Uncountability: Intuition x formal.

[Zuhair]

In a prior post I raised the question of how can there be UNCOUNTABLY
many reals each two of which are ONLY discriminated from each other on
Finite basis, and yet we have only COUNTABLY many distinct Finite
initial segments of reals?

What is meant by discrimination on finite basis is the following:

For all reals r1,r2 ( r1 is discriminated 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)

In other words we know that the capacity for finite discrimination is
Countable, so how come we have uncountably many reals that are
finitely discriminated? It looks as if we are having more reals than
what we could have?

The argument that since we can only distinguish Countably many finite
initial segments of reals then we must have countably many reals so
distinguished is what I call as the 'distinguishability argument'. It
is an argument of Intuition so far as no formalization of it has been
put so far.

The intuitive background of this is found at the head post of that
posting, see below link:

https://groups.google.com/group/sci.logic/browse_thread/thread/9f925ab94b369fad?hl=en

I raised four possible responses to this:

(1) To say that the formal proof of Cantor is clear and exact in
formal terms, but the distinguishability argument is clear on
intuitive level but has not been verified in formal terms, so
accordingly we have the option of saying that Infinity do not copy
intuitions derived from the finite world, and deem the result as just
counter-intuitive but not paradoxical. I think this is the standard
approach.

(2) To say that the distinguishability argument is so clear and to
accept it as a proved result despite the possibility of verifying it
at formal level or not, and also maintaining that Cantor's proof is
very clear and valid, and so we deduce that we have a genuine paradox
(antinomy) that resulted from assuming having completed infinity, and
thus we must reject having completed infinity. That's what WM is
saying

(3) To consider uncountability of the reals to be FALSE by
interpreting the universal quantifier in Cantor's argument to range
over *elements* of the universe of discourse, and thus it doesn't
cover all available functions (which are subsets of the universe of
discourse), so there is a bijection between the reals and the naturals
that is simply missing from being among the *elements* of the universe
of discourse. So the reality of that matter is that the set of all
reals is countable but discourse misses the necessary bijection. This
is the interpretation of having a countable model of a theory that
proves the existence of uncountably many objects. It is a consequence
of Skolem's arguments. The justification for that is that if consider
the universal quantifier in Cantor's argument to range over subsets of
the universe of discourse, then we are at second order logic which is
not known to support a proof system and so it can barely be seen as a
kind of logic or something that we prove things after. So we are
better with at first order. If we use Henkin semantics to explain the
second order quantifier then we'll end up by the same argument of
Skolem. So Henkin semantics for second order or first order semantics
both ensure having a countable model of any theory, and since proof
theory by ordinal analysis depends on constructiveness within
countable limits, then we only need to stipulate the existence of
countable models since it is those the ones that are both provable and
economic in the sense of reductionist approach reminiscent of Ockam's
razor, that is if we can do the job with less so why demand more. And
since there is no logical argument to force us to adhere to
uncountability without assuming it first, so why adhere to?

(4) To adopt a very radical perspective about sets, a perspective that
only admits sets that are definable in a parameter free manner to
exist, and accordingly to consider countability of all finite initial
segments of reals to be FALSE, i.e. to say that we have uncountably
many finite initial segments of reals and as well we have uncountably
many reals. This clearly preserves congruity of the argument, but it
requires justification. The idea is that the alleged bijection between
the finite initial segments of reals and the set N of all naturals is
NOT parameter free definable and so this bijection does not exist, and
it is false to say that it is. This claim only accepts infinite sets
to exist if there is a parameter free formula after which membership
of those sets is determined, so if there is non then it doesn't accept
the existence sets that are not parameter free definable.

MY PERSONAL RESPONSE:
I personally favor the standard response which is the first response.
However I think that further insights to address the intuitive
challenge presented by the distinguishability argument is warranted.

I'm not sure if the fourth response is possible, any proof of the
alleged bijection (between the naturals and the set of all initial
segments of parameter free definable reals) being parameter free
definable would abolish that response. However even if a proof is
presented of the alleged bijection being not parameter free definable,
still there is a strong sense to compel us to agree to the set of all
finite initial segments of reals being countable, and to get rid of
that just for the sake of parameter free definability doesn't seem to
be plausible at all. So all in all the fourth response is not
plausible that if not impossible.

The second response is too radical and it upgrades 'counter-
intuitiveness' to the level of antinomy without putting forwards any
formal proof, and thus it is just an assertion that have some
intuitive appeal but it is unbaked so far by any rigorous formal
proof.

The third response though rises within many contexts of formal
theories in first order languages, however it seems to be an
artificial midway fix between intuition and formality.

Zuhair

[WM]

On 28 Dez., 09:21, Zuhair <zaljo...@gmail.com> wrote:
> In a prior post I raised the question of how can there be UNCOUNTABLY
> many reals each two of which are ONLY discriminated from each other on
> Finite basis, and yet we have only COUNTABLY many distinct Finite
> initial segments of reals?
>
> What is meant by discrimination on finite basis is the following:
>
> For all reals r1,r2 ( r1 is discriminated 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)
>
> In other words we know that the capacity for finite discrimination is
> Countable, so how come we have uncountably many reals that are
> finitely discriminated? It looks as if we are having more reals than
> what we could have?
>
> The argument that since we can only distinguish Countably many finite
> initial segments of reals then we must have countably many reals so
> distinguished is what I call as the 'distinguishability argument'. It
> is an argument of Intuition so far as no  formalization of it has been
> put so far.

The set of finite paths in the Binary Tree is countable. Every digit
that could be used to distinguish two real numbers belongs to a finite
initial segment of the string of digits. Every digit of the diagonal
of a Cantor-list belongs to a finite initial segment too.

There is no intuition.

>
> MY PERSONAL RESPONSE:
> I personally favor the standard response which is the first response.
> However I think that further insights to address the intuitive
> challenge presented by the distinguishability argument is warranted.

Do you need a formal proof that n in g(n) =/= f(n,n) belongs to a
finite initial segment of |N?

Regards, WM

[Zuhair]

Just write the whole formal proof of yours if you have any, put it in
symbols as how Cantor put his proof in exact formal language, write it
as proofs are written in the standard manner, If you manage to do that
then you will have a genuine paradox (antinomay), if you cannot then
you just have an intuitive gesture.

Zuhair

[George Greene]

On Dec 28, 3:21 am, Zuhair <zaljo...@gmail.com> wrote:
> In a prior post I raised the question of how can there be UNCOUNTABLY
> many reals each two of which are ONLY discriminated from each other on
> Finite basis, and yet we have only COUNTABLY many distinct Finite
> initial segments of reals?

You have NOT raised this question. This IS NOT a question.
It is inherently NOT SURPRISING that there are more reals than
there are initial segments of reals, given that the initial segments
ALL MUST be finite. OBVIOUSLY, THERE ARE MORE infinite things than
finite things. No matter how large an initial segment of an infinite
list you may consider, there are ALWAYS INFINITELY many things after
it
and ALWAYS ONLY FINITELY many before it.

These things are simply not in any kind of intuitive tension.
There is nothing surprising happening here.
THERE *IS*NO* question FOR you to raise.

>
> What is meant by discrimination on finite basis is the following:

Oh, SHUT UP. EVERYBODY ALREADY KNOWS what is meant by "discrimination
on a finite basis".
Every index in a real IS finite. Therefore, "a finite basis" IS THE
ONLY basis AVAILABLE,
for objects that are structured like this. If you had to go
infinitely far then that would
imply that they were THE SAME< SO FAR, out TO that infinitely far,
which WOULD MAKE THEM THE SAME.

It is of course POSSIBLE to devise infinite structures where NOT
everything is only finitely
far away from the beginning; there are ordinals bigger than omega or
than N.
BUT THOSE ARE NOT what we are talking about.

[George Greene]

On Dec 28, 3:21 am, Zuhair <zaljo...@gmail.com> wrote:

> In a prior post I raised the question of how can there be UNCOUNTABLY
> many reals each two of which are ONLY discriminated from each other on
> Finite basis, and yet we have only COUNTABLY many distinct Finite
> initial segments of reals?

I repeat, this IS NOT counter-intuitive.
What arguably IS counter-intuitive is one level LOWER than this.
You don't have to go to the power-set of N, to the reals, to get
your intuition contradicted. The problem is with N itself, not
with the reals or with N's powerset. BEFORE the list became square,
BEFORE you had infinitely many decimal-places per real AND an infinite
number of reals, YOU FIRST had N.

If we were to just WRITE OUT each element of N,
0,
1,
2,
3,
4,
5,
....,
...

Then you could ask the following question: how can this list (or
column)
be INFINITELY long when each ELEMENT of it IS FINITE??

You get the SAME tension between the finite vs. infinite cardinalities
AS the tension you are trying to highlight between the countably and
uncountably infinite ones.
But YOUR problem is, you have NOT yet conceded THAT THIS PROPERTY OF N
is "counter-intuitive". If you don't have any problem with this (and
of course
most people don't), then you don't have ANY REASONABLE INTELLECTUAL
GROUNDS
for having a problem with the same phenomenon lifted one power.

[Graham Cooper]

On Dec 29, 9:40 am, George Greene <gree...@email.unc.edu> wrote:
> On Dec 28, 3:21 am, Zuhair <zaljo...@gmail.com> wrote:
>
> > In a prior post I raised the question of how can there be UNCOUNTABLY
> > many reals each two of which are ONLY discriminated from each other on
> > Finite basis, and yet we have only COUNTABLY many distinct Finite
> > initial segments of reals?
>
> You have NOT raised this question.  This IS NOT a question.
> It is inherently NOT SURPRISING that there are more reals than
> there are initial segments of reals, given that the initial segments
> ALL MUST be finite.  OBVIOUSLY, THERE ARE MORE infinite things than
> finite things.  No matter how large an initial segment of an infinite
> list you may consider, there are ALWAYS INFINITELY many things after
> it
> and ALWAYS ONLY FINITELY many before it.
>
> These things are simply not in any kind of intuitive tension.
> There is nothing surprising happening here.
> THERE *IS*NO* question FOR you to raise.
>
>
>
> > What is meant by discrimination on finite basis is the following:
>
> Oh, SHUT UP.

YOU SHUT UP YOU F*NG MONSTER

YOU ARE A LIAR AND CON

AND A GUTLESS TWERP

WHO HIDES BEHIND HIS DESK AND TAKES POT SHOTS ALL DAY

THEN RUNS AND HIDES WHENEVER FACED WITH A QUESTION

NOW... ARE YOU GETTING BACK TO YOUR LIE THAT YOU MADE

ABOUT YOUR CLAIMS ABOUT PATHS THROUGH T???


Herc

[Zuhair]

On Dec 29, 2:40 am, George Greene <gree...@email.unc.edu> wrote:
> On Dec 28, 3:21 am, Zuhair <zaljo...@gmail.com> wrote:
>
> > In a prior post I raised the question of how can there be UNCOUNTABLY
> > many reals each two of which are ONLY discriminated from each other on
> > Finite basis, and yet we have only COUNTABLY many distinct Finite
> > initial segments of reals?
>
> You have NOT raised this question.  This IS NOT a question.
> It is inherently NOT SURPRISING that there are more reals than
> there are initial segments of reals, given that the initial segments
> ALL MUST be finite.  OBVIOUSLY, THERE ARE MORE infinite things than
> finite things.  No matter how large an initial segment of an infinite
> list you may consider, there are ALWAYS INFINITELY many things after
> it
> and ALWAYS ONLY FINITELY many before it.
>
> These things are simply not in any kind of intuitive tension.
> There is nothing surprising happening here.
> THERE *IS*NO* question FOR you to raise.
>
>
>
> > What is meant by discrimination on finite basis is the following:
>
> Oh, SHUT UP.  EVERYBODY ALREADY KNOWS what is meant by "discrimination
> on a finite basis".

You can yell as LOUD as you may, but there is an argument that you
simply didn't get. Next time read FULLY what the other is presenting
and then try to respond, you didn't even address what I've said. And I
think also you need to learn some manners when speaking to others, and
try stop this loud voice of yours, and using words like contempt and
the alike, you have some good knowledge but the way you write is
DISGUSTING.

The argument rests upon an intuitive generalization (that might be
considered false by some) that if we don't have further distinct nodes
after some level of the infinite binary tree, then the number of paths
will not increase as we move up from that level (it means the number
of paths at any level higher than that level is not bigger than the
number of paths at that level) this is of course a finite level
argument, the generalization is to go generalize that to the
following: For any Size the number of all paths of the CIBT of that
size if no distinct nodes exist at a level corresponding to that Size
in the infinite binary tree then it would be equal to the number of
paths of the Size below it. Now INFINITY here is considered as a size
criterion and FINITENESS is considered as the size criterion below it,
so the total number of infinitely long paths of the CIBT must be the
same as the total number of all finitely long paths of the CIBT,
because there is no distinct nodes occurring at any level in the CIBT
that corresponds to INFINITY (because there is no infinite level in
the CIBT) and since FINITENESS is the level below INFINITY. I call it
the 'distinguishability argument'

It is an argument of Intuition, it is not a formal argument at all.

Zuhair

[WM]

On 29 Dez., 00:40, George Greene <gree...@email.unc.edu> wrote:

>
> You have NOT raised this question.  This IS NOT a question.
> It is inherently NOT SURPRISING that there are more reals than
> there are initial segments of reals, given that the initial segments
> ALL MUST be finite.  OBVIOUSLY, THERE ARE MORE infinite things than
> finite things.

Obviously there are not more infinite things, because each infinite
thing needs a finite name in order to be named. And things that cannot
be named and cannot be proven to exist (as individual entites) do not
belong to mathematics - but ast most to matheology.

Regards, WM

[WM]

On 29 Dez., 00:45, George Greene <gree...@email.unc.edu> wrote:
> On Dec 28, 3:21 am, Zuhair <zaljo...@gmail.com> wrote:
>
> > In a prior post I raised the question of how can there be UNCOUNTABLY
> > many reals each two of which are ONLY discriminated from each other on
> > Finite basis, and yet we have only COUNTABLY many distinct Finite
> > initial segments of reals?
>
> I repeat, this IS NOT counter-intuitive.

It is, whether you recognize it or not.

> What arguably IS counter-intuitive is one level LOWER than this.
> You don't have to go to the power-set of N, to the reals, to get
> your intuition contradicted.  The problem is with N itself, not
> with the reals or with N's powerset.  BEFORE the list became square,

The list is not square in the sense that it is as long as broad. A
list of aleph_0 lines and aleph_0 columns can be square in the sense
that there is a diagonal existing. But if the number of lines is
doubled, there is again a square list of aleph_0 lines and aleph_0
columns, but it has no diagonal if the first list had one.
Therefore there is no diagonal in infinite lists.

> BEFORE you had infinitely many decimal-places per real AND an infinite
> number of reals, YOU FIRST had N.
>
> If we were to just WRITE OUT each element of N,
> 0,
> 1,
> 2,
> 3,
> 4,
> 5,
> ....,
> ...
>
> Then you could ask the following question: how can this list (or
> column)
> be INFINITELY long when each ELEMENT of it IS FINITE??

Yes,this is an important question. It is rarely put because usually
with respect to this question infinity is regarded as potential only.
Only by frequently intermingling potential and actual infinity, set
theory could exist for 100 years.

>
> You get the SAME tension between the finite vs. infinite cardinalities
> AS the tension you are trying to highlight between the countably and
> uncountably infinite ones.
> But YOUR problem is, you have NOT yet conceded THAT THIS PROPERTY OF N
> is "counter-intuitive".  If you don't have any problem with this (and
> of course
> most people don't), then you don't have ANY REASONABLE INTELLECTUAL
> GROUNDS
> for having a problem with the same phenomenon lifted one power.

As I told you above, this point can be veiled by intermingling
potential and actual infinity. In the Binary Tree even rather blind
formalists can recognize (sometimes, like Zuhair just is about to do)
that the problem is persisting and not avoided by intermingling
infinities. So there is a big difference and a very reasonable
intettlectual ground to use the CIBT.

Regards, WM

[fom]

Nice.

I tried to redirect the question to the fact
that diagonal arguments are about type
differences and that the uncountability
problem only arises with respect to an
additional assertion. Oh well...

This brings type difference to the fore
at the simplest level.

[fom]

Fine.

Prove the existence of the number one.

[WM]

On 28 Dez., 09:21, Zuhair <zaljo...@gmail.com> wrote:

> (1) To say that the formal proof of Cantor is clear and exact in
> formal terms,

Where did Cantor publish his formal proof? Source please?

Regards, WM

[WM]

On 30 Dez., 09:36, fom <fomJ...@nyms.net> wrote:

> I tried to redirect the question to the fact
> that diagonal arguments are about type
> differences and that the uncountability
> problem only arises with respect to an
> additional assertion.

That is nonsense. All real numbers including the anti-díagonal are of
same level. And all are identified, according to Cantor, by digits (or
nodes - in the Binary Tree).
Everything not defined by nodes in the CIBT is ill-defined.

Regards, WM

[WM]

On 28 Dez., 19:27, Zuhair <zaljo...@gmail.com> wrote:

> Just write the whole formal proof of yours if you have any, put it in
> symbols as how Cantor put his proof in exact formal language,

Give the quote please. Where is the text from which you think that
Cantor argued in a manner that differs from mine. Thank you.

Regards, WM

[WM]

On 30 Dez., 10:43, fom <fomJ...@nyms.net> wrote:

>
> Prove the existence of the number one.

You assume that I understand what you wrote. In fact I do. That proves
the existence of our idea of the number one. And since numbers are
nothing than ideas, this is the proof of the existence of 1.

If you are not really ill in your head (perhaps spoilt by too much
"logic") then you recognize that 1 contrary to uncountably many real
"numbers" exists.

Regards, WM

[fom]

On 12/30/2012 10:04 AM, WM wrote:
> On 30 Dez., 10:43, fom <fomJ...@nyms.net> wrote:
>
>>
>> Prove the existence of the number one.
>
>
> You assume that I understand what you wrote. In fact I do. That proves
> the existence of our idea of the number one. And since numbers are
> nothing than ideas, this is the proof of the existence of 1.

Prove that you have not
conflated the existence of
an idea of a number with
the existence of an idea
of a name.

[fom]

Cantor is not the only author of
foundational material.

Type distinction is fundamental
to Russell's constructions. You
may be comforted that he says in
the second edition of Principia
Mathematica that he cannot reproduce
certain results from Cantor's work.

It is true, that all real numbers
are the same logical type. That
is precisely the case when they are
defined to be fundamental sequences
inheriting the arithmetical relations
of rational numbers whose representative
fundamental sequences form a dense
subset.

Since you deny these things, please
give your definition of a real
number.

[forbi...@gmail.com]

From http://en.wikipedia.org/wiki/Georg_Cantor

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article,[29] "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").[37] This article was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).

[fom]

That would be every word every uttered or scrawled
by the man.

If you wish to alter that fact, present definitions,
axioms, and theorems in a fashion recognizably
similar to that found in textbooks or use
standard mathematical terminology as expected
by publishers of journal articles.

[WM]

Impossible. Numbers are nothing but names (that belong to a system
governed by certain rules. In my opinion these rules have to be copied
from and checked by reality. But many mathematicians assert that the
rules can be defined just as the mood or fancy takes them.)

Regards, WM

[WM]

On 31 Dez., 04:59, forbisga...@gmail.com wrote:
> On Sunday, December 30, 2012 4:20:28 AM UTC-8, WM wrote:
> > On 28 Dez., 09:21, Zuhair <zaljo...@gmail.com> wrote:
>
> > > (1) To say that the formal proof of Cantor is clear and exact in
>
> > > formal terms,
>
> > Where did Cantor publish his formal proof? Source please?
>
> > Regards, WM
>

> Fromhttp://en.wikipedia.org/wiki/Georg_Cantor

>
> The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article,[29]

Where did Cantor publish his *formal* proof? (Nowhere and never.)

Regards, WM

[WM]

On 31 Dez., 04:58, fom <fomJ...@nyms.net> wrote:
> On 12/30/2012 6:22 AM, WM wrote:
>
> > On 30 Dez., 09:36, fom <fomJ...@nyms.net> wrote:
>
> > That is nonsense. All real numbers including the anti-díagonal are of
> > same level. And all are identified, according to Cantor, by digits (or
> > nodes - in the Binary Tree).
> > Everything not defined by nodes in the CIBT is ill-defined.
>
> Cantor is not the only author of
> foundational material.

But here I investigate his ideas.

>
> Type distinction is fundamental
> to Russell's constructions.  You
> may be comforted that he says in
> the second edition of Principia
> Mathematica that he cannot reproduce
> certain results from Cantor's work.

On another occasion he destroys this ray of hope and says with respect
to Cantor: "The solution of the difficulties which formerly surrounded
the mathematical infinite is probably the greatest achievement of
which our age has to boast."

>
> It is true, that all real numbers
> are the same logical type.  That
> is precisely the case when they are
> defined to be fundamental sequences
> inheriting the arithmetical relations
> of rational numbers whose representative
> fundamental sequences form a dense
> subset.
>
> Since you deny these things, please
> give your definition of a real
> number.

I assume for the discussion being that each real number of the unit
unterval is represented by an infinite path in the Binary Tree.

Regards, WM

[WM]

Do you know what "formal" means? Do you know that Cantor did not use
formal language? Do you know that did not need axioms?

Probably not.

Regards, WM

[fom]

Thank you.

Good answer.

[fom]

That was precisely why I referenced standards
of publication. Cantor met the standards of
his day.

But, as for the standards of Cantor's day. It
certainly pre-dated formal languages.