The Distinguishability argument of the Reals.

[Zuhair]

The distinguishability argument is a deep intuitive argument about the
question of Countability of the reals. It is an argument of mine, it
claims that the truth is that the reals are countable. However it
doesn't claim that this truth can be put in a formal proof.

The idea is that we cannot have more objects than what we can
distinguish.

The argument originated with discussions about the Infinite binary
tree, and it is built on the following observations and generalization
and consideration:

Observation 1:

In any finite binary tree if we change the labeling of all nodes
beyond a specific level in such a manner that all of those receive the
same label, then the number of paths that can be distinguished by the
labels of their nodes will not increase beyond that of the last
unaltered level.

Example: The infinite binary tree with Two levels below the root node.

0
/ \
0 1
/ \ | \
0 1 0 1

Now lets alter the last level (i.e. Level 2):

0
/ \
0 1
/ \ | \
0 0 0 0

Now the number of paths at level 1 (the unaltered paths) is 2, those
are:

0-0
0-1

The number of paths at the altered level which is level 2 would be
also 2, those are:

0-0-0
0-1-0

Now lets add another level with fixed labeling with 0, this would be:

0
/ \
0 1
/ \ | \
0 0 0 0
/\ /\ /\ /\
00 00 00 00

The number of paths at level 4 would be also 2, those are:

0-0-0-0
0-1-0-0


Generalization: From the above observation we can make the following
intuitive generalization_ That in the case of ANY binary tree the
total number of paths distinguishable by labels of their nodes of Size
n Will be equal to the total number of paths distinguishable by labels
of their nodes of Size m where m > n
iff distinct labeling of nodes seize to exist after nodes at the end
of paths of size n.

Observation 2:

The complete Infinite binary tree have all its nodes labeled
distinctly occurring at end of FINITELY long paths. And accordingly No
discrimination by labeling of nodes occurs at the end of some
infinitely long path, so there is not discrimination by labeling that
occurs at INFINITE level, all distinct labeling do occur at FINITE
level only.

Consideration: We Consider FINITE and INFINITE to be kinds of gross
(semi-quantitative) Size criteria where INFINITE size criterion is
bigger than FINITE size criterion, i.e. INFINITE > FINITE.

Now from Generalization, Observations 2 and Consideration we arrive at
the following:

RESULT 1:

The number of Infinitely long paths of the complete infinite binary
tree is the same as the number of the finitely long paths of the
complete infinite binary tree.

Observation 3:
The total number of FINITE paths of the complete Infinite binary tree
that are distinguishable by labeling of their nodes is COUNTABLE.

From Result 1 and Observation 3, we reach at:

Result 2
The number of all INFINITE paths and thus ALL paths of the complete
binary tree is COUNTABLE.

Observation 4:
Each real is identified with a distinguishable path by labeling of its
nodes of the complete infinite binary tree.

From Result 2 and Observation 4 we arrive finally at:

FINAL CONCLUSION:

The number of all reals is COUNTABLE.
QED

Zuhair

[George Greene]

On Jan 1, 1:19 pm, Zuhair <zaljo...@gmail.com> wrote:
> The distinguishability argument is a deep intuitive argument about the
> question of Countability of the reals. It is an argument of mine, it
> claims that the truth is that the reals are countable. However it
> doesn't claim that this truth can be put in a formal proof.

Are you just STUPID or what? Not only can your argument not be "put
in" a formal proof,
IT CAN BE REFUTED by a formal proof! It can be DISPROVED!
Suppose the reals WERE countable. Suppose you HAD proved that AND
your proof
was sound and correct. THEN IT WOULD *FOLLOW*FROM* your proof of the
countability of the reals
that the anti-diagonal of your enumeration both WAS AND WASN'T
assigned some counting NUMBER.
THAT IS A CONTRADICTION, DUMBASS.

[George Greene]

On Jan 1, 1:19 pm, Zuhair <zaljo...@gmail.com> wrote:

> The idea is that we cannot have more objects than what we can
> distinguish.

Any two reals are distinguishable by a very trivial algorithm,
if they really ARE two. If they are NOT distinguishable then they are
NOT two,
so THAT case DOESN'T MATTER.
But if, instead of confirming that two are two, you really WERE IN
DOUBT
as to whether two were or were not ONE, were or were not EQUAL, then
by THE USUAL algorithm,
that would only be semi-decidable, since if they WERE equal, that
algorithm would not halt.

With purely finitary means, WE CAN *distinguish* ALL OF THEM, HOWEVER
MANY THAT IS.

[George Greene]

On Jan 1, 1:19 pm, Zuhair <zaljo...@gmail.com> wrote:

> The argument originated with discussions about the Infinite binary
> tree, and it is built on the following observations and generalization
> and consideration:
>
> Observation 1:
>
> In any finite binary tree if we change the labeling of all nodes

Then "we" are blithering idiots because IT IS THE EDGES AND NOT the
nodes
that get labeled.
Your obsession with thinking there is something to be gained by
labeling nodes
is really embarrassing. To you.

[Zuhair]

You are just so STUPID enough not to discriminate between Intuitive
argumentation and Formal ones. This is an argument at INTUITIVE level,
and I already said that I'm not claiming to have a formal argument,
nor do I think it can be formalized. Of course IF this could be
formalized then we'd arrive at a formal paradox, i.e. a full genuine
paradox or an antinomy, by then we'd reject the concept of completed
infinity that led to this paradox. But I'm not claiming that. What I'm
claiming is that at INTUITIVE level the distinguishability argument
gives the impression that the Reals are Countable. However at FORMAL
level we have Cantor's diagonal argument proving Uncountability of the
Reals. So Cantor's argument is to be understood as COUNTER-INTUITIVE
result and not as a genuine paradox. Baboon head.

[Zuhair]

On Jan 1, 9:41 pm, George Greene <gree...@email.unc.edu> wrote:
> On Jan 1, 1:19 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > The idea is that we cannot have more objects than what we can
> > distinguish.
>
> Any two reals are distinguishable by a very trivial algorithm,
> if they really ARE two.  If they are NOT distinguishable then they are
> NOT two,
> so THAT case DOESN'T MATTER.

I think you were DRUNK when you wrote that.

> But if, instead of confirming that two are two, you really WERE IN
> DOUBT
> as to whether two were or were not ONE, were or were not EQUAL, then
> by THE USUAL algorithm,
> that would only be semi-decidable, since if they WERE equal, that
> algorithm would not halt.
>

SIMPLY remarks of no significance but to Drunk people like you. Even
Baboons think better.

> With purely finitary means, WE CAN *distinguish* ALL OF THEM, HOWEVER
> MANY THAT IS.

Yes that is correct. Which is trivial but nevertheless the best that
your Baboon head can think of.

[Zuhair]

Yes of course, indeed. You are really messed up. I told you that this
is not essential to this kind of argumentation, but your Baboon head
cannot just grasp matters when they go a little bit above concrete
thinking.

Zuhair

[Ross A. Finlayson]

Decorum, gentlemen, hew to it.

http://www.kipling.org.uk/poems_if.htm

The intuituinistic is in a sense formal: with a formal allowance for
that the expectations of the intuition are in a sense axioms, or here
truisms. The intuituionistic is essentially the formal, though it
alludes to the development in passing instead of its explicit
development. Then, where it may rely on the development as implicit,
it is from truisms and first principles.

Then an intuitionist might find that the infinite is infinite, and
that's enough to be infinite.

Count the integers, is there count not yet an integer?

Are there, infinite integers?

In Boucher's F, Paris and Kirby with nonstandard yet countable
integers, points at infinity in the compact natural integers, there
are already theories with infinite integers, as an example, of
intuitionistic thinking, supported by axiomatics and symbolic logic,
to the formal.

Regards,

Ross Finlayson

[forbi...@gmail.com]

So, here's the thing...

My intuition tells me there is no largest natural number
but all of them are distinguishable. There is exactly
one natural number that isn't a successor to another.

My intuition tells me that every node in the CIBT has
two child nodes even if I don't name them and this
means not naming them doesn't prevent them from existing
in the same way that not naming the successors between any
natural number and any other natural number doesn't mean
they don't exist.

For instance I don't have to name the successors between
200 and 100^100^100 and I don't need to write down the
natural number that is 100^100^100 to know it exists.

So, when considering the CIBT, that is the tree starting
with a node with value zero and all nodes having two children
nodes one valued zero and the other one where the top level
node representing two to the zeroeth power and each child
level represent either the inclusion or exclusion of two raised
to one less power as a component of a number that is the sum of
all included components in the infinite path, one can find all
numbers in the range [0,1] and some numbers can be found more
than once in the tree.

My intuition tells me that the number of paths is not defined
by the number of nodes because each node has an infinite number
of paths passing through it and each path either represents a
unique real number or one of two distinct paths representing
the same number.

No node completly defines a real number but rather a range of
real numbers. This is different from the rationals where each
rational represents a particular real number and many pairs of
natural numbers represent the same rational number so one can't say
anthing thing about the countability of the reals from the
CIBT or any node where one can say something about the countability
of irrationals from the set of rationals and about the countability
of the reals from the union of the set of rationals and any finite
sets of irrationals.

[WM]

On 1 Jan., 19:19, Zuhair <zaljo...@gmail.com> wrote:
> The distinguishability argument is a deep intuitive argument about the
> question of Countability of the reals. It is an argument of mine, it
> claims that the truth is that the reals are countable. However it
> doesn't claim that this truth can be put in a formal proof.

The distinguishability argument is neither deep nor intuitive. And is
not an argument of yours since you do not even understand its
implications. It is simply the basis of the axiom of extensionality.
How should we distinguish elements if they could not be distinguished?

we arrive finally at:
>
> FINAL CONCLUSION:
>
> The number of all reals is COUNTABLE.

Of course this would be the result if "countable" was a sensible
notion.

Regards, WM

[WM]

On 1 Jan., 19:38, George Greene <gree...@email.unc.edu> wrote:
> On Jan 1, 1:19 pm, Zuhair <zaljo...@gmail.com> wrote:
>
> > The distinguishability argument is a deep intuitive argument about the
> > question of Countability of the reals. It is an argument of mine, it
> > claims that the truth is that the reals are countable. However it
> > doesn't claim that this truth can be put in a formal proof.
>
> Are you just STUPID or what?  Not only can your argument not be "put
> in" a formal proof,
> IT CAN BE REFUTED by a formal proof!

Yes, hence the formal theory is disproved. Matheology does in fact not
belong to science. One of one points where Virgil is right.

Regards, WM

[WM]

On 1 Jan., 20:56, Zuhair <zaljo...@gmail.com> wrote:

> What I'm
> claiming is that at INTUITIVE level the distinguishability argument
> gives the impression that the Reals are Countable. However at FORMAL
> level we have Cantor's diagonal argument proving Uncountability of the
> Reals.

Have you meanwhile found out where Cantor published his formal proof?

Regards, WM

[WM]

Not uncountably many however. Not even actually infinitely many.
All you can distinguish up to a finite level n is 2^n. And you do
never leave a finite level.

Regards, WM

[WM]

No, it is wrong. All you can distinguish up to a finite level n is
2^n.
But of course it is only then wrong, if somebody believes in actual
infinitely many or even more. Every sober brain can in fact
distinguish all of them, namely 2^n up to level n.

Regards, WM

[WM]

On 1 Jan., 19:43, George Greene <gree...@email.unc.edu> wrote:

> Then "we" are blithering idiots because IT IS THE EDGES AND NOT the
> nodes
> that get labeled.
> Your obsession with thinking there is something to be gained by
> labeling nodes
> is really embarrassing.  To you.

A node is nothing but the end of an edge. It is the same as the bit in
a binary representation or a digit in a decimal representation of a
number.

In kindergarden, they show it by pictures like this:

This binary tree
0.
0 1
0101
contains the same numbers represented by sequences of node nodes that
here given in usual representation:
0.00
0.01
0.10
0.11

Therefore your remark is completely irrelevant for the proof that
Cantor failed.

Regards, WM

[WM]

On 2 Jan., 00:22, forbisga...@gmail.com wrote:

> For instance I don't have to name the successors between
> 200 and 100^100^100 and I don't need to write down the
> natural number that is 100^100^100 to know it exists.

You could not write down more than 10^50 real numbers as long as you
are tied to earth.

> My intuition tells me that the number of paths is not defined
> by the number of nodes because each node has an infinite number
> of paths passing through it

How could you distinguish these infinitely many paths if not by nodes?
As they all extend into the infinite, they all must cross one and the
same level (as soon as they exist). Hence they must be distinct on
this level by nodes, i.e., at least one node per path. You may say
that there is no level where all of them exist. That is true. Even
more true is: Nowhere an infinite number of paths will exist! At no
finite level! And there are no infinite levels! But if an infinte
number of paths would exist, as you in your splendid stupidity assume,
then they would not exist without at least as many different nodes
existing to distinguish them. - And this paragraph does not contain
any intuition.

If a child closes its eyes because it is afraid, then this is
understandable. If a man closes his brain because he is afraid to lose
his pet theory, then he is childish - and should at least refrain from
pretending to understand logic or matematics.

Regards, WM

[WM]

On 1 Jan., 19:41, George Greene <gree...@email.unc.edu> wrote:

> But if, instead of confirming that two are two, you really WERE IN
> DOUBT
> as to whether two were or were not ONE, were or were not EQUAL, then
> by THE USUAL algorithm,
> that would only be semi-decidable,

SEMI-DECIDABLE! That sounds as if intelligent and knowledgeable people
created the language of matheology - and not a gang of incompetent
bunglers.

No real number can be given by an infinite string of digits! No
reasonable person has ever tried that. Therefore no reasonable person
will ever try to distinguish two infinite strings. Everything that can
be accomplished in order to define a real number is using a finite
word like 0.111... or 1/9 or pi or crt(3).

Semi-decidable in this framework is a word that shows that its creator
tried to appear as a knowledgeable man but is in fact an amateur like
that penpusher who scribbled in the holy bible that God created the
whale among the fishes. Typical cases of pretending knowledge that is
more apparent than real.

Regards, WM

[Virgil]

In article
<8f627aa7-a0d2-4554...@17g2000vba.googlegroups.com>,

A set is countable if and only if there is a surjection from the set of
naturals to that set.

This definition is perfectly sensible in ZFC even if WM claims that it
is not sensible in his WMytheology.

And ZFC is considerably more sensible than WMytheology.
--

[Bill Taylor]

On Jan 2, 7:19 am, Zuhair <zaljo...@gmail.com> wrote:

> The distinguishability argument is a deep intuitive argument about the
> question of Countability of the reals. It is an argument of mine, it
> claims that the truth is that the reals are countable. However it
> doesn't claim that this truth can be put in a formal proof.

I have a deep, intuitive argument that 3 equals 7.
However, this truth cannot be put in a formal proof.

Unfortunately, the informal argument is just
too long to put into a sig-line.

-- Blethering Bill

** The informal argument that 3 equals 7 involves noticing that if

[Ralf Bader]

WM wrote:

> On 1 Jan., 19:19, Zuhair <zaljo...@gmail.com> wrote:
>> The distinguishability argument is a deep intuitive argument about the
>> question of Countability of the reals. It is an argument of mine, it
>> claims that the truth is that the reals are countable. However it
>> doesn't claim that this truth can be put in a formal proof.
>
> The distinguishability argument is neither deep nor intuitive.

It is not even an argument, just question-begging.

> And is
> not an argument of yours since you do not even understand its
> implications. It is simply the basis of the axiom of extensionality.
> How should we distinguish elements if they could not be distinguished?
>
> we arrive finally at:
>>
>> FINAL CONCLUSION:
>>
>> The number of all reals is COUNTABLE.
>
> Of course this would be the result if "countable" was a sensible
> notion.

You even know what the result would be if non-sensible notions involved
were sensible. Mückenheim, you are either the Greatest Genius Of All Times
or one of the greatest idiots.

[Virgil]

In article <kc2tp8$f8o$1...@news.m-online.net>,

And there is sufficient evidence to eliminate the former possibility.
--

[Zuhair]

On Jan 3, 6:26 am, Ralf Bader <ba...@nefkom.net> wrote:
> WM wrote:
> > On 1 Jan., 19:19, Zuhair <zaljo...@gmail.com> wrote:
> >> The distinguishability argument is a deep intuitive argument about the
> >> question of Countability of the reals. It is an argument of mine, it
> >> claims that the truth is that the reals are countable. However it
> >> doesn't claim that this truth can be put in a formal proof.
>
> > The distinguishability argument is neither deep nor intuitive.
>
> It is not even an argument, just question-begging.
>

Call it what may you, what is there is:

(1) ALL reals are distinguishable on finite basis

(2) Distinguishability on finite basis is COUNTABLE.

So we conclude that:

"The number of all reals distinguishable on finite basis must be
countable".

Since ALL reals are distinguishable on finite basis, then:

"The number of all reals is countable".

Because generally speaking no set contain more elements than what it
CAN have. So you cannot distinguish more reals than what you CAN
distinguish. Since all reals are distinguished by finite initial
segments of them, and since we only have COUNTABLY many such finite
initial segments, then for the first glance it seems that there ought
to be COUNTABLY many reals so distinguished. This is what our
intuition would expect!

Nobody can say that this simple and even trivial line of thought have
no intuitive appeal. Definitely there is some argument there, at least
at intuitive level.

However Cantor's arguments all of which are demonstrated by explicit
and rigorous formal proofs have refuted the above-mentioned intuitive
gesture, however that doesn't make out of Cantor's argument an
intuitive one, no, Cantor's argument remains COUNTER-INTUITIVE, it had
demonstrated a result that came to the opposite of our preliminary
intuitive expectation.

Zuhair

[gus gassmann]

On 03/01/2013 5:31 AM, Zuhair wrote:

> Call it what may you, what is there is:

> (1) ALL reals are distinguishable on finite basis
> (2) Distinguishability on finite basis is COUNTABLE.

What does this mean? If you have two _different_ reals r1 and r2, then
you can establish this fact in finite time. The set of reals that are
describable by finite strings over a finite character set is countable.
However, not all reals have that property.

> So we conclude that:
>
> "The number of all reals distinguishable on finite basis must be
> countable".
>
> Since ALL reals are distinguishable on finite basis, then:

You seem to use "distinguishable" in two different ways.

Seeing your argument reminds me of the old chestnut about cats: A cat
has three tails. Proof: No cat has two tails. A cat has one tail more
than no cat. QED.

[Zuhair]

On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote:
> On 03/01/2013 5:31 AM, Zuhair wrote:
>
> > Call it what may you, what is there is:
> > (1) ALL reals are distinguishable on finite basis
>
>  > (2) Distinguishability on finite basis is COUNTABLE.
>
> What does this mean? If you have two _different_ reals r1 and r2, then
> you can establish this fact in finite time. The set of reals that are
> describable by finite strings over a finite character set is countable.
> However, not all reals have that property.

I already have written the definition of that in another post, and
this post comes in continuation to that post, to reiterate:

r1 is distinguished from r2 on finite basis <->
Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th
digit
of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)

>
> > So we conclude that:
>
> > "The number of all reals distinguishable on finite basis must be
> > countable".
>
> > Since ALL reals are distinguishable on finite basis, then:
>
> You seem to use "distinguishable" in two different ways.
>
> Seeing your argument reminds me of the old chestnut about cats: A cat
> has three tails. Proof: No cat has two tails. A cat has one tail more
> than no cat. QED.
>
>

There is nothing of that. The intuitive argument of mine here is clear
as far as its presentation is concerned, I didn't mention the
definition of "distinguishability on finite basis" because it is well
known (actually I was asked to SHUT UP by one of the posters because I
mentioned explicitly the definition of it?) and because this topic
actually comes as a continuation to earlier threads on this topic
presented by myself to this Usenet only recently.

Zuhair

[Jesse F. Hughes]

Zuhair <zalj...@gmail.com> writes:

> On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote:
>> On 03/01/2013 5:31 AM, Zuhair wrote:
>>
>> > Call it what may you, what is there is:
>> > (1) ALL reals are distinguishable on finite basis
>>
>>  > (2) Distinguishability on finite basis is COUNTABLE.
>>
>> What does this mean? If you have two _different_ reals r1 and r2, then
>> you can establish this fact in finite time. The set of reals that are
>> describable by finite strings over a finite character set is countable.
>> However, not all reals have that property.
>
> I already have written the definition of that in another post, and
> this post comes in continuation to that post, to reiterate:
>
> r1 is distinguished from r2 on finite basis <->
> Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th
> digit
> of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)

So, what does statement (2) mean and what sort of argument can you give
in its favor?

It seems like you mean something like:

Let S be any set such that every pair of distinct elements r, s in S
are distinguishable on finite basis. Then S is countable.

I don't see why that should be true at all. It seems plainly obvious to
me, for instance, that the set of paths in the full binary tree are an
example of such a set S, but that there are uncountably many of them.

I know you said that this "argument" of yours may not be capable of
being expressed as a formal proof, but you have to give me *some* reason
to think that (2) is at least plausible.

--
"The Hammer is not force. It is absolute power. The Hammer is from Idea Space.
That's the real world. Here is the magical realm.
You are creatures in that realm, who do not quite understand.
But it doesn't matter. There is a story to be told..." James S. Harris, poet.

[gus gassmann]

On 03/01/2013 8:58 AM, Zuhair wrote:
> On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote:
>> On 03/01/2013 5:31 AM, Zuhair wrote:
>>
>>> Call it what may you, what is there is:
>>> (1) ALL reals are distinguishable on finite basis
>>
>> > (2) Distinguishability on finite basis is COUNTABLE.
>>
>> What does this mean? If you have two _different_ reals r1 and r2, then
>> you can establish this fact in finite time. The set of reals that are
>> describable by finite strings over a finite character set is countable.
>> However, not all reals have that property.
>
> I already have written the definition of that in another post, and
> this post comes in continuation to that post, to reiterate:
>
> r1 is distinguished from r2 on finite basis <->
> Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th
> digit
> of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)

Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*

reals r1 and r2, then you can establish this fact in finite time.

However, if you are given two different descriptions of the *SAME* real,
you will have problems. How do you find out that NOT exist n... in
finite time?

Moreover, being able to distinguish two reals at a time has nothing at
all to do with the question of how many there are, or how to distinguish
more than two. Your (2) uses a _different_ concept of distinguishability.

[WM]

On 3 Jan., 04:26, Ralf Bader <ba...@nefkom.net> wrote:
> WM wrote:
> > On 1 Jan., 19:19, Zuhair <zaljo...@gmail.com> wrote:
> >> The distinguishability argument is a deep intuitive argument about the
> >> question of Countability of the reals. It is an argument of mine, it
> >> claims that the truth is that the reals are countable. However it
> >> doesn't claim that this truth can be put in a formal proof.
>
> > The distinguishability argument is neither deep nor intuitive.
>
> It is not even an argument, just question-begging.

It is prerequisite when dealing with numbers. And it is exactly what
Cantor applied, He distinguished numbers at finite places.

>
> > And is
> > not an argument of yours since you do not even understand its
> > implications. It is simply the basis of the axiom of extensionality.
> > How should we distinguish elements if they could not be distinguished?
>
> > we arrive finally at:
>
> >> FINAL CONCLUSION:
>
> >> The number of all reals is COUNTABLE.
>
> > Of course this would be the result if "countable" was a sensible
> > notion.
>
> You even know what the result would be if non-sensible notions involved
> were sensible. Mückenheim, you are either the Greatest Genius Of All Times
> or one of the greatest idiots.

It is not necessary to be a genius in order to recognize blocked
brains. Every average psychiatrist knows that. And it is not
necessary to be an idiot in order to believe that Cantor by
distinguishing real nunbers "proved" the existence of
indistinguishable numbers. But, of course, it helps.

Regards, WM

[WM]

On 3 Jan., 13:23, gus gassmann <g...@nospam.com> wrote:
> On 03/01/2013 5:31 AM, Zuhair wrote:
>
> > Call it what may you, what is there is:
> > (1) ALL reals are distinguishable on finite basis
>
>  > (2) Distinguishability on finite basis is COUNTABLE.
>
> What does this mean? If you have two _different_ reals r1 and r2, then
> you can establish this fact in finite time. The set of reals that are
> describable by finite strings over a finite character set is countable.
> However, not all reals have that property.

Perhaps not all, but all that can be distinguished in a Cantor list.

>
> > So we conclude that:
>
> > "The number of all reals distinguishable on finite basis must be
> > countable".
>
> > Since ALL reals are distinguishable on finite basis, then:
>
> You seem to use "distinguishable" in two different ways.

But you don't dare to say what these differences are.

>
> Seeing your argument reminds me of the old chestnut about cats: A cat
> has three tails. Proof: No cat has two tails. A cat has one tail more
> than no cat. QED.
>

Unfortunately this joke has nothing to do with the question whether
cats exist or whether matheologians can be intelligent.

Regards, WM

[Zuhair]

On Jan 3, 4:52 pm, gus gassmann <g...@nospam.com> wrote:
> On 03/01/2013 8:58 AM, Zuhair wrote:
>
>
>
>
>
>
>
>
>
> > On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote:
> >> On 03/01/2013 5:31 AM, Zuhair wrote:
>
> >>> Call it what may you, what is there is:
> >>> (1) ALL reals are distinguishable on finite basis
>
> >>   > (2) Distinguishability on finite basis is COUNTABLE.
>
> >> What does this mean? If you have two _different_ reals r1 and r2, then
> >> you can establish this fact in finite time. The set of reals that are
> >> describable by finite strings over a finite character set is countable.
> >> However, not all reals have that property.
>
> > I already have written the definition of that in another post, and
> > this post comes in continuation to that post, to reiterate:
>
> > r1 is distinguished from r2 on finite basis <->
> > Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th
> > digit
> > of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)
>
> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*
> reals r1 and r2, then you can establish this fact in finite time.
> However, if you are given two different descriptions of the *SAME* real,
> you will have problems. How do you find out that NOT exist n... in
> finite time?
>

That is irrelevant (or at least that's how it appears to me) ANY two
distinct (i.e. different) reals can be distinguished in finite time
and any two reals that are distinguished in finite time are distinct.
That's all what we need. The question of revealing the identity of
some real in finite time is another matter, and my argument do not
involve it at all, so it is irrelevant.

> Moreover, being able to distinguish two reals at a time has nothing at
> all to do with the question of how many there are, or how to distinguish
> more than two. Your (2) uses a _different_ concept of distinguishability.

(2) simple refers to how many finite initial segments of reals can be
distinguishable? i.e. what is the total number of such segments.
Clearly we have COUNTABLY many finite initial segments of reals. In
other words we cannot distinguish more than COUNTABLY many finite
initial segments of reals. Of course all of those are distinguished in
finite time no doubt.

I think that distinguishability in (2) is the same distinguishability
in (1) it has exactly the same definition.

We have only COUNTABLY many finite initial segments of reals that we
can distinguish of course on finite basis, that's what is available,
we don't have more.

Now every Two distinct reals are distinguishable on FINITE basis. But
we have only Countably many finite initial segments of reals available
for us to distinguish reals by, so how come we've ended up with
UNCOUNTABLY many reals, what is the source of the excess in the number
of reals, how can we distinguish more than what is available for us to
distinguish. This is like saying that we have Three SEEDS, and Each
Two distinct TREES grown from planting the three seeds must have Two
distinct seeds where each Tree have grown from one seed, and then one
comes and say that planting the three seeds had resulted in FOUR
Trees? This is an example of a product outnumbering the potential of
production?

By the way I might be wrong of course, I'll be glad to have anyone
spot my error, my analogies might simply be misleading.

Zuhair

[WM]

On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote:

> Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*
> reals r1 and r2, then you can establish this fact in finite time.
> However, if you are given two different descriptions of the *SAME* real,
> you will have problems. How do you find out that NOT exist n... in
> finite time?

Does that in any respect increase the number of real numbers? And if
not, why do you mention it here?

>
> Moreover, being able to distinguish two reals at a time has nothing at
> all to do with the question of how many there are, or how to distinguish

> more than two. Your (2) uses a _different_ concept of distinguishability.-

Being able to distinguish a real from all other reals is crucial for
Cantor's argument. "Suppose you have a list of all real numbers ..."
How could you falsify this statement if not by creating a real number
that differs observably and provably from all entries of this list?

Regards, WM

[WM]

On 3 Jan., 17:30, Zuhair <zaljo...@gmail.com> wrote:

>
> I think that distinguishability in (2) is the same distinguishability
> in (1) it has exactly the same definition.

Of course it is. To distinguish or not to distinguish, that is the
question.

>
> We have only COUNTABLY many finite initial segments of reals that we
> can distinguish of course on finite basis, that's what is available,
> we don't have more.

And Cantor does not pretend to use more in his argument. So whether
there are indistinguishable reals is completely irrelevant for his
argument and the set of reals considered by him.

> By the way I might be wrong of course, I'll be glad to have anyone
> spot my error,

You will not find anybody to do so. The matheologians only blather
irrelevant nonsense because it is obvious to every sober brain, that
Cantor was in error. But it seems to be so deeply inplanted in most
mathematicians brains that they are incapable of thinking the
opposite. I enjoy every semester the experience that 40 young and very
bright student understand immediately. Their only advantage is that
most of them never heard about Cantor until one week before I tell
them the truth.

Regards, WM

[Virgil]

In article
<93f76682-c212-43a1...@a15g2000vbf.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 3 Jan., 04:26, Ralf Bader <ba...@nefkom.net> wrote:
> > WM wrote:
> > > On 1 Jan., 19:19, Zuhair <zaljo...@gmail.com> wrote:
> > >> The distinguishability argument is a deep intuitive argument about the
> > >> question of Countability of the reals. It is an argument of mine, it
> > >> claims that the truth is that the reals are countable. However it
> > >> doesn't claim that this truth can be put in a formal proof.
> >
> > > The distinguishability argument is neither deep nor intuitive.
> >
> > It is not even an argument, just question-begging.
>
> It is prerequisite when dealing with numbers. And it is exactly what
> Cantor applied, He distinguished numbers at finite places.

Cantor may have distinguished numerals at finite places, but numerals
are not numbers. "1.2" and "0.5" are different as numerals but the
represent the same number.

> >
> > You even know what the result would be if non-sensible notions involved

> > were sensible. M�ckenheim, you are either the Greatest Genius Of All Times

> > or one of the greatest idiots.

And, considering the obvious flaws in his attempts at proofs, he is NOT
the greatest genius.
--

[Virgil]

In article
<bfa6347b-2078-4418...@a15g2000vbf.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 3 Jan., 13:23, gus gassmann <g...@nospam.com> wrote:
> > On 03/01/2013 5:31 AM, Zuhair wrote:
> >
> > > Call it what may you, what is there is:
> > > (1) ALL reals are distinguishable on finite basis
> >
> >  > (2) Distinguishability on finite basis is COUNTABLE.
> >
> > What does this mean? If you have two _different_ reals r1 and r2, then
> > you can establish this fact in finite time. The set of reals that are
> > describable by finite strings over a finite character set is countable.
> > However, not all reals have that property.
>
> Perhaps not all, but all that can be distinguished in a Cantor list.

The thing is that every such list by its own existence proves the
existence of reals not listed in it.

> >
> > > So we conclude that:
> >
> > > "The number of all reals distinguishable on finite basis must be
> > > countable".
> >
> > > Since ALL reals are distinguishable on finite basis, then:
> >
> > You seem to use "distinguishable" in two different ways.
>
> But you don't dare to say what these differences are.

It is WM who does not dare, as it would reveal his errors.

> >
> > Seeing your argument reminds me of the old chestnut about cats: A cat
> > has three tails. Proof: No cat has two tails. A cat has one tail more
> > than no cat. QED.
> >
> Unfortunately this joke has nothing to do with the question whether
> cats exist or whether matheologians can be intelligent.

WM certainly cannot be.
--

[Virgil]

In article
<a60601d5-24a2-4501...@ci3g2000vbb.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 3 Jan., 14:52, gus gassmann <g...@nospam.com> wrote:
>
> > Exactly. This is precisely what I wrote. IF you have TWO *DIFFERENT*
> > reals r1 and r2, then you can establish this fact in finite time.
> > However, if you are given two different descriptions of the *SAME* real,
> > you will have problems. How do you find out that NOT exist n... in
> > finite time?
>
> Does that in any respect increase the number of real numbers? And if
> not, why do you mention it here?

It shows that WM considerably oversimplifies the issue of
distinguishing between different reals, or even different names for the
same reals.

> >
> > Moreover, being able to distinguish two reals at a time has nothing at
> > all to do with the question of how many there are, or how to distinguish
> > more than two. Your (2) uses a _different_ concept of distinguishability.-
>
> Being able to distinguish a real from all other reals is crucial for
> Cantor's argument. "Suppose you have a list of all real numbers ..."
> How could you falsify this statement if not by creating a real number
> that differs observably and provably from all entries of this list?

Actually, all that is needed in the diagonal argument is the ability
distinguish one real from another real, one pair of reals at a time.
--

[Virgil]

In article
<6f0337ce-7dee-4af5...@d4g2000vbw.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 3 Jan., 17:30, Zuhair <zaljo...@gmail.com> wrote:
>
> >
> > I think that distinguishability in (2) is the same distinguishability
> > in (1) it has exactly the same definition.
>
> Of course it is. To distinguish or not to distinguish, that is the
> question.
> >
> > We have only COUNTABLY many finite initial segments of reals that we
> > can distinguish of course on finite basis, that's what is available,
> > we don't have more.
>
> And Cantor does not pretend to use more in his argument. So whether
> there are indistinguishable reals is completely irrelevant for his
> argument and the set of reals considered by him.

Thus you validate Cantor's diagonal argument regardless of whether ther
are any indistinguishable reals at all.

>
> > By the way I might be wrong of course, I'll be glad to have anyone
> > spot my error,
>
> You will not find anybody to do so. The matheologians only blather
> irrelevant nonsense because it is obvious to every sober brain, that
> Cantor was in error.

Except that a great many brains far more sober than WM's has ever been
disagree with him on that issue.

That WM has a bee in his bonnet does not mean its buzzing need bother
anyone else.

> But it seems to be so deeply inplanted in most
> mathematicians brains that they are incapable of thinking the
> opposite. I enjoy every semester the experience that 40 young and very
> bright student understand immediately. Their only advantage is that
> most of them never heard about Cantor until one week before I tell
> them the truth.

Poor sods may take years recovering from having WM's bees inserted into
their bonnets.

It is a shame that the German Educational system allows such evils to
occur.
--

[Virgil]

In article <87ehi29...@phiwumbda.org>,

Query: While JSH clearly has somewhat less connection to reality than
WM, does WM's greater power to infect his students make him more to a
threat to sanity?
--

[fom]

On 1/3/2013 6:58 AM, Zuhair wrote:
> On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote:
>> On 03/01/2013 5:31 AM, Zuhair wrote:
>>
>>> Call it what may you, what is there is:
>>> (1) ALL reals are distinguishable on finite basis
>>
>> > (2) Distinguishability on finite basis is COUNTABLE.
>>
>> What does this mean? If you have two _different_ reals r1 and r2, then
>> you can establish this fact in finite time. The set of reals that are
>> describable by finite strings over a finite character set is countable.
>> However, not all reals have that property.
>
> I already have written the definition of that in another post, and
> this post comes in continuation to that post, to reiterate:
>
> r1 is distinguished from r2 on finite basis <->
> Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th
> digit
> of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)

There is a difference between saying ALL REALS are
distinguishable and what you have written above. What
you have written asserts that EVERY PAIR OF DISTINCT REALS
have discernible representations. It simply says
that the representation of the real numbers to which it
refers can serve as canonical names.

Dedekind cuts define all reals.

Cantor fundamental sequences define all reals.

The Euclidean algorithm does not define any reals.

You may, as WM does, deny uses of a completed infinity.

You may, as Skolem does, question the meaningfulness of
nondenumerability with respect to finitely-generated
countable languages.

But you need to recognize that the problem here
is a poorly constructed statement.

[fom]

YES!

It sure is nice to see the problem in the definition.

[fom]

On 1/3/2013 10:30 AM, Zuhair wrote:

> By the way I might be wrong of course, I'll be glad to have anyone
> spot my error, my analogies might simply be misleading.

All right.

Why did Dedekind make his investigations?

Why did Bolzano feel compelled to prove the
intermediate value theorem?

Why was Cauchy careful to not say that the
fundamental sequences converged into the
space from which their elements had been
given?

I realize that you are not talking about
those subjects. But you are taking them
to the garbage heap -- along with every
plausible piece of mathematics that uses
the completeness axiom for the real numbers.

You cannot prove the fundamental theorem
of algebra without results from analysis.
It requires the existence of irrational
roots for polynomials and the intermediate
value theorem. So, you are tossing
algebra onto the same heap with analysis.


Now, there is a circularity in the topology
of real numbers. If you want to have

x=y

it must satisfy the axioms of a metric
space. But those axioms are too
strong.

Go get yourself a copy of "General Topology"
by Kelley and read about uniformities and
the metrization lemma for systems of relations.

What you will find is that the metric space
axioms (the important direction associated
with pseudometrics) depend on the least upper
bound principle.

One can simply view it as fundamental sequences
being grounded by cuts. It is not circular
in that sense. It simply makes Dedekind prior
to Cantor.

Before you continue with this mess, you should
take some time to learn what it means for two
real numbers to be equal to one another.

It is not the Euclidean algorithm.

[fom]

One canonical name from another canonical name.

[Virgil]

In article
<6302ee90-f0a2-4be5...@c16g2000yqi.googlegroups.com>,

Zuhair <zalj...@gmail.com> wrote:

> Since all reals are distinguished by finite initial
> segments of them,

Some reals are distinguished by finite initial segments of their decimal
representations, most are not.

Sqrt(2) cannot be distinguished from ALL other reals by any finite
initial segment of its decimal representation, but it can be so
distinguished by the decimal representation of its square.

And there are at least countably many other reals for which the same is
true.
--

[Virgil]

In article <cM-dnQvN5M9NtnvN...@giganews.com>,

Just so!
--

[forbi...@gmail.com]

On Thursday, January 3, 2013 8:30:09 AM UTC-8, Zuhair wrote:
> ANY two
> distinct (i.e. different) reals can be distinguished in finite time
> and any two reals that are distinguished in finite time are distinct.

Actually, I wonder about that assertion.
Between any two rationals lie an infinite set of rationals and
irrationals. An irrational number has no repeating sequence.
How do you go about distinguishing two irrationals whose distinguisability
lies beyond any arbitrary depth of the binary tree?

I found this for those feeling lucky:
http://www.stanford.edu/~nitsche/cgi-bin/numbers.pl?googolplex

[Zuhair]

On Jan 4, 5:33 am, Virgil <vir...@ligriv.com> wrote:
> In article
> <6302ee90-f0a2-4be5-9dbb-c1f999c3a...@c16g2000yqi.googlegroups.com>,

>
>  Zuhair <zaljo...@gmail.com> wrote:
> > Since all reals are distinguished by finite initial
> > segments of them,
>
> Some reals are distinguished by finite initial segments of their decimal
> representations, most are not.
>

r is distinguishable on finite basis iff For Every real x. ~x=r ->
Exist n: d_n of r =/= d_n of x.

As far as I know every real is so distinguishable.

In your version you changed the quantifier order, your version is
speaking about the following:

r is distinguishable on finite basis iff Exist n. For Every real x.
~x=r -> d_n of r =/= d_n of x.

Of course all reals are to be represented by *INFINITE* binary decimal
expansions, so 0.12 is represented as 0.120000...

So we are not speaking about the same distinguishability criterion.

Zuhair

[Zuhair]

On Jan 4, 6:31 am, forbisga...@gmail.com wrote:
> On Thursday, January 3, 2013 8:30:09 AM UTC-8, Zuhair wrote:
> > ANY two
> > distinct (i.e. different) reals can be distinguished in finite time
> > and any two reals that are distinguished in finite time are distinct.
>
> Actually, I wonder about that assertion.

By the way I'm treating reals as Infinite binary digit sequences, so
I'm really not making modifications to cover the case where two such
sequence would come to refer to the same real like in 0.0111... and
0.1000..., I'm really not taking this issue into concern because it
can be dispensed with, since the main element in the diagonal argument
is actually uncountability of those sequences.

> Between any two rationals lie an infinite set of rationals and
> irrationals.  An irrational number has no repeating sequence.
> How do you go about distinguishing two irrationals whose distinguisability
> lies beyond any arbitrary depth of the binary tree?
>

Distinguishability of reals on finite basis is applicable to ALL the
reals whether rational they were or not, it doesn't matter, ANY reals
x,y are said to be distinguishable on finite basis, I again emphasis
ANY.

Zuhair

[Zuhair]

Dear fom I'm not against Uncountability, I'm not against Cantor's
argument. I'm saying that Cantor's argument is CORRECT. All what I'm
saying is that it is COUNTER-INTUITIVE as it violates the
Distinguishability argument which is an argument that comes from
intuition excerised in the FINITE world. That's all.

Zuhair

[Zuhair]

On Jan 4, 6:31 am, forbisga...@gmail.com wrote:

> On Thursday, January 3, 2013 8:30:09 AM UTC-8, Zuhair wrote:
> > ANY two
> > distinct (i.e. different) reals can be distinguished in finite time
> > and any two reals that are distinguished in finite time are distinct.
>
> Actually, I wonder about that assertion.
> Between any two rationals lie an infinite set of rationals and
> irrationals.  An irrational number has no repeating sequence.
> How do you go about distinguishing two irrationals whose distinguisability
> lies beyond any arbitrary depth of the binary tree?

It doesn't matter irrationals are also distinguishable in finite time
for any irrational r1 in order for it to be distinguished from any
irrational r2 (or any real) you must have some natural n such that the
n_th digit of the infinite decimal expansion of r1 is distinct from
the n_th digit of the infinite decimal expansion of r2.

Zuhair

[Virgil]

In article
<3c133339-6b4c-4f74...@t5g2000vba.googlegroups.com>,

Exactly!
--

[Zuhair]

On Jan 4, 10:22 am, Virgil <vir...@ligriv.com> wrote:
> In article

> <3c133339-6b4c-4f74-937f-804bdaad3...@t5g2000vba.googlegroups.com>,

which mean that your objection is irrelevant to my argument. I think
that the argument that I've presented shows some COUNTER-INTUITIVENESS
to uncountability, that's all.

Zuhair

[WM]

On 4 Jan., 10:54, Zuhair <zaljo...@gmail.com> wrote:
> On Jan 4, 10:22 am, Virgil <vir...@ligriv.com> wrote:
>
>
>
>
>
> > In article
> > <3c133339-6b4c-4f74-937f-804bdaad3...@t5g2000vba.googlegroups.com>,
>
> >  Zuhair <zaljo...@gmail.com> wrote:
> > > On Jan 4, 5:33 am, Virgil <vir...@ligriv.com> wrote:
> > > > In article
> > > > <6302ee90-f0a2-4be5-9dbb-c1f999c3a...@c16g2000yqi.googlegroups.com>,
>
> > > > Zuhair <zaljo...@gmail.com> wrote:
> > > > > Since all reals are distinguished by finite initial
> > > > > segments of them,
>
> > > > Some reals are distinguished by finite initial segments of their decimal
> > > > representations, most are not.

Those are not different numbers. Such objects cannot appear in any
Cantor list as entries or diagonal

> > > Of course all reals are to be represented by *INFINITE* binary decimal
> > > expansions, so 0.12 is represented as 0.120000...

It is impossible to represent any real number by an infinite expansion
that is not defined by a finite word.

>
> > > So we are not speaking about the same distinguishability criterion.

There is no other criterion.

>
> which mean that your objection is irrelevant to my argument. I think
> that the argument that I've presented

that you have parroted without understanding its implications

> shows some COUNTER-INTUITIVENESS
> to uncountability, that's all.

Have you ever seen that Cantor's argument works without
distinguishability? Why must b_n =/= a_nn? Never wondered why that is
required at a finite n?Anybody who pretends that there are numbers
that cannot be distinguished is outside of mathematics and even
outside of Cantor's argument and its implications.

Regards, WM

[gus gassmann]

Exactly. The only reals that matter to Cantor's argument are the
*countably* many that are assumed to have been written down. There is no
need (nor indeed an effective way) to distinguish the constructed
diagonal from *all* the potential numbers that could have been
constructed that are not on the list, either. Any *one* number not on
the list shows that the list is incomplete and thus establishes the
uncountability of the reals.

[WM]

On 3 Jan., 22:36, Virgil <vir...@ligriv.com> wrote:

> > It is prerequisite when dealing with numbers. And it is exactly what
> > Cantor applied, He distinguished numbers at finite places.
>
> Cantor may have distinguished numerals at finite places, but numerals

So Cantor "proved" that there are uncountably many numerals?
Why then does anybody believe in uncountably many numbers?

> are not numbers. "1.2" and "0.5" are different as numerals but the
> represent the same number.

Another remarkable observation.


Regards, WM

[WM]

On 4 Jan., 01:35, fom <fomJ...@nyms.net> wrote:

> Dedekind cuts define all reals.

Every cut is defined by a finite word. The set of definable cuts is
the set of cuts and is countable.

> Cantor fundamental sequences define all reals.

No infinite definition defines anything.

> You may, as WM does, deny uses of a completed infinity.

I do not deny it, but show that it is self-contradictory.

Regards, WM

[WM]

On 4 Jan., 04:31, forbisga...@gmail.com wrote:
> On Thursday, January 3, 2013 8:30:09 AM UTC-8, Zuhair wrote:
> > ANY two
> > distinct (i.e. different) reals can be distinguished in finite time
> > and any two reals that are distinguished in finite time are distinct.
>
> Actually, I wonder about that assertion.

Are you sure to post in a group that is approriate for you?

> Between any two rationals lie an infinite set of rationals and
> irrationals.  An irrational number has no repeating sequence.
> How do you go about distinguishing two irrationals whose distinguisability
> lies beyond any arbitrary depth of the binary tree?

How do you distinguish numbers that cannot be distinguished? And why
would you call them distinct?

Regards, WM

[WM]

On 4 Jan., 13:36, gus gassmann <g...@nospam.com> wrote:
> On 03/01/2013 5:53 PM, Virgil wrote:
>
>
>
>
>
> > In article

> > <a60601d5-24a2-4501-a28b-84a7b1e53...@ci3g2000vbb.googlegroups.com>,

Therefore they cannot interfere with Cantor's argument and cannot
result from his procedure.

> Any *one* number not on
> the list shows that the list is incomplete and thus establishes the
> uncountability of the reals

No, it establishes the incompleteness of infinity or the infinity of
incompleteness.

Cantor's list establishes the uncountability of distinguishable and
hence constructable reals. Why should nonconstructable and hence
nondistinguishable reals matter in Cantor's argument?

Cantor proves the uncountability of a countable set. For some people
that has an effect like a drug.

Regards, WM

[Jesse F. Hughes]

Zuhair <zalj...@gmail.com> writes:

> Dear fom I'm not against Uncountability, I'm not against Cantor's
> argument. I'm saying that Cantor's argument is CORRECT. All what I'm
> saying is that it is COUNTER-INTUITIVE as it violates the
> Distinguishability argument which is an argument that comes from
> intuition excerised in the FINITE world. That's all.

But you've neither explained the meaning of your second premise nor
given any indication why it is plausible.

--
Jesse F. Hughes

"How lucky we are to be able to hear how miserable Willie Nelson could
imagine himself to be." -- Ken Tucker on Fresh Air

[fom]

Fair enough.

Much of what you have been posting seems
to be "confused" between the naive finitism
such as WM is asserting and the kind of
finitism that leads to constructive mathematics.

It seems, however, that you are merely
"investigating" matters.

At the heart of matters is the question of
how mathematics seems to have the explanatory
force that it has. Perhaps it is geometry. But
when we go to represent the geometric form
arithmetically, we are confronted with the
fact that a system of names is different from
the ostensive dubbing of a name.

Descartes introduced the problem. Every
piece of analytic geometry presupposes a
consistent global labeling of points.
Newton and Leibniz made it apparent with
the calculus.

The modern calculus gets around the foundational
problem posed by differentials with the
"little-oh" mechanism that ignores errors
by virtue of the fact that errors approach
zero at least as quickly as the differential
takes its value. It is somewhat humorous
since Berkeley's analysis of Newton's calculus
asserted that its effectiveness was analogous
to bad bookkeeping in which an error in one
set of books is countered by another error
in a second set of books. History seems to
have forgotten that even though calculus has
banished the infinitesimal.

In any case, have fun. I will leave you
to your "investigating".

[fom]

On 1/4/2013 5:41 AM, WM wrote:
> On 4 Jan., 10:54, Zuhair <zaljo...@gmail.com> wrote:
>> On Jan 4, 10:22 am, Virgil <vir...@ligriv.com> wrote:
>>
>>
>>
>>
>>
>>> In article
>>> <3c133339-6b4c-4f74-937f-804bdaad3...@t5g2000vba.googlegroups.com>,
>>
>>> Zuhair <zaljo...@gmail.com> wrote:
>>>> On Jan 4, 5:33 am, Virgil <vir...@ligriv.com> wrote:
>>>>> In article
>>>>> <6302ee90-f0a2-4be5-9dbb-c1f999c3a...@c16g2000yqi.googlegroups.com>,
>>
>>>>> Zuhair <zaljo...@gmail.com> wrote:
>>>>>> Since all reals are distinguished by finite initial
>>>>>> segments of them,
>>
>>>>> Some reals are distinguished by finite initial segments of their decimal
>>>>> representations, most are not.
>
> Those are not different numbers. Such objects cannot appear in any
> Cantor list as entries or diagonal
>
>>>> Of course all reals are to be represented by *INFINITE* binary decimal
>>>> expansions, so 0.12 is represented as 0.120000...
>
> It is impossible to represent any real number by an infinite expansion
> that is not defined by a finite word.
>>
>>>> So we are not speaking about the same distinguishability criterion.
>
> There is no other criterion.

In logic, discernibility is taken to be with
respect to properties. I personally have a problem
with too much work with parameters (symbols taken to
have the characteristic of definite names but actually
varying over whatever one purports to be speaking
about) and too little work with names that actually
resolve to truth. So, arguing with parameters
ranging over properties does little more than
unfold the circularity of defining an object into
an infinite hierarchy.

Your position seems to be that since the names determine
the model which, in turn, determines the truth, then the
names are the only criterion.

But, the use/mention distinction associated with
names leads to a similar hierarchy:

oo

'oo'

''oo''

'''oo'''

and so on.

<snip>

[fom]

On 1/4/2013 10:27 AM, WM wrote:
> On 3 Jan., 22:36, Virgil <vir...@ligriv.com> wrote:
>
>>> It is prerequisite when dealing with numbers. And it is exactly what
>>> Cantor applied, He distinguished numbers at finite places.
>>
>> Cantor may have distinguished numerals at finite places, but numerals
>
>
> So Cantor "proved" that there are uncountably many numerals?
> Why then does anybody believe in uncountably many numbers?
>

Descartes.

He put the numbers on the geometry.

When the geometry failed to be canonical, they
put the geometry on the numbers.

[fom]

On 1/4/2013 10:29 AM, WM wrote:
> On 4 Jan., 01:35, fom <fomJ...@nyms.net> wrote:
>
>> Dedekind cuts define all reals.
>
> Every cut is defined by a finite word. The set of definable cuts is
> the set of cuts and is countable.
>
>> Cantor fundamental sequences define all reals.
>
> No infinite definition defines anything.

No infinite definition is finitely realizable.

The problem is the use and interpretation of "all".

Dedekind and Cantor speak of "systems." It was
Russell and Wittgenstein who tried to ground
systems so that "all" had a more definite conception.

Russell did not confine his logic by the introduction
of names (it was, in fact, designed that way so that
one could speak of non-existents without presupposition
failure).

Wittgenstein was a finitist. To my knowledge, he is the
earliest author to point out that Cantor's proof was as
much an indictment of the use of "all" as it was a
proof of an uncountable infinity.

Neither Russell or Wittgenstein (or Skolem, for that
matter) has given a system that is useful for the
exercise of empirical science. Computational models
are obscuring that fact, but even a modest introduction
to numerical analysis explains the role of classical
mathematics behind those models.

That is the pragmatic problem. The theoretical problem
is that mathematicians are confronted with the science
of mathematics as a logical system. If a completed
infinity is ground for a system of names reflecting
geometric completeness, then its investigation is an issue.

>
>> You may, as WM does, deny uses of a completed infinity.
>
> I do not deny it, but show that it is self-contradictory.

That may be. Your proofs, however, lie with the nature
of models and not with the nature of how a deductive
calculus relates to definitions and axioms. In that
sense you are not speaking of self-contradiction. Rather,
you speak of the ill-foundedness of trees having
infinite branches.

To be honest, I prefer your contemptuousness for
it over the kind of crap that was published in the
popular book "Goedel, Escher, Bach"

[fom]

That is what is special about topological
completeness. The identity criterion is not
from the system itself. Rather, it is from
the underlying system.

So, the trichotomy of the rationals becomes the
basis for trichotomy of the reals. The identity
criterion for the reals depends on the identity
criterion of the rationals.

But, you even have problems with the decimal
expansions of the rationals. Fortunately, they
can be expressed differently.

Still, they cannot all be expressed.

[fom]

Correct. The diagonal argument is not the definition
of the reals.

>
>> Any *one* number not on
>> the list shows that the list is incomplete and thus establishes the
>> uncountability of the reals
>
> No, it establishes the incompleteness of infinity or the infinity of
> incompleteness.

The latter of the two statements is a better choice. The rationals
are not complete. So much so, in fact, that they are a set
of measure zero.

But, wait. A set of measure zero presumes a sigma algebra generated
from the open sets of the topology (or the compact sets if you
prefer).

>
> Cantor's list establishes the uncountability of distinguishable and
> hence constructable reals.

Constructible real has a definite sense that you
do not abide by. You should talk of nameable reals and
Frege's notion of definite symbols.

> Why should nonconstructable and hence
> nondistinguishable reals matter in Cantor's argument?
>
> Cantor proves the uncountability of a countable set. For some people
> that has an effect like a drug.

Cantor proves that names isolated from the
systemic relation of their definition are
subject to local finiteness conditions.

[Zuhair]

On Jan 4, 8:13 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Zuhair <zaljo...@gmail.com> writes:
> > Dear fom I'm not against Uncountability, I'm not against Cantor's
> > argument. I'm saying that Cantor's argument is CORRECT. All what I'm
> > saying is that it is COUNTER-INTUITIVE as it violates the
> > Distinguishability argument which is an argument that comes from
> > intuition excerised in the FINITE world. That's all.
>
> But you've neither explained the meaning of your second premise nor
> given any indication why it is plausible.
>

I did but you just missed it.

My second premise is that finite distinguishability is countable.

What I meant by that is that we can only have countably many
distinguishable finite initial segments of reals. And this has already
been proved. There is no plausibility here, this is a matter that is
agreed upon.

Zuhair

[Virgil]

In article
<8915ce43-f582-4e69...@r4g2000vbi.googlegroups.com>,

Except that WM 's "proofs" are always invalid!
--

[Virgil]

In article
<00fe6649-4559-43ed...@r14g2000vbd.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 3 Jan., 22:36, Virgil <vir...@ligriv.com> wrote:
>
> > > It is prerequisite when dealing with numbers. And it is exactly what
> > > Cantor applied, He distinguished numbers at finite places.
> >
> > Cantor may have distinguished numerals at finite places, but numerals
>
>
> So Cantor "proved" that there are uncountably many numerals?

Actually, Cantor proved that the set of real numbers was not countable
in the sense that no mapping from N to R can be surjective, which is
what, by definition, being countable requires.

> Why then does anybody believe in uncountably many numbers?

Because counting is provably unable to exhaust them.


Does WM have some definition of countability of a set other than
existence of surjection from N to the set in question?

If so, he should present his alternative ASAP, because as long as that
definition holds, the reals remain provably uncountable.
--

[Virgil]

In article <kc6idc$jc$1...@Kil-nws-1.UCIS.Dal.Ca>,

But try getting WM to see it!
--

[Jesse F. Hughes]

Zuhair <zalj...@gmail.com> writes:

> On Jan 4, 8:13 pm, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>> Zuhair <zaljo...@gmail.com> writes:
>> > Dear fom I'm not against Uncountability, I'm not against Cantor's
>> > argument. I'm saying that Cantor's argument is CORRECT. All what I'm
>> > saying is that it is COUNTER-INTUITIVE as it violates the
>> > Distinguishability argument which is an argument that comes from
>> > intuition excerised in the FINITE world. That's all.
>>
>> But you've neither explained the meaning of your second premise nor
>> given any indication why it is plausible.
>>
> I did but you just missed it.
>
> My second premise is that finite distinguishability is countable.
>
> What I meant by that is that we can only have countably many
> distinguishable finite initial segments of reals. And this has already
> been proved. There is no plausibility here, this is a matter that is
> agreed upon.

Sure, there's only countably many finite sequences over {0,...,9}, if
that's what you mean, but I don't see what that has to do with whether R
is countable or not.

I thought your error involved something else, namely the following
equivocation on distinguishability of a set S.

Any pair of reals is finitely distinguishable. That is,

(Ax)(Ay)(x != y -> (En)(x_n != y_n))

where x_n is the n'th digit of x.

Now, there are two possible definitions of distinguishability for a set
S.

A set S is pairwise distinguishable if each pair of (distinct)
elements is finitely distinguishable.

A set S is totally distinguishable if there is an n in N such that for
all x, y in S, if x != y then there is an m <= n such that x_m != y_m.

Clearly, the set of reals is pairwise distinguishable but not totally
distinguishable. But so what? I see no reason at all to think that it
*is* totally distinguishable. The fact that each pair of reals is
distinguishable gives no reason to think that the set of all reals is
totally distinguishable.


--
"Philosophy, as a part of education, is an excellent thing, and there
is no disgrace to a man while he is young in pursuing such a study;
but when he is more advanced in years, the thing becomes ridiculous
[like] those who lisp and imitate children." -- Callicles, in Gorgias

[Virgil]

In article
<2466aecd-5309-4468...@f19g2000vbv.googlegroups.com>,

According to standard mathematics, a set is "countable" if and only if
there is surjection from N to that set. And until WM can produce such a
surjection, none of his claims that the reals are countable count.

>
> > Any *one* number not on
> > the list shows that the list is incomplete and thus establishes the
> > uncountability of the reals
>
> No, it establishes the incompleteness of infinity or the infinity of
> incompleteness.

And until WM can produce a surjection from N to R, none of his claims
that the reals are countable show that the definition of countability is
met.

>
> Cantor's list establishes the uncountability of distinguishable and
> hence constructable reals. Why should nonconstructable and hence
> nondistinguishable reals matter in Cantor's argument?

Until WM can produce a surjection from N to R, none of his claims that
the reals are countable show that the definition of countability can be
met.

>
> Cantor proves the uncountability of a countable set. For some people
> that has an effect like a drug.

And until WM has produced a surjection from N to R, he is just blowing
hot air.
--

[Virgil]

In article
<af7faea0-149c-4151...@b11g2000yqh.googlegroups.com>,

And until WM can produce a surjection from N to R, none of his claims
that the reals are countable show that the definition of countability

for R can be met.

Thus R remains uncounted even in Wolkenmuekenheim.
--

[gus gassmann]

I try not getting to see WM.

[Ross A. Finlayson]

On Jan 4, 4:36 am, gus gassmann <g...@nospam.com> wrote:
> On 03/01/2013 5:53 PM, Virgil wrote:
>
>
>
>
>
>
>
>
>
> > In article

> > <a60601d5-24a2-4501-a28b-84a7b1e53...@ci3g2000vbb.googlegroups.com>,

No, they are assumed to have expansions and need not be written.

Consider the function that is the limit of functions f(n,d) = n/d, n =
0, ..., d; n, d E N. In binary, its expansions would only start .000
{m many zeros} 000..., the only antidiagonal is .111... = 1.0 =
f(d,d).

The antidiagonal is in the range of the function from the naturals, at
the end.

The reals there are distinguished from one another by their constant
monotone progression.

Regards,

Ross Finlayson

[Virgil]

In article
<7850ae29-08d9-49ef...@m4g2000pbd.googlegroups.com>,

"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:

> Consider the function that is the limit of functions f(n,d) = n/d, n =
> 0, ..., d; n, d E N.

You mean the zero function?

For every n, the limit of f(n,d) as d -> oo is 0, so your limit function
would have to be the zero function: f(n,oo) = 0 for all n.
--

[WM]

On 4 Jan., 18:57, fom <fomJ...@nyms.net> wrote:
> On 1/4/2013 5:41 AM, WM wrote:

> >>>> So we are not speaking about the same distinguishability criterion.
>
> > There is no other criterion.
>
> In logic, discernibility is taken to be with
> respect to properties.

Can't a number be considered to be a property?

>
> Your position seems to be that since the names determine
> the model which, in turn, determines the truth, then the
> names are the only criterion.

The model determines the truth if the rules which have to be obeyed by
the names are taken from observation of reality.

Regards, WM

[WM]

On 4 Jan., 19:33, fom <fomJ...@nyms.net> wrote:
> On 1/4/2013 10:29 AM, WM wrote:
>
> > On 4 Jan., 01:35, fom <fomJ...@nyms.net> wrote:
>
> >> Dedekind cuts define all reals.
>
> > Every cut is defined by a finite word. The set of definable cuts is
> > the set of cuts and is countable.
>
> >> Cantor fundamental sequences define all reals.
>
> > No infinite definition defines anything.
>
> No infinite definition is finitely realizable.

A definition has to be realizable.

> Wittgenstein was a finitist.  To my knowledge, he is the
> earliest author to point out that Cantor's proof was as
> much an indictment of the use of "all" as it was a
> proof of an uncountable infinity.

Poincaré was also very early.

>
> Neither Russell or Wittgenstein (or Skolem, for that
> matter) has given a system that is useful for the
> exercise of empirical science.  Computational models
> are obscuring that fact, but even a modest introduction
> to numerical analysis explains the role of classical
> mathematics behind those models.

Classical mathematics has been introduced by many from Pythagoras to
Kronecker and Weierstraß, but not by Cantor.

>
> That is the pragmatic problem.  The theoretical problem
> is that mathematicians are confronted with the science
> of mathematics as a logical system.

Mathematics and logic as a science recognize that there are only
countably many names, or better, that there are only potentially
infinitely many names.

> If a completed
> infinity is ground for a system of names reflecting
> geometric completeness, then its investigation is an issue.
>

If this investigation shows that completed infinity is self-
contradictory, then a science should accept this result. And if the
identity of numbers defined by different definitions cannot be proved
(or if nobody is interested like in case of 1+1+1 and 3, for
instance), then this fact will not increase the number of definable
numbers. Then there are simply less numbers than different
definitions, finite definitions, of course. That is a result of logic.

Geometry is as "incomplete" as arithmetic.

Regards, WM

[WM]

On 4 Jan., 19:47, fom <fomJ...@nyms.net> wrote:
>  The rationals
> are not complete.  So much so, in fact, that they are a set
> of measure zero.
>
> But, wait.

No reasonable reason to wait. "Measure zero" is a nonsense expression
because it presupposes aleph_0 as a meaningful notion. There are
convergent sequences, but the limit is never assumed, not "after
aleph_0 steps".

>  A set of measure zero presumes a sigma algebra generated
> from the open sets of the topology (or the compact sets if you
> prefer).
>
>
>
> > Cantor's list establishes the uncountability of distinguishable and
> > hence constructable reals.
>
> Constructible real has a definite sense that you
> do not abide by.  You should talk of nameable reals and
> Frege's notion of definite symbols.

Constructible reals are a subset of nameable reals. (Every
prescription for a construction is a name.) Therefore I do not make an
error.

Regards, WM

[WM]

On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Sure, there's only countably many finite sequences over {0,...,9}, if
> that's what you mean, but I don't see what that has to do with whether R
> is countable or not.

So you are blind. Let me try to make you see.

> I thought your error involved something else, namely the following
> equivocation on distinguishability of a set S.
>
> Any pair of reals is finitely distinguishable.  That is,

How do you get a pait of reals?

>
>   (Ax)(Ay)(x != y -> (En)(x_n != y_n))
>
> where x_n is the n'th digit of x.
>
> Now, there are two possible definitions of distinguishability for a set
> S.
>
>   A set S is pairwise distinguishable if each pair of (distinct)
>   elements is finitely distinguishable.

A, I see. Distinct elements must be distingusihable.

>
>   A set S is totally distinguishable if there is an n in N such that for
>   all x, y in S, if x != y then there is an m <= n such that x_m != y_m.

Nonsense. A real number need not be given by a string of digits. In
most cases that is even impossible. Given is a finite definition like
"pi". And this is distinct from all other real numbers.

>
> Clearly, the set of reals is pairwise distinguishable but not totally
> distinguishable.  But so what?  I see no reason at all to think that it
> *is* totally distinguishable.  The fact that each pair of reals is
> distinguishable gives no reason to think that the set of all reals is
> totally distinguishable.

In order to have a pair of distinct elements (reals), they must be
finitely defined such that none of them has more than one and only one
meaning. Already if you have only one "given real", it must be defined
such that it is observably and provably different from *all* other
reals, not only from a second "given real".

If these reals were not distinguishable from all other reals, then
they would not be "given". But in order to have a given real number,
it must have a finite definition. How else should an infinite string
of digits be obtained if not from a finite formula?

Regards, WM

[WM]

On 4 Jan., 22:38, Virgil <vir...@ligriv.com> wrote:

> According to standard mathematics, a set is "countable" if and only if
> there is  surjection from N to that set.

If and only if? Do you agree that a subset of a countable set is
countable?

The set of all finite definitions is countable.
The set of all finitely defined reals cannot be put in bijection with |
N, but it is a subset of all finite definitions, isn't it?

Regards, WM

[WM]

On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> Clearly, the set of reals is pairwise distinguishable but not totally
> distinguishable.  But so what?

A good question. A set distinguishable by such an n would necessarily
be finite. Do you think that anybody, and in particular Zuhair, claims
that |R is finite? Or did you miss this implication?

A set S of infinite strings of digits (the God of matheology may
present the strings without defining them in another way) is finitely
distinguishable if for all x, y in S, if x != y then there is an m in |
N (i.e., a finite index) such that x_m != y_m.

Regards, WM

[forbi...@gmail.com]

On Saturday, January 5, 2013 4:56:52 AM UTC-8, WM wrote:
> In order to have a pair of distinct elements (reals), they must be
> finitely defined such that none of them has more than one and only one
> meaning. Already if you have only one "given real", it must be defined
> such that it is observably and provably different from *all* other
> reals, not only from a second "given real".

Why can't I just stipulate two distinct reals x and y?
If I have a flock of geese I can assert their distinguishability
without distinguishing them. In fact I might not be able to distinguish
them until I've marked them in some way but marking them only proves
they were distinguishable even before I did so. The same logic applies
to number even though the set of them is unbound. I can make a claim
such as "x isn't the largest integer" and it is true for all integers.

You keep talking about the largest human definable integer. I have no
idea how big it is but I can easily talk about the factorial of a
googolplex. I can tell you a googolplex raised to a googolplex power
is larger than the factorial of a googolplex.

[fom]

On 1/5/2013 5:41 AM, WM wrote:
> On 4 Jan., 18:57, fom <fomJ...@nyms.net> wrote:
>> On 1/4/2013 5:41 AM, WM wrote:
>
>>>>>> So we are not speaking about the same distinguishability criterion.
>>
>>> There is no other criterion.
>>
>> In logic, discernibility is taken to be with
>> respect to properties.
>
> Can't a number be considered to be a property?

Usually logicians try to keep grammatical properties
separate from "material" properties. Analysis of
the paradoxes led to the use of formalized languages,
in part, to prevent some self-reference attributable
to the ability of natural languages to intermingle
such references. So, I did not consider the property
of the names in my response. On the other hand, it
is in the canonicity of a given system of names that
reflects their use as unique identifiers. But, at
least where Leibniz identity of indiscernibles may
be in play, the non-grammatical properties should
be justifying the grammatical distinctions.

>>
>> Your position seems to be that since the names determine
>> the model which, in turn, determines the truth, then the
>> names are the only criterion.
>
> The model determines the truth if the rules which have to be obeyed by
> the names are taken from observation of reality.

Reality is subjective.

One of the goals of science is a version of truth
that may be considered objective. What if a
completed infinity is a necessary consequence of
the objectivity of science?

[fom]

On 1/5/2013 6:56 AM, WM wrote:

> Nonsense. A real number need not be given by a string of digits. In
> most cases that is even impossible. Given is a finite definition like
> "pi". And this is distinct from all other real numbers.

"pi" is certainly a finite string used as a name.

Please offer a (Russellian) description that attaches it
to an idea of number.

And, since you make provability an issue in these matters,
please show that your description is uniquely distinguishing
pi from other numbers.

[fom]

On 1/5/2013 9:46 AM, forbi...@gmail.com wrote:
> On Saturday, January 5, 2013 4:56:52 AM UTC-8, WM wrote:
>> In order to have a pair of distinct elements (reals), they must be
>> finitely defined such that none of them has more than one and only one
>> meaning. Already if you have only one "given real", it must be defined
>> such that it is observably and provably different from *all* other
>> reals, not only from a second "given real".
>
> Why can't I just stipulate two distinct reals x and y?
> If I have a flock of geese I can assert their distinguishability
> without distinguishing them.

There is a difference between applying logic to a given linguistic
analysis and using logic to ground a linguistic synthesis. What you
do here is apply logic under the assumption of sufficient identity
criteria.

These questions arise more significantly in descriptive metaphysics.
That logicism has so profoundly extinguished discussion of these
matters is a testament to bad manners among analyical philosophers.

I found this paper interesting

http://philpapers.org/rec/ARADCA-2

but the actual paper seems to have been taken offline because
it is now published. Here is another by the same author
that still seems available,

http://istvanaranyosi.net/resources/Rev%20solo.pdf

The received paradigm comes from Russell and Wittgenstein
for the most part. Wittgenstein saw no logical reason for
ascribing to the identity of indiscernibles. Ontologically,
one could consider objects that are not distinguishable on
the basis of properties. But, note that when Wittgenstein
had been introducing his arguments to this effect, he said
that names were like geometric points.

So, in a sense, Wittgenstein has simply obfuscated the
hard issues.

The problem is that a demonstrative science is based on
definitions. So, what one must deal with are
descriptively-defined names. Although Russellian description
theory dominated for nearly forty years, Strawson challenged
the assertions of Russell in "On Referring". In Strawson's
book "Individuals" you will find, once again, the fundamental
geometric basis for numerical identity in contrast to the
qualitative identity upon which description theory is
necessarily based.

In the end, Cantor's intersection theorem is the best
implementation of Leibniz identity of indiscernibles as
explained in "Discourse on Metaphysics." The community of
analytic philosophy has a great deal to say about this
principle without ever providing the part of the quote
where Leibniz makes reference to the way in which geometers
apply specific differences (metric topologies, in my
estimation) or to the summary of Aquinas which motivated
the general principle ("an individual is a lowest species")

You can find a good reference to the treatment of identity
in Morris' "Understanding Identity Statements". There are
actually three distinct aspects at work all at the same
time.

When I write

x=x

it is an assertion of ontological invariance. It is also
an assertion of semantic consistency. As a signifying symbol,
the axiom asserts that it will be uniformly interpreted under
any satisfaction map.

The historical problem is the informative identity,

x=y

Under the received paradigm, the problems with this
expression are ignored. Eventually, they come back
with Tarski's satisfaction maps. No one ever talks
about satisfaction maps failing to model identity
statements. You will find mention of this in another
critic of Russellian description theory. Look for
"On Constrained Denotation" by Abraham Robinson. He
makes plain the relationship between model-theoretic
identity and the use of descriptions to introduce
names.

Tarski, himself, introduced an axiom for informative
identity:

AxAy(x=y <-> Ez(x=z /\ z=y))

This appears when he begins to treat logic in an
algebraic fashion in "Cylindrical Algebras" by
Tarski and Monk.

> In fact I might not be able to distinguish
> them until I've marked them in some way but marking them only proves
> they were distinguishable even before I did so.

True. The problem lies with "...until I've
marked them in some way..."

You have yet to prove anything. I do not say that
critically. Rather, I am trying to convey the difference
between applying logic under the presupposition of
adequate identity criteria and using logic to define
a system with adequate identity criteria.

> The same logic applies
> to number even though the set of them is unbound.

Sadly, no. The problem with number is worse because
it is complicated by description theory... unless
you know where I can purchase the latest model of
zero. I can drive across town to buy a goose dinner.

> I can make a claim
> such as "x isn't the largest integer" and it is true for all integers.
>

Thank goodness for prime numbers and increase by units.

> You keep talking about the largest human definable integer. I have no
> idea how big it is but I can easily talk about the factorial of a
> googolplex. I can tell you a googolplex raised to a googolplex power
> is larger than the factorial of a googolplex.
>

No. WM keeps talking about nameability and incorrectly
uses terms like "define" and "construct". Virgil keeps
pointing out all of the other terms he is using incorrectly,
as well.

[Jesse F. Hughes]

WM <muec...@rz.fh-augsburg.de> writes:

> On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
>> Sure, there's only countably many finite sequences over {0,...,9}, if
>> that's what you mean, but I don't see what that has to do with whether R
>> is countable or not.
>
> So you are blind. Let me try to make you see.

Sorry, but I'm not in the mood to discuss this topic with you. I was
discussing Zuhair's ideas with Zuhair. Your own incoherent thoughts are
a separate matter entirely.

And shame on you for pretending that you are competent to teach
mathematics.

--
"Do you know why I'm tall?" "Why?"
"Because I eat apples." "Do you know why I'm short?"
"Why?" "Because I'm a kid."
--Quincy P. Hughes (age almost 4) bests his father.

[Jesse F. Hughes]

WM <muec...@rz.fh-augsburg.de> writes:

> On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
>> Clearly, the set of reals is pairwise distinguishable but not totally
>> distinguishable.  But so what?
>
> A good question. A set distinguishable by such an n would necessarily
> be finite. Do you think that anybody, and in particular Zuhair, claims
> that |R is finite? Or did you miss this implication?

After posting, I came to that conclusion as well. Thus, I've no idea
what Zuhair means when he says that distinguishability implies
countability. (He said that it means the set of finite initial segments
is countable, but since this does not contradict the uncountability of
R, I don't know why he thinks this is paradoxical.)

Again, let's let Zuhair clarify precisely what his argument is. I just
don't see it.

--
"This is based on the assumption that the difference in set size is what
makes the important difference between finite and infinite sets, but I think
most people -- even the mathematicians -- will agree that that probably
isn't the case." -- Allan C Cybulskie explains infinite sets

[Virgil]

In article
<5c682c83-3374-4ccb...@u19g2000yqj.googlegroups.com>,

The "god" of WMytheology, namely WM himself, will insist that every such
potential string must have a finite definition in order to be thought
about at all.
>
> Regards, WM

Actually, real math suggests that there are reals so incredibly
inaccessible that finding the digit sequence representing one of them is
not possible.

In which case such numbers may well not be "finitely distinguishable".
--

[Virgil]

In article
<b7fc4126-ed54-4740...@h2g2000yqa.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 4 Jan., 22:38, Virgil <vir...@ligriv.com> wrote:
>
> > According to standard mathematics, a set is "countable" if and only if
> > there is �surjection from N to that set.
>
> If and only if? Do you agree that a subset of a countable set is
> countable?

Since a surjection from N to a superset of S trivially implies a
surjection to S, it also implies that any subset of a countable set is
also countable. at least outside of WMythology. What the rules are for
goes on inside or WMythology is only accessible to WM himself.

>
> The set of all finite definitions is countable.
> The set of all finitely defined reals cannot be put in bijection with |
> N, but it is a subset of all finite definitions, isn't it?

There is nothing in any definition of a complete ordered field of reals
that requires each of its members to be finitely definable.

At least not outside WMytheology.
--

[WM]

On 5 Jan., 22:26, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:

> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> >> Sure, there's only countably many finite sequences over {0,...,9}, if
> >> that's what you mean, but I don't see what that has to do with whether R
> >> is countable or not.
>
> > So you are blind. Let me try to make you see.
>
> Sorry, but I'm not in the mood to discuss this topic with you.

It is not a matter of mood, as anybody can see by your attempt to have
the reals restricted to 2^n elements, it is a matter of distinguished
incompetence.

Regards, WM

[WM]

On 5 Jan., 16:46, forbisga...@gmail.com wrote:
> On Saturday, January 5, 2013 4:56:52 AM UTC-8, WM wrote:
> > In order to have a pair of distinct elements (reals), they must be
> > finitely defined such that none of them has more than one and only one
> > meaning. Already if you have only one "given real", it must be defined
> > such that it is observably and provably different from *all* other
> > reals, not only from a second "given real".
>
> Why can't I just stipulate two distinct reals x and y?

You can do so, but you cannot do so by using an infinite string
(because a stipulation needs to be stipulated in finite time).

You can stipulate the number SUM 1/n! for n > 745
or, when talking with a mathematician,
sqrt(65745667).

> If I have a flock of geese I can assert their distinguishability
> without distinguishing them.

Objects of reality exist as distinct objects by their space-time
corrdinates already. But objects of thought need to be thought as
distinct objects.

> In fact I might not be able to distinguish
> them until I've marked them in some way but marking them only proves
> they were distinguishable even before I did so.  The same logic applies
> to number even though the set of them is unbound.  I can make a claim
> such as "x isn't the largest integer" and it is true for all integers.

But that truth does not distinguish integers.

>
> You keep talking about the largest human definable integer.

That is my private interest. It has nothing to do with the ongoing
discussion about definability.

>  I have no
> idea how big it is but I can easily talk about the factorial of a
> googolplex.  I can tell you a googolplex raised to a googolplex power
> is larger than the factorial of a googolplex.

There is, in my opinion, no largest human definable integer. But there
is an upper bound to the complexity of definitions (not only of
humans). Compare your pocket calculator. You can use vast numbers,
like 10^50 but you cannot use a number like 12312312312 with more than
digits.

The universe is in a similar way limited like a pocket calculator. But
again: This has nothing to do with the ongoing discussion about
definability and the fact that there is no uncountable set of
distinguishable elements.

Regards, WM

[WM]

On 5 Jan., 17:21, fom <fomJ...@nyms.net> wrote:

> Reality is subjective.

Not everything. It is not subjective to say that tomorrow sun will
rise and the dropped stone will fall down. Many aspects of reality are
sure to 100.000 % - and that is enough for most cases. These aspects
include that 2 + 2 = 4 and other primitives of mathematics like tables
up to 10 (and larger).

>
> One of the goals of science is a version of truth
> that may be considered objective.  What if a
> completed infinity is a necessary consequence of
> the objectivity of science?

It is not, since completed infinity can be excluded by logic.

Regards, WM

[WM]

On 5 Jan., 17:47, fom <fomJ...@nyms.net> wrote:
> On 1/5/2013 6:56 AM, WM wrote:
>
> > Nonsense. A real number need not be given by a string of digits. In
> > most cases that is even impossible. Given is a finite definition like
> > "pi". And this is distinct from all other real numbers.
>
> "pi" is certainly a finite string used as a name.

But it allows to calculate a potentially infinite string. i.e., a
finite string with more than any given number of digits.

>
> Please offer a (Russellian) description that attaches it
> to an idea of number.
>
> And, since you make provability an issue in these matters,
> please show that your description is uniquely distinguishing
> pi from other numbers.

I show it by what you think when reading my pi.
Mathematics is discourse, sending and receiving messages. And I am
sure that you would determine the same sequence of digits as I, if we
used an appropriate formalism, for instance the slowly converging
Gregory-Leibniz.

Regards, WM

[Virgil]

In article
<bba43ced-8d33-44b5...@4g2000yqv.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 4 Jan., 22:36, "Jesse F. Hughes" <je...@phiwumbda.org> wrote:
>
> > Sure, there's only countably many finite sequences over {0,...,9}, if
> > that's what you mean, but I don't see what that has to do with whether R
> > is countable or not.
>
> So you are blind. Let me try to make you see.

Better blind like Pontryagin then seeing like WM.

> > I thought your error involved something else, namely the following
> > equivocation on distinguishability of a set S.
> >
> > Any pair of reals is finitely distinguishable. �That is,
>
> How do you get a pait of reals?

Have them delivered!
> >
> > � (Ax)(Ay)(x != y -> (En)(x_n != y_n))

> >
> > where x_n is the n'th digit of x.
> >
> > Now, there are two possible definitions of distinguishability for a set
> > S.
> >
> > � A set S is pairwise distinguishable if each pair of (distinct)
> > � elements is finitely distinguishable.
>
> A, I see. Distinct elements must be distingusihable.

If you meant "distinguishable", he did not say that.

"If" does not require that there are any distinct elements that are
finitely distinguishable.

> >
> > � A set S is totally distinguishable if there is an n in N such that for

> > � all x, y in S, if x != y then there is an m <= n such that x_m != y_m.
>
> Nonsense. A real number need not be given by a string of digits.

Then sets containing reals which cannot be given by strings of digits
will not be "totally distinguishable".

So it is WM's nonsense in being able to understand what is being said.

> In
> most cases that is even impossible. Given is a finite definition like
> "pi". And this is distinct from all other real numbers.

Pi can be given, at need, to any finite number of digits, so can be
distinguished in this way from any other real which can be, at need,
given to any finite number of digits.

> >
> > Clearly, the set of reals is pairwise distinguishable but not totally
> > distinguishable. �But so what? �I see no reason at all to think that it
> > *is* totally distinguishable. �The fact that each pair of reals is
> > distinguishable gives no reason to think that the set of all reals is
> > totally distinguishable.

That presumes that every real can be, at need, expressed to any finite
number of digits, which need not be the case.

>
> In order to have a pair of distinct elements (reals), they must be
> finitely defined such that none of them has more than one and only one
> meaning. Already if you have only one "given real", it must be defined
> such that it is observably and provably different from *all* other
> reals, not only from a second "given real".

Since real numbers can have all sorts of extremely complicated
definitions, it may be extermely difficult, or even impossible, to prove
that two such definitions define different numbers.

>
> If these reals were not distinguishable from all other reals, then
> they would not be "given".

That assumes that all definitions easily lead to some method of
comparing, like decimal representations, but real numbers can be defined
in lots of ways which make finding their decimal approximations so
difficult as to be effectively impossible beyond a very few digits of
accuracy.

Then such real numbers cannot be distinguished from any but sufficiently
distant real numbers.

> But in order to have a given real number,
> it must have a finite definition. How else should an infinite string
> of digits be obtained if not from a finite formula?

Finite formulae may involve increasingly impractical complexity beyond
a very few digits of accuracy.
--

[WM]

On 5 Jan., 22:36, Virgil <vir...@ligriv.com> wrote:

> Actually, real math suggests that there are reals so incredibly
> inaccessible that finding the digit sequence representing one of them is
> not possible.
>
> In which case such numbers may well not be "finitely distinguishable".
> --

And they cannot be used in any way in a discourse or in mathematics or
spring off a Cantor-list.

So, why talk about them at all?

Regards, WM

[Virgil]

In article
<bb7be2f4-a0c3-4538...@b8g2000yqh.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 4 Jan., 19:47, fom <fomJ...@nyms.net> wrote:
> >  The rationals
> > are not complete.  So much so, in fact, that they are a set
> > of measure zero.
> >
> > But, wait.
>
> No reasonable reason to wait. "Measure zero" is a nonsense expression
> because it presupposes aleph_0 as a meaningful notion. There are
> convergent sequences, but the limit is never assumed, not "after
> aleph_0 steps".

So WM would throw out all of measure theory with his bathwater.

>
> >  A set of measure zero presumes a sigma algebra generated
> > from the open sets of the topology (or the compact sets if you
> > prefer).
> >
> >
> >
> > > Cantor's list establishes the uncountability of distinguishable and
> > > hence constructable reals.
> >
> > Constructible real has a definite sense that you
> > do not abide by.  You should talk of nameable reals and
> > Frege's notion of definite symbols.

WM should not talk at all, since he would throw out so much of standard
mathematics in order to keep his WMytheology "pure".

>
> Constructible reals are a subset of nameable reals. (Every
> prescription for a construction is a name.) Therefore I do not make an
> error.

Non-Sequitur. It is plainly obvious to everyone except WM that he makes
errors all the time, and then squirms wildly to cover his ass.
>
> Regards, WM
--

[Virgil]

In article
<64e948d9-cb53-4798...@d10g2000yqe.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 4 Jan., 19:33, fom <fomJ...@nyms.net> wrote:
> > On 1/4/2013 10:29 AM, WM wrote:
> >
> > > On 4 Jan., 01:35, fom <fomJ...@nyms.net> wrote:
> >
> > >> Dedekind cuts define all reals.
> >
> > > Every cut is defined by a finite word. The set of definable cuts is
> > > the set of cuts and is countable.
> >
> > >> Cantor fundamental sequences define all reals.
> >
> > > No infinite definition defines anything.
> >
> > No infinite definition is finitely realizable.
>
> A definition has to be realizable.

Why, when you ignore so many of them when they are?

>
> > Wittgenstein was a finitist. �To my knowledge, he is the
> > earliest author to point out that Cantor's proof was as
> > much an indictment of the use of "all" as it was a
> > proof of an uncountable infinity.
>

> Poincar� was also very early.

> >
> > Neither Russell or Wittgenstein (or Skolem, for that
> > matter) has given a system that is useful for the
> > exercise of empirical science. �Computational models
> > are obscuring that fact, but even a modest introduction
> > to numerical analysis explains the role of classical
> > mathematics behind those models.
>
> Classical mathematics has been introduced by many from Pythagoras to

> Kronecker and Weierstra�, but not by Cantor.

One can only have something introduced to the world once.

> >
> > That is the pragmatic problem. �The theoretical problem
> > is that mathematicians are confronted with the science
> > of mathematics as a logical system.
>
> Mathematics and logic as a science recognize that there are only
> countably many names, or better, that there are only potentially
> infinitely many names.

But a lot more things that, at least in theory, need names, but,
in practice, don't.
>
> >�If a completed

> > infinity is ground for a system of names reflecting
> > geometric completeness, then its investigation is an issue.
> >
> If this investigation shows that completed infinity is self-
> contradictory, then a science should accept this result.

But mathematics is NOT a science. Its truths and values are in no way
dependent on physical experimentation or scientific observations of the
physical world

>
> Geometry is as "incomplete" as arithmetic.

But not as incomplete as WMytheology.
--

[Virgil]

In article
<49c23543-93cb-4fa8...@z8g2000yqo.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 4 Jan., 18:57, fom <fomJ...@nyms.net> wrote:
> > On 1/4/2013 5:41 AM, WM wrote:
>
> > >>>> So we are not speaking about the same distinguishability criterion.
> >
> > > There is no other criterion.

There are others for those who do not live in Wolkenmuekenheim.

> >
> > In logic, discernibility is taken to be with
> > respect to properties.
>
> Can't a number be considered to be a property?

In his Wolkenmuekenheim, WM can consider anything he pleases.

> >
> > Your position seems to be that since the names determine
> > the model which, in turn, determines the truth, then the
> > names are the only criterion.
>
> The model determines the truth if the rules which have to be obeyed by
> the names are taken from observation of reality.

Mathematics takes very little directly from reality, and even that is
taken in highly abstracted form.
>
> Regards, WM
--

[Virgil]

In article <VJudnfwSTauIznXN...@giganews.com>,

If so, it would totally confirm WM's lack of objectivity!
--

[Virgil]

In article
<559dcb41-5aea-496d...@v7g2000yqv.googlegroups.com>,

WM <muec...@rz.fh-augsburg.de> wrote:

> On 5 Jan., 17:47, fom <fomJ...@nyms.net> wrote:
> > On 1/5/2013 6:56 AM, WM wrote:
> >
> > > Nonsense. A real number need not be given by a string of digits. In
> > > most cases that is even impossible. Given is a finite definition like
> > > "pi". And this is distinct from all other real numbers.
> >
> > "pi" is certainly a finite string used as a name.
>
> But it allows to calculate a potentially infinite string. i.e., a
> finite string with more than any given number of digits.
> >
> > Please offer a (Russellian) description that attaches it
> > to an idea of number.
> >
> > And, since you make provability an issue in these matters,
> > please show that your description is uniquely distinguishing
> > pi from other numbers.
>
> I show it by what you think when reading my pi.

To be as sure as you claim, ,you would have to be a mindreader and read
the mind of someone your have probably never met and don't know where to
find.

> Mathematics is discourse, sending and receiving messages.

So is Morse code.
--

[Virgil]

In article
<553bf1fb-5267-47c9...@u19g2000yqj.googlegroups.com>,

Because if they are not there, some of the necessary properties of the
real number field are also not there.
--

[Virgil]

In article
<8614b98b-c517-4ae3...@w3g2000yqj.googlegroups.com>,

It is also a matter of WM's undistingished but nearly universal
incompetence.
--

[Ross A. Finlayson]

On Jan 4, 10:20 pm, Virgil <vir...@ligriv.com> wrote:
> In article
> <7850ae29-08d9-49ef-8c7b-e8979e037...@m4g2000pbd.googlegroups.com>,
>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:
>
> > Consider the function that is the limit of functions f(n,d) = n/d, n =
> > 0, ..., d; n, d E N.
>
> You mean the zero function?
>
> For every n, the limit of f(n,d) as d -> oo is 0, so your limit function
> would have to be the zero function: f(n,oo) = 0 for all n.
> --


No, none of those is the zero function, and each d->oo has it so that
d/d = 1.

Here the range of the elements are defined by their constant monotone
difference, the sum of which, is one.

This is a way to define real numbers, between zero and one, as:
between zero and one.

Simply enough divide how many integers you can count by how many there
are, it ranges from zero to one.

Regards,

Ross Finlayson