 3, ...) goes on forever or approaches infinity. I agree that
the individually named lower numbers of the sequences have a finite
number of digits, but if sequences are infinite in extent how can one
be certain that the number representing 0.333... is not included in
the infinite set representing this function. After all, drawing an
inference from the naturals with respect to finite vs infinite to the
decimals may be tenuous since the representation of numbers differ
significantly for decimals and naturals. The set of naturals has no
representation (of which I am aware) beyond the "..." for large n
approaching infinity but has no difficulty representing any number
near zero (the finites). The set of decimalic expressions have
representations of numbers that show sequences approaching or
extending toward infinity (0.3, 0.33, 0.333, ..., 0.333...) where a
terminus can be specified, but the numbers in between the lower
finites and the terminus representing a real number cannot be
specified.
Thus for naturals we get representations of sequences like (1, 2, 3,
..., n, ...) where the commas separate the elements; but for decimals
we can represent sequences using the (0.3, 0.33, 0.333, ...,
0.333...)
> >then not only do the finite
> >decimalic numbers between 0 -1 approach inifintesimally close to the
> >repeating rational and irrational decimalic numbers
>
> Correct. For any real in [0, 1], there is a (possibly infinite) sublist of
> your list that converges to it as a limit. BUT, NO INDIVIDUAL element of
> that sublist EQUALS an irrational or repeating rational, since EACH
> INDIVIDUAL element ends with an infinite sequence of 0 digits.
>
You are the one that is specifying that each individual element ends
with an infinite sequence of zeroes. I could equally well argue that
each individual element of the naturals starts with an infinite
sequence of 0 digits to the right of what you call the number n. In
that case, they all have infinitely many digits but only finite value
where the rightmost zeroes are being replaced one by one with the
nonzero numeral digits as the finite value of n increases. I am not
sure why you feel it necessary to add the infinite sequence of 0
digits, since I agree that each named element is a finite truncated
decimal (not a repeating rational, nor an irrational).
> >but the "actual"
> >rational and irrational decimalic numbers are included as expressed by
> >the use of the "..." in 0.333...
>
> You're misplacing the "...". The list 0.3, 0.33, 0.333, ... is infinite,
> but each element only has a finite number of 3's (just as every natural N is
> finite). NO element is exactly 0.33....
>
This may be the crucial misunderstanding I continue to hold. I don't
understand how the infinite list does not contain this element if it
is in fact infinite.
<snip>
> >Did I miss anything?
>
> Yes, mainly the idea of an infinite SET (or list) containing only FINITE
> elements.
>
Thank you, Barb for your detailed comments. Generally these comments
have helped me better understand the issues relating to this topic.
Your statement "Infinite SET of finite ELEMENTS..." that you suggested
I learn is compelling - in particular in conjunction with the concept
of convergence. I believe I am gaining an understanding of your
perspective. Nevertheless, I still am having difficulty with the
concept of an infinte set or the number of digits associated with some
elements of an infinite set. Perhaps you could help my understanding
here also.
I am not sure how best to talk about an infinite set. A set whose
number of elements are infinite, or where the number of elements
approaches infinity, or where the number of elements increases without
bound, or where the number of elements are infinite in extent. Posts
on this site have cautioned against using the concept of actual
infinity (?), so let me know if I should avoid using any of these
phrases.
Consider the following infinite sets that have a one to one
correspondence:
N = {1, 2, 3, ...} = {1, 2, 3, ..., n, ...}
and the set of positive integer numbers containing only numeral 2s:
#2# = {2, 22, 222, ...}.
#2# = {2, 22, 222, ...}; and the set corresponding to the number of
digits of the nth element in #2# = d
d = {1, 2, 3, ...}.
#2# is a subset of N.
The set N = the set d.
(1) Every named (and nameable by accepted representations) element in
each of these sets is finite.
(2) By induction and/or one to one correspondence, each set has an
infinite number of elements.
(3) However, in the set #2# the number of digits associated with each
element is exactly provided by the corresponding number n associated
with set d = set N.
(4) In order for set #2# to have an infinite number of elements by
induction or by one to one correspondence, there must also be an
infinite number of digits associated with this set;
If this sounds too much like actual infinity, I could easily restate
them using other language to describe the number of elements in the
infinite set, such as the number of elements (and digits for d)
approaches infinity.
(1) Every named (and nameable by accepted representations) element in
each sets is finite.
(2) By induction and/or one to one correspondence, each set is an
infinite set and therefore the number of elements in each set
approaches infinity.
(3) However, in the set #2# the total number of digits associated with
each element is exactly provided by the corresponding number n
associated with set d = set N.
(4) Using induction or by one to one correspondence, in order for the
number of elements in set #2# to approach infinity, the number of
digits associated with the elements for this set must also approach
infinity.
Similar arguments concerning the number of digits and their
representation can also be made by extension for the decimalic
expansions associated with the sets N = {1, 2, 3, ...}, with Dec2 =
{0.2, 0.22, 0.222, ...}, and d = {1. 2, 3, ...}.
In the three sets N, Dec2, and d, the "..." represents the infinite
number of elements by induction or by one to one correspondence
associated with these being infinite sets. Unlike N and d, Dec2 has a
possible alternate representation of the concept that there are an
infinite number of elements by using the terminus representation
"0.222..." in conjunction with a central '...' as a substitute for the
trailing *...*.
Dec2= {0.2, 0.22, 0.222, ...} = {0.2, 0.22, 0.222, ..., 0.222...}.
Are these not equvalent ways of representing that the set Dec2
contains an infinite number of elements each representing decimalic
numbers that contain only *2*s? I believe such a representation is
appropriate for this set - since this representation makes clear that
not only does the set Dec2 contain an infinite number of elements
utilizing only the numerals 2 in the decimalic digits, but that the
set Dec2 also contains an infinite number of digits. If you prefer,
this representation makes clear that not only do the number of
elements utilizing only the numerals 2 in the decimalic digits of set
Dec2 approach infinity, but that the number of digits of the
succesive elements in set Dec2 approaches infinity.
N = {1, 2, 3, ...} = {1, 2, 3, ..., n, ...}
Dec2 = {0.2, 0.22, 0.222, ...} = {0.2, 0.22, 0.222, ..., 0.222...}
d = {1, 2, 3, ...} = {1, 2, 3, ..., n, ...}
These are all easily recognized representations for infinite sets.
The question becomes how does the number represented by 0.222... = 2/9
differ from the trailing *...* nature of the representation that there
are an infinite number of elements in the set Dec2 = {0.2, 0.22,
0.222, ...} as well as the seemingly exact 0.222... representation for
the fact that there are an infinite number of elements as well as an
infinite number of digits in Dec2 = {0.2, 0.22, 0.222, ..., 0.222...}?
Since each element in Dec2 represents an actual decimalic number,
how do these two numbers each represented by 0.222... differ?
2/9 = 0.222... Counting digits : 1, 2, 3, ...
Dec2 = {0.2, 0.22, 0.222, ...} Counting digits on successive elements
:
1, 2, 3, ...
Dec2 = {0.2, 0.22, ..., 0.222...} Counting digits of representation
for the terminus number 0.222... which combined with the internal
"..." represents that the set Dec2 is an infinite set : 1, 2, 3, ...
Since each and every element, of the ordered sets #2# and Dec2, has
exactly the same number of digits as the corresponding natural number
n, and since the number of elements in the set of naturals is
infinite, then the number of digits associated with the infinite
succession of elements in sets #2# and Dec2 must also be infinite.
If this statement is true, then for Dec2, such an element is exactly
given by the number "0.222... " . It is particulrly difficult for
me to understand how this "0.222..." =/= 0.222... = 2/9.
Am I mixing up representations and numbers?
If set #2# or Dec2 do not have an infinite number of digits, they
cannot have an infinite number of elements. Since each and every
element, of the ordered sets #2# and Dec2, has exactly the same number
of digits as the corresponding natural number n; and if the set of
naturals has an infinite number of elements, then there must be an
element in set #2# and Dec2 with an infinite number of digits or if
you prefer - the number of digits associated with the individual
elements of set #2# and Dec2 approaches infinity as the number of
elements increases successively. (Is this the actual infinity that I
should avoid?) If there is not an infinite number of digits
associated with set #2# or Dec2 then the total number of digits
associated with set #2# must be finite, then set #2# or Dec2 must
only be finite and terminate (a contradiction with item (2) above
where it is an inifinite set with an infinite number of elements), and
consequently there must also not be an infinite number of elements in
the set of naturals N since they are in one to one correspondence.
Clearly this must not be the case.
N = {1, 2, 3, ...} = {1, 2, 3, ..., n, ...}
#2# = {2, 22, 222, ...} = {2, 22, 222, ..., *22...22*, ...} =
{2, 22, 222, ..., *...222*} = #2#
Dec2 = {0.2, 0.22, 0.222, ...} = {0.2, 0.22, 0.222, ..., *0.22...22*,
...} =
{0.2, 0.22, 0.222, ..., 0.222...} = Dec2
Note, N does not have representation beyond the "..." to represent the
infinite succession of numbers; whereas #2# and Dec2 each have
possible alternate nonstandard representations that I enclosed with *
*; and where Dec2 has a widely used representation for the infinite
succession of numbers in Dec2.
Where did I go wrong?
This same type of reasoning is what I tried to apply to the function
previously stated and was why I thought the naturals mapped the reals.
Don Whitehurst
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