countability of real numbers? What am I missing?
Saijanai
fraction sequence. I assert it lists all real numbers [0,1] (allowing
for duplicates):
{0.0, 0.1},
{0.00, 0.01, 0.10, 0.11},
{0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111},
...
Tonico
Hey, wonderful! Now, just for the sake of fun, please do tell us what
binary fraction from the ones you list represents...I dunno...say, the
number 1/sqrt(2)? Or the numer 1/e?
You know what? Forget the above: you wrote: "I assert it lists all
real numbers in [0,1]...", so what about a little proof for your
assertion?
Tonio
Saijanai
The proof should be self-evident in the same way that listing the
Naturals as 1, 2, 3, 4,... obviously doesn't miss any.
What I'm claiming (and I'm not claiming I'm correct, only that I don't
see the problem) is that the usual issues like Cantor's
Diagonalization don't refute it. This is almost certainly because I
don't understand Cantor's Diagonalization, but regardless, I don't see
where the problem is.
My own guess is that I'm conflating lexical ordering with numerical
ordering but I'm not sure if that is relevant. An order is an order
and a countable sequence is a countable sequence, regardless of how
you arrive at them.
L.
Torsten Hennig
I only see rational numbers in your enumeration ;
where are the irrational ones ?
Best wishes
Torsten.
Saijanai
wrote:
The sequence is never-ending, so you can follow an arbitrarily long
trail from level to level to create a Cauchy-Sequence that corresponds
to any real, including the irrationals. My assertion is that you don't
miss anything in the ordering, because the initial squence-ordering
isn't numerical, just lexical.
I'm not sure if that's cheating or not. I suspect it is, but I'm only
auditing elementary level number theory/set theory/analysis lectures
online right now and I'm no doubt missing something (or just don't
understand the lectures I've already seen in the first place).
L.
Tonico
> On May 18, 6:34 am, Tonico <Tonic...@yahoo.com> wrote:
>
>
>
>
>
> > On May 18, 4:27 pm, Saijanai <saija...@gmail.com> wrote:
>
> > > OK, all you set/number theorists, what is wrong with this binary
> > > fraction sequence. I assert it lists all real numbers [0,1] (allowing
> > > for duplicates):
> > > {0.0, 0.1},
> > > {0.00, 0.01, 0.10, 0.11},
> > > {0.000, 0.001, 0.010, 0.011, 0.100, 0.101, 0.110, 0.111},
> > > ...
>
> > Hey, wonderful! Now, just for the sake of fun, please do tell us what
> > binary fraction from the ones you list represents...I dunno...say, the
> > number 1/sqrt(2)? Or the numer 1/e?
>
> > You know what? Forget the above: you wrote: "I assert it lists all
> > real numbers in [0,1]...", so what about a little proof for your
> > assertion?
>
> > Tonio
>
> The proof should be self-evident in the same way that listing the
> Naturals as 1, 2, 3, 4,... obviously doesn't miss any.
>
> What I'm claiming (and I'm not claiming I'm correct, only that I don't
> see the problem) is that the usual issues like Cantor's
> Diagonalization don't refute it. This is almost certainly because I
> don't understand Cantor's Diagonalization, but regardless, I don't see
> where the problem is.
Problems? For one, and from what you wrote, I can see only rational
numbers, and even that of a very special form. For example, the
rational number 0.10101010....(continuing in this fashion) doesn't
appear in your list according to the model you wrote...
Tonio
>
> My own guess is that I'm conflating lexical ordering with numerical
> ordering but I'm not sure if that is relevant. An order is an order
> and a countable sequence is a countable sequence, regardless of how
> you arrive at them.
>
> L.-
Saijanai
My assertion is that the sequence doesn't miss any real and by
extension, you can get arbitrarily close to any real by going out far
enough. The relevant issue is countability, at least to me. This gives
a countable sequence of numbers that doesn't miss any reals. The fact
that the ordering isn't numerical is not relevant to the claim that
the sequence is ordered and doesn't miss numbers. Again, I think its
an issue with conflating different definitions of ordering, but the
fact that a specific number, repeating or non-repeating, doesn't
appear in some finite sublist isn't relevant to the claim that no
number is "missed".
I'm side-stepping the issues, not countering them. The more
interesting question is: does the sidestep make the question
meaningless or is there some deeper thingie going on.
Lawson
Torsten Hennig
The point is that it's not enough that you can form
a Cauchy sequence from the numbers of your
enumeration to reach every real number in [0;1].
An enumeration (x_n) of the reals in [0;1] would mean:
for a specified real number x in [0;1] you must be able to name the index K for which x_K = x.
Best wishes
Torsten.
Saijanai
OK, I understand what you're saying, but is it a valid issue? I'm not
claiming to be able to give the index or a specific real only that my
enumeration doesn't miss any.
L.
Chip Eastham
While you can give a countable sequence
of numbers that come "arbitrarily close"
to any real number in [0,1], this is a
weaker result than actually listing _all_
real numbers in [0,1], which is what it
would mean for the real numbers to be
countable.
What you describe with "arbitrarily close"
is that the real numbers contain a countable
dense subset (in the sense of the metric
topology), which is well-known. (For historical
reasons this property is termed "separable" by
topologists.)
Again, it does not conflict with the equally
well-known theorem that the real numbers are
uncountable.
regards, chip
Chip Eastham
Your use of "doesn't miss any" seems disingenuous.
Either every specific real number in [0,1] is in
the list you gave or there's some real number in
[0,1] that is not in the list. As discussed up-
thread, countability is not about coming close.
Your list misses almost everything in [0,1], as
apparently you only hit the finite (terminating)
binary fractions. As pointed out, you don't get
to even such rational numbers as 1/3 no matter
how far out the list is taken, because 1/3 has
no terminating binary representation.
regards, chip
William Hughes
Without actually looking at your list, Karnac says you
only have finite decimal expansions in your list.
Karnac is right again.
It is well known that the finite sequences are countable.
If you want to list an infinite decimal then
approximating it is not good enough.
It has to be in the list.
Cantor's argument will not work with finite decimals,
because the anti-diagonal produced is not a finite decimal.
- William Hughes
Aage Andersen
"Saijanai" > OK, all you set/number theorists, what is wrong with this
I don't see 0.111111111111111111111111111.......... in your list.
Aage
W^3
<6b027f1d-3ba8-4f9f...@y18g2000prn.googlegroups.com>,
Saijanai <saij...@gmail.com> wrote:
Your list doesn't include x = .010101010.... Yes, from your list you
can find a sequence converging to x. And this might be interesting to
you. But it's irrelevant to the point at hand: Yours is not a list of
all real numbers because x is a real number and it is nowhere in your
list.
James Burns
>>The point is that it's not enough that you can form
>>a Cauchy sequence from the numbers of your
>>enumeration to reach every real number in [0;1].
>>An enumeration (x_n) of the reals in [0;1] would mean:
>>for a specified real number x in [0;1] you must be able
>>to name the index K for which x_K = x.
> OK, I understand what you're saying, but is it a valid
> issue? I'm not claiming to be able to give the index
> or a specific real only that my enumeration doesn't
> miss any.
The question rests upon what you (or we) mean by
"doesn't miss any". The complete binary expansion
for 1/3 doesn't show up on your list anywhere, although
matches to the initial piece of it, for arbitrarily
long initial pieces, do show up.
If that counts as "showing up", then, yes, every real
"shows up". This is well known, but it is not the sense
used in comparing the sizes of sets. If two sets have
the same cardinality, then there is a bijection between
them. You will have natural numbers (your index)
corresponding to all those approximations to 1/3,
but no natural number corresponding to 1/3 itself.
We know that there is no such natural number, because
every real in your list has the form k/2^m, for
some naturals k and m.
Do you accept different size infinities, such as
the more general Cantor's proof shows? That is a much
simpler proof, with less room for wiggling around to
look for a counter-example.
For any set S, and any function f: S -> P(S), from
S to the powerset of S, f is not a bijection, therefore
S and P(S) do not have the same cardinality.
Proof: Let Y = the subset of S {x| x not in f(x)}.
There is no y in S such that f(y) = Y.
Notice that you can biject binary sequences of {0,1} with
the subsets of the naturals, .10101... <-> {1,3,5,...}
If you take care of cases like 0.1000... = 0.0111...
(messy but quite doable), you will have shown that
the reals are a larger infinity than the naturals.
Jim Burns
Transfer Principle
Ah, a countability of R thread. This is the perfect opportunity
for me to practice new habits of what to say in this situation.
I've resolved to acknowledge at least three possibilities to
describe the OP, Saijanai:
Case 1. Saijanai wishes to describe a theory other than the
classical reals (a theory in which only the numbers that are
in his list count as reals in the proposed theory).
Case 2. Saijanai knows that there exist uncountably many
classical reals, but doesn't like this fact.
Case 3. Saijanai believes that his list actually contains every
classical real. This is the only case for which it would be
accurate to call Saijanai wrong.
In the past, I've usually assumed Case 1 without considering
the other cases. Now I must consider all three cases.
This thread is only four hours old. As the thread progresses,
we can find out which case is most likely.
Tonico
Uh? Wasn't enough that he explicitly talked about Cauchy sequences
from his list's numbers that can approximate every number in [0,1]
INSTEAD of actually having all the real numbers in his list, as he was
told by several posters, to conclude that he's wrong? He thought that
having Cauche seq. does the trick, and this means, imfho, that he's
wrong....what are you waiting for to deduce this? Perhaps during time
he'll change in a subtle way his stand so that you'll say he wasn't
wrong but...whatever? What has time to do with this at all??
How much will you wait to deduce I'm wrong if I start a thread
claiming that 2 + 2 = 5 is true within the usual field axioms of the
real numbers? 8 hours, 16 hours, 2 days..?
Tonio
James Burns
Here is my opportunity to demonstrate a technique for
distinguishing between the three cases you list, or
between other possibilities, uhm, possibly overlooked.
A-HEM! Saijanai, which of Transfer Principle's three
cases is the case? Or is there some other description
of your beliefs about reals and countability that would
be more appropriate?
This technique doesn't always work. It may be that we
will not hear from Saijanai again, soon or ever. Often
posters get what they want and go back to real life.
While we wait, I will bet that Saijanai knows, or at
least strongly suspects, that he(?) is making an error
somewhere but wants to understand where specifically
the mistake is -- to improve his understanding of
cardinality, the reals, etc, etc.
See, for example, Saijanai:
<I'm not sure if that's cheating or not. I suspect
<it is, but I'm only auditing elementary level number
<theory/set theory/analysis lectures online right now
<and I'm no doubt missing something (or just don't
<understand the lectures I've already seen in the
<first place).
<62540561-1418-4c09...@31g2000prc.googlegroups.com>
It's an interesting question for you what to do,
if I am right. Then, Saijanai would be right that
he is wrong. Would you disagree and tell him he
is right, or agree that he is wrong?
(This sort of thing is not a problem for me.
I'll tell him which arguments are right or wrong,
and let Saijanai worry about what he /wants/ to be
right or wrong.)
Jim Burns
Transfer Principle
> Here is my opportunity to demonstrate a technique for
> distinguishing between the three cases you list, or
> between other possibilities, uhm, possibly overlooked.
I dis acknowledge that the list isn't necessarily
exhaustive, when I wrote:
> > I've resolved to acknowledge _at_least_ three possibilities to
> > describe the OP, Saijanai:
(emphasis added)
> A-HEM! Saijanai, which of Transfer Principle's three
> cases is the case? Or is there some other description
> of your beliefs about reals and countability that would
> be more appropriate?
Direct asking sometimes works, but I only like to ask
when I can avoid making it sound like an _interrogation_
or giving a poster the third degree. In particular, when
others are already asking the OP questions, I usually
avoid doing so.
Asking vs. interrogating...
> It's an interesting question for you what to do,
> if I am right. Then, Saijanai would be right that
> he is wrong. Would you disagree and tell him he
> is right, or agree that he is wrong?
Burns and Tonio, admittedly, do give evidence suggesting
that the OP is in Case 3. Therefore, I have decided to
agree with them that Case 3 is the most likely.
And so, even though I do find a theory in which the reals
with finite binary expansions are the only reals to be an
interesting theory, Case 3 tells us that such a theory
would have nothing to do with Saijanai. It would be
considered wrong ("pandering") of me to attempt to write
such a theory in this thread.
Case 3 tells us that Saijanai is wrong, but there are
already enough posters telling him so in this thread, and
so for me to join them would be redundant. I'd rather not
post at all than _repeat_ that he is wrong.
Therefore, I must say good-bye to this thread. This is my
final post in this thread.
christian.bau
> Case 3. Saijanai believes that his list actually contains every
> classical real. This is the only case for which it would be
> accurate to call Saijanai wrong.
The others fall in the category "this is so bad, it's not even
wrong".
What the original poster doesn't realise is that he has a sequence
with countably many elements, but the number of subsequences that are
Cauchy sequences is uncountable. And yes, the (uncountably many)
Cauchy sub-sequences _do_ cover all the reals in [0, 1]; but that
obviously does nothing to show the reals are countable.
Tim Little
Every number in this list is of the form k/2^N for some natural
numbers k and N. That doesn't even include all the rationals, let
alone the reals.
For example: if 1/3 is somewhere in the list, you should be able to
tell me at least one pair of (k,N) values it has.
- Tim
Tim Little
> extension, you can get arbitrarily close to any real by going out
> far enough.
Getting arbitrarily close is not enough.
Definition: X is "countable" iff there exists f:N->X such that for all
x in X, there exists n in N with f(n) = x. For each x there must
exist *specific* n where f(n) is *equal to* x.
> The relevant issue is countability, at least to me. This gives a
> countable sequence of numbers that doesn't miss any reals.
The range of the sequence is "dense" in the interval (in the sense of
the usual topology), but fails to include most of them.
> Again, I think its an issue with conflating different definitions of
> ordering,
No, it's an issue with what it means for a sequence to include a
real number.
- Tim
Mike Terry
news:0df6bdc8-51e2-4390...@t14g2000prm.googlegroups.com...
It seems to me that nearly all of these threads fall into Case 3, and can be
subclassified as
Case 3a. OP does not understand the standard meaning of "list" (or
"sequence")
Case 3b. OP does not understand what it means for a number "to be in" a
list/sequence.
Case 3c. Other error.
(Mostly I think 3a or 3b is the issue. Hard to tell which since they're
obviously related. No doubt you would say that in these cases the OP is
intending to be using some non-standard definition of list/sequence? :-)
Mike.
cwldoc
case 4. Saijanai simply ignores everything posters have said, repeating the same nonsense over and over. He probably had no intention from the outset of being enlightened in any way! Posters seem naive about his intentions, thinking that he just needs more instruction, or perhaps they suspect the truth but just give him the benefit of the doubt. Then they keep posting more clarifications, apparently still being surprised that their insights are not heeded! I don't know whether to be amused or upset at having wasted my time reading this post!
Ronald Bruck
By "listing" every real number, we mean it must ACTUALLY APPEAR
somewhere in the list--not just that it can be constructed as some sort
of limit of items in the list.
If the list is L, then it can be enumerated L1, L2, L3, ... Thus every
real number must appear EXACTLY ONCE.
So, for example, I don't see the number 1/3 in your list. In binary,
that would be 0.010101..., which has infinitely digits and so is at NO
level in your list. Your list only enumerates the dyadic fractions
m/2^n, where 0 <= m < 2^n.
But I don't expect you to understand this. You're either a troll,
deliberately provoking responses, or you don't understand standard
nomenclature.
--Ron Bruck
If one wishes to patent a number--some have actually done this--I claim
it would be best to patent 0 or 1, or better yet, 0 AND 1. For no
other number can be written in binary without using at least one of
these.
The net effect of this observation is that we probably all owe a lot in
royalties to either AT&T or IBM.
You object, But people have been using these for many years already. I
reply, That doesn't seem to stop companies from ignoring prior art in
many other patent applications.
Jesse F. Hughes
> If one wishes to patent a number--some have actually done this[...]
Er, can you give a few details?
--
One these mornings gonna wake | Ain't nobody's doggone business how
up crazy, | my baby treats me,
Gonna grab my gun, kill my baby. | Nobody's business but mine.
Nobody's business but mine. | -- Mississippi John Hurt
Ronald Bruck
<je...@phiwumbda.org> wrote:
> Ronald Bruck <br...@math.usc.edu> writes:
>
> > If one wishes to patent a number--some have actually done this[...]
>
> Er, can you give a few details?
Well, there's this article:
<http://www.theregister.co.uk/2003/11/26/first_integer_patented/>
But I note that in "other stories" cited on the same page, it is
reported that a man in England ("of no stable address") was convicted
of sodomy with a donkey and a horse. Oh, those English!
The article is certainly tongue-in-cheek. In another article they
quote (concerning a lawsuit supposedly filed against the company which
patented the first integer), "The rumour, which started in the Usenet
newsgroup sci.math.research but rapidly spread to
rec.pets.cats.anecdotes, states that the ’Wron number and two other
‘large’ integers together ganged up on an unwilling smaller (but
technically oversize) integer and forced it to indulged in a Fermatic
practices with them".
I doubt that the US patent office would allow such patents. But then,
I doubted that they would permit patents of parts of the human genome.
My, but we live in interesting times.
-- Ron Bruck
cbr...@cbrownsystems.com
> Ronald Bruck <br...@math.usc.edu> writes:
> > If one wishes to patent a number--some have actually done this[...]
>
> Er, can you give a few details?
Perhaps something like this fits the bill...
http://en.wikipedia.org/wiki/Illegal_prime
Cheers - Chas