A new point of view on Cantor's diagonalization arguments

Click For Summary
The discussion revolves around a critique of a new perspective on Cantor's diagonalization and cardinality presented in a linked PDF. The main contention is that the alephs discussed in the article do not align with conventional mathematical definitions, leading to inconsistencies in cardinality claims. Participants argue that the proposed system fails to adhere to established mathematical conventions, rendering it incapable of serving as a substitute for cardinality. Additionally, the conversation touches on the implications of unique structural properties in infinite sets and the nature of mappings between them. Ultimately, the debate highlights fundamental disagreements about the interpretation of infinity and cardinality in mathematics.
  • #151
Hurkel !

I ask you about the futher
so way did you answer me
about the past ?

Moshek
 
Physics news on Phys.org
  • #152
Originally posted by moshek
Matt:

Wittgenstaein said : "..Mathematics have no foundation in Set theory or in any other theory. Mathematics is depend only on our living.." .

maybe this is way Organic ask us what is the real meening of the 3 points in :

{1,2,3,... }

Or about the very similarity of Rashel paradox and Cantor diagolazation metod to prove that R in uncontable.

I found something about Gowers , thank you !
I understand that his new idea about mathematics deal with the connection between mathematics and Physics.

Well i defiantly agree with that direction.

Do you know the Michel Atiya ( Also a well known fields medal) said in his lecture at the conference about the unity of mathematics at Harvard university ( I was there)
that we are waiting to a new Newton that will broke the Enigma between mathematics and Physics

So do you think that maybe Gowers
is our new Newton ?

Thank you
Moshek
No.

Gowers prefers to deal with elementarily stated problems, that are therefore hard to solve.

If you want a mathematical physicist then Baez, Dolan, Atiyah (still. Mathematically I am his partial grandson as I was supervised in my phD for a year by one of his students), Witten, Grothendieck (if he's stoppped doing biology), Orlov or Lusztig seem more likely.

Perhaps Hurkyl doesn't desal with the futher because there's no such word as futher. Do you mean future or father?Mathematics is precisely what we make it, I agree, which seems to conflict with Organic's apparently previously held belief that the natural numbers do not exist independently of set theory.
 
  • #153
Originally posted by Organic
Matt,

[2]*[3]=[6] but because my nanural number is a structural/quantitative product, there is more than one solution to [6], which are dependend in the inner structure of [2] and [3], for example:

[2]=(1,1) , ((1),1)
[3]= (1,1,1) , ((1,1),1) , (((1),1),1)

Therefore:

[2]*3=((1,1),(1,1),(1,1)) or (((1),1),((1),1),((1),1))
[3]*2=((1,1,1),(1,1,1)) or (((1,1),1),((1,1),1)) or ((((1),1),1),(((1),1),1))
which justifies my assertion that the is no unique element that represents [n] and the multiplication is therefore not well defined - something you riduculed in that reposted message.
 
  • #154
Matt :

Do you know that Alein Connes the great mathematician who invent Non -commutate Geometry ( also a fields medal) end with his interesting lecture the conference "100 to Hilbert" at August 2000 at U.C.L.A when he said that we need today a new understanding in mathematics that is base on Geometry and not on logic ?

Organic:

Thank you for sharing with us
your very interesting noncomutative arithmetic.


Moshek



Who was killed by a solders and said:

Bring me one point so i can change the whole world .
 
  • #155
Matt :

Good to know that !

I am Sorry for my English mistake in my question to Hurkel
yes you are right i mean future. ( of mathematics)

Well your Grandfather gave the best and maybe the only lecture that was relate really to the question of the unity of mathematics ( as was posed by Hilbert at the end of his lecture).

Please ask him if i quote him correctly about a change in the paradigm of mathematics !.

And if he said also :

"We are waiting today to a new Newton ?

Maybe you know another possibility if it is not Gowers ?

Thank you
Moshek
 
  • #156
Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

We can clearly see this here:

http://www.geocities.com/complementarytheory/ETtable.pdf


Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.


----------------------------------------------------------------------------

The big paradigm's shift is QM and not SR, please read this:

http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

http://www.geocities.com/complementarytheory/ETtable.pdf

More details can be found here:

http://www.geocities.com/complementarytheory/POV.pdf

Man is no longer an observer but a participator, which its influence must be included in any explored system.

It mean that we cannot ignore our cognition's abilities to create Math language anymore, as I clearly show here:

http://www.geocities.com/complementarytheory/count.pdf
 
Last edited:
  • #157
Dear Organic:

We all know already that you are not so good in what is called mathematics...


But :

Your gentle treatment of number with there inherent duality is seeing the world in the eyes of a child.

Like Einstein did to relativity theory it is the only way to create a unmoors change with the paradigm.

So you share with us by this
a positive intepatatino to Gödel theorem.

My Best wishes
Moshek
 
  • #158
The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?
 
  • #159
Matt:

well It's all about modeling of the world, like Newton did !
Einstein put us in the world but still use Newton mathematics.

Did you heard or read Wolfram Book "new kind of science"


Please Ask Michel Atiya about redefinition of the 6 problem of Hilbert ! maybe Organic is trying to share with us his solution to this very difficult problem.

as you know the 6,8,16 problems
are still opme in the list.

Thank you
Moshek





p.s I study now his very nice book on commutative algebra of Atiya and Mecdonalds.
 
  • #160
Matt,

Instead of reading the details of my previous post you take some name and play with it.

Please show me some influence on basic thing like the natural number, by one of these theories.
 
  • #161
Dear Matt:

We can say that the origin of the possibility to a paradigm change in mathematic was at the end of Hilbert famous lecture. Unfortanly that math world relate mostly to the famous 23 list of open problem which Hilbert himself said that the are only example.
This was 5 years before Einstein appears to the world. Wittgenstaein pose already about this possibility also when he talk about develop anew dimension to mathematics in the geometry of the Klein bottle. In the Epilog of the nice book of Ian Stewat “Nature Numbers” he also talk about he need to invent new –mathematic which will be flexible and rigid together. Alein Connes talk about a new understanding that may come to mathematic beside the regular logic. And Michel Atiya lecture about the need to broke the enigma of the connection between mathematics and physics

“We are looking for a new Newton today..” he said at the conference about the unity of mathematics in Honored of Israel Gelfand 90 birthday.

Yes Hurkal is right !

A similar change happened already but it was ~2,500 years ago when Hipasus of metapontum discover that irrational number exist. Which Contradict Pythagoras “Every thing is Number”.

I ma sure you know the legend what his teacher did to him after that.

Organic share with us here his discovery about Number when time is Vanish.
So you must hold the quantity and the structure always together.



Take care
Moshek
 
Last edited:
  • #162
Organic would need to demonstrate the the natural numbers are not sufficient for the job they are used for for that parallel to hold here.

The greeks thought that all numbers on their number line were rational originally. This is not true, therefore their assumption is wrong.

Where do we go wrong with the definition that the natural numbers are the basic counting blocks of finite sets? What has gone wrong?

Sure we can add all the extra structure to it you want, but it doesn't alter the basic facts.

You could argue the 'inventing' the negative integers, then the rationals, then the reals and then the complexes are all paradigm shifts, but they don't mean the naturals are incomplete within themselves.

We dont' assume those are all the arithmetical objects that exist.

As I keep telling organic he's just playing around with basic operads and monads, if he wants to add extra structure to them that's fine, but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers. That is not the objection to what he's doing. The objection is that he's making wild unsubstantiated and plainly false claims about things he doesn't understand.
 
  • #163
Matt:

You Ignore by your question to Organic
all my background that i share with you here
as i promise you few hour ago!

Moshek


I will let Organic to answer your question about his numbers.
 
  • #164
Matt,

First, you have to show me that you understand what I am doing.

For example, it is the second time that you write general things about my work instead of reading and and reply detailed comments on this:

Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

We can clearly see this here:

http://www.geocities.com/complementarytheory/ETtable.pdf


Matt,

Natural number system of Standard Math is not simple but trivial, and my attitude is to give it the power of simplicity instead of the weakness of triviality.


----------------------------------------------------------------------------

The big paradigm's shift is QM and not SR, please read this:

http://plato.stanford.edu/entries/qm-copenhagen/#4

This paradigm's shift, does not exist in the basis of Standard Math language, because Boolean Logic or Fuzzy Logic are private cases of what I call Complementary Logic, that an overview of it can be found here: http://www.geocities.com/complementarytheory/BFC.pdf

Through my point of view Natural numbers are complementary elements, based on discreteness(particle-like)-continuum(wave-like) associations.

The information structure of the standard Natural numbers, is only a private case of these associations, for example:

http://www.geocities.com/complementarytheory/ETtable.pdf

More details can be found here:

http://www.geocities.com/complementarytheory/POV.pdf

Man is no longer an observer but a participator, which its influence must be included in any explored system.

It mean that we cannot ignore our cognition's abilities to create Math language anymore, as I clearly show here:

http://www.geocities.com/complementarytheory/count.pdf


-------------------------------------------------------------------------------
The quantum shift is mathematical: quantum groups, quantum cohomology, quantized universal enveloping algebras, quantum chaos, even quantum mechanics (I learned that in a maths degree...), I've even been learning about quantum linear algebra. And then there's quantum computing and quantum information theory, both well established and mathematical. And did I mention quantum Kac-Moody Lie algebras? Deformation Theory? q-schur algebras, quantum binomial coefficients?
Please show me some influence on basic thing like the natural number, by one of these theories.
 
  • #165
but he's already using the naturals inside his definition to define his "new naturals" there is nothing there to suggest these new objects should replace the natural numbers.
So, you don't understand that the stantard natural number is a trivial private case of infinitely many structural/quantitative information's forms that ignored by Standard Math paradigm.

Do you get it?

A NUMBER is first of all an information's form, and to understand this we MUST explore our cognition's abilities to define this information's form, as I do here:

http://www.geocities.com/complementarytheory/count.pdf
 
Last edited:
  • #166
Originally posted by Organic Each one of these structural-quantitative products is unique, therefore can be used as a building-block for much more interesting and richer information form, then your “quantitative-only" unique [n] result, which is nothing but a private-case of no-redundancy-no-uncertainty structural-quantitative product of my number system.

This would only hold true if I had ever said that the Natural Numbers are the only thing in mathematics. We have N, simply, N, the set of natural numbers. Then there is (N,+) which is a semi-ring. Then the is (N,<) the ordinals. And who can forget (N,+,<) the ordered semi-ring. On top of that one might do (N[-1],+), ie Z under adding formal inverses. Not forgettiing the ordering too. What about localizing and getting a field, Q, and completing to get R, and closing to get C? Then there's the division rings and Wedderburns' structure theorem over that.

Of course there is the general theory of semi-rings etc that 'has N as a private case of a larger language', in you words.

There is a whol rich tapestry of objects that have N, Z, Q or R as the simplest version. This apparent paradigm shift is already there.

If you incidentally knew about the quantized structures above and the rigorous framework behind quantum mechanics, then perhaps you'd understand the analogies.
 
  • #167
Matt,
Two critical things you ignore when you define N members:

1) A research of your ability to count:
http://www.geocities.com/complementarytheory/count.pdf

2) that natural numbers are first of all information forms, therefore
Their minimal existence must start form here:
Code: 
(1*4)              ={1,1,1,1} <------------- Maximum symmetry-degree, 
((1*2)+1*2)        ={{1,1},1,1}              Minimum information's 
(((+1)+1)+1*2)     ={{{1},1},1,1}            clarity-degree
((1*2)+(1*2))      ={{1,1},{1,1}}            (no uniqueness) 
(((+1)+1)+(1*2))   ={{{1},1},{1,1}}
(((+1)+1)+((+1)+1))={{{1},1},{{1},1}}
((1*3)+1)          ={{1,1,1},1}
(((1*2)+1)+1)      ={{{1,1},1},1}
((((+1)+1)+1)+1)   ={{{{1},1},1},1} <------ Minimum symmetry-degree,
                                            Maximum information's  
                                            clarity-degree                                            
                                            (uniqueness)

After these two "MUST HAVE" steps, we can continue to develop the next numbers in the number system.
 
  • #168
Why are these must have steps? We can define the natural numbers without them, so they obviously aren't must have are they? Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?
 
  • #169
Matt,

The answer is very simple.

Any number is first of all an information form, therefore any aspect of information form MUST be researched by us, where our cognition’s abilities to research information MUST be included too.

Form this point of view, redundancy AND uncertainty cannot be ignored, and through this approach(which is not an extra approach but the MINIMAL approach to understand the natural number concept) we can clearly show that the standard natural numbers are only a one and only one private case of verity of information forms, which are ordered by their vagueness degrees from maximum vagueness to minimum vagueness when a given quantity remains unchanged.

Man is no longer an observer but a participator, which its influence must be included in any explored system.

The above is the QM paradigm shift that is not understood yet by the current community of pure mathematicians.

For example: Be aware that what you call a function is first of all a reflection of your memory.


Or are you forgetting the DEFINITION of the natural numbers (as a set, just a set, with no other structure)?
A set is only a framework that helps us to explore our ideas, no less no more.

When there is a paradigm shift this framework is chaneged too.
 
Last edited:
  • #170
It would be nice if you told us what the uncertainty and redundancy of a number is. So that we know what we MUST be aware of.
 
  • #172
Originally posted by Organic
The answer is here:

http://www.geocities.com/complementarytheory/count.pdf

Ah, of course! The answer to my question is in another pdf! This one doesn't even mention the words uncertainty and redundancy. In what way is that an answer?

Do you mean to imply that numbers are uncertain because if we have set of identical objects we cannot distinguish between them? I don't see why that makes numbers uncertain. I see why it makes identifying identical objects impossible, but that has nothing to do with quantity. Suppose I just had one bead. I blink, I see an identical bead in the same place. is it the same bead? I do that for more than 1 bead.. what has quantity got to do with it?
 
  • #173
From your response it looks that you did not read the pdf file.

Please reply if you have some technical problems to open the pdf.

Thank you,

Organic
 
  • #174
I opened the pdf link you gave and didn't see the word uncertainty mentioned once, please tell me which line it is on. Or, as the pdf is only a page long, try stating in plain simple English what you mean the the uncertainty of a number, or its redundancy. It's another simple request. Actually it's the same simple request isn't it? A simple paragraph starting:

The redundancy of a number is...

or perhaps it ought to start

A number is redundant if...or even,

given a number n, and some object related to n, then the object is redundant if...
 
  • #175
Dear Matt,

You are going too far.

This pdf is not a technical paper, but a simple test that answer to the question:

what are the minimal conditions that give us the ability to count?

Please read it again from this point of view, and don't search for any definitions there, just try to understand this simple test.

Thank you.

Organic
 
  • #176
I don't see the arguement

Cantor's Diagonalization Argument

Theorem-The interval [0,1] is not countably infinite.
Proof:-The proof by contradiction proceeds as follows:

Assume (for the sake of argument) that the interval [0,1] is countably infinite. We may then enumerate all numbers in this interval as a sequence, ( r1, r2, r3, ... ) We already know that each of these numbers may be represented as a decimal expansion. We arrange the numbers in a list (they do not need to be in order). In the case of numbers with two decimal expansions, like 0.499 ... = 0.500 ..., we chose the one ending in nines. Assume, for example, that the decimal expansions of the beginning of the sequence are as follows:

r1 = 0 . 5 1 0 5 1 1 0 ...
r2 = 0 . 4 1 3 2 0 4 3 ...
r3 = 0 . 8 2 4 5 0 2 6 ...
r4 = 0 . 2 3 3 0 1 2 6 ...
r5 = 0 . 4 1 0 7 2 4 6 ...
r6 = 0 . 9 9 3 7 8 3 8 ...
r7 = 0 . 0 1 0 5 1 3 5 ...
...
We shall now construct a real number x in [0,1] by considering the kth digit after the decimal point of the decimal expansion of rk.

r1 = 0 . 5 1 0 5 1 1 0 ...
r2 = 0 . 4 1 3 2 0 4 3 ...
r3 = 0 . 8 2 4 5 0 2 6 ...
r4 = 0 . 2 3 3 0 1 2 6 ...
r5 = 0 . 4 1 0 7 2 4 6 ...
r6 = 0 . 9 9 3 7 8 3 8 ...
r7 = 0 . 0 1 0 5 1 3 5 ...
...

The digits we will consider are indicated in bold and illustrate why this is called the diagonal proof. From these digits we define the digits of x as follows.
if the kth digit of rk is 5 then the kth digit of x is 4
if the kth digit of rk is not 5 then the kth digit of x is 5
For the example above this will result in the following decimal expansion.

x = 0 . 4 5 5 5 5 5 4 ...

The number x is a real number (we know that all decimal expansions represent real numbers) in [0,1] (clearly). Hence we must have rn = x for some n, since we have assumped that ( r1, r2, r3, ... ) enumerates all real numbers in [0, 1]. However, because of the way we have chosen 4's and 5's as digits in step (6), x differs in the nth decimal place from rn, so x is not in the sequence ( r1, r2, r3, ... ). This sequence is therefore not an enumeration of the set of all reals in the interval [0,1]. This is a contradiction.

Hence the assumption that the interval [0,1] is countably infinite must be false.

Q.E.D.

It is a direct corollary of this result that the set R of all real numbers is uncountable. If R were countable, we could enumerate all of the real numbers in a sequence, and then get a sequence enumerating [0, 1] by removing all of the real numbers outside this interval. But we have just shown that this latter list cannot exist. Alternatively, we could show that [0, 1] and R are the same size by constructing a bijection between them.
 
Last edited:
  • #177
Cantormath,

Welcome to Physics Forums! And you're right, there is no argument.

What you've stumbled upon here is a piece of crackpottery from the old days when we adopted an "anything goes" attitude in the Theory Development Forum. We've since tightened things up so that we only allow things that make sense. :smile:

So feel free to stick around and enjoy the Forums. Don't worry about this thread, because the orginal poster isn't even here anymore.
 

Similar threads

B One thing I don't understand about Cantor's diagonal argument
2
Replies
55
Views
8K
Does Cantor's Diagonal Argument Apply to the Rationals?
Replies
4
Views
3K
I Understanding Cantor's Diagonalization Proof: A Brief Explanation
Replies
17
Views
3K
Stationary points classification using definiteness of the Lagrangian
Replies
4
Views
2K
A new point of view on Math language
Replies
4
Views
5K
Cantor's Diagonalization Proof of the uncountability of the real numbers
2
Replies
93
Views
20K
Cantor diagonalization argument
Replies
2
Views
3K
I Effective Hamiltonian to Rotational Term Values
Replies
0
Views
3K
Is the Cantor diagonal argument conclusive?
Replies
49
Views
11K
Is the Real-Line a Fractal? Weak Points Examined
Replies
4
Views
2K

Hot Threads

Recent Insights

Back
Top