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.
  • #121
You cannot translate my ideas by your mathematical tools

The reason for this is because you do not convey your ideas effectively, and you are self-contradictory.

(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")



2) After you read all what I wrote can you answer to this?:

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Can you translate it into math-speak?
 
Physics news on Phys.org
  • #122
Hurkyl :

I there is in mathematics
a definition to "definition" ?

Thank you

Moshek
 
  • #123
I there is in mathematics
a definition to "definition" ?


I'm sure there are mathematical theories which have a class of objects called definitions, but in general the term "definition" is meta-mathematical, not mathematical, so to answer your question literally, the answer (in general) is no.
 
  • #124
Dear Hurkl!

I was glad for your kind answer to me !

Since you are a matematition
but you can't defind "definition"
Way you ask it from Organic ?

Thank you
Moshek
 
  • #125
I'm not asking him to define definition, I'm asking him to provide one. :smile:


I'm probably somewhat more abstract than "mainstream" mathematics, but I would define "definition" metamathematically as simply one or more (precise) mathematical statements initially taken to be true.


The point, then, is that we can carry out logical deductions from the definitions to derive theorems.
 
  • #126
No Hurkl !

The point is only a point !

Let me ask you my question in anothr way:

Do you believe that what happened to physice at 1905
chould Hapend also to mathematics in one day?

Thank you
Moshek
 
  • #127
It happened to mathematics first. :smile:
 
  • #128
when ?
 
  • #129
It happened to mathematics first.

If you speak about non-Euclidian geometry then you right but what about QM and Bohr's complementary attitude?

(e.g. admitting that the set of real numbers satisfies the definition of "uncountable", yet in the same breath you assert that the set of real numbers is not "uncountable")
No, N and R are enumerable by my definitions because my aleph0 is not your aleph0, no more no less.

Also I see that you don't understand my post about the difference between Cantor's aleph0 and my aleph0.

So, here it is again, but please this time stop on each part of it and please ask me about it, if you think that you don't understand it, thank you:

Dear Hurkyl (this time please answer to this post),


My basic approach about the infinity concept is that redundancy and uncertainty are naturally involved, no more no less.

Therefore aleph0 is a notation that stands for general and flexible "cloud like" thing.

Cantor's thing is a frozen one, mine is not.

I think that my approach is much more interesting and fruitful than Cantor's approach.

Please let me put these two different approaches "on the table" and I would like to examine them together with you.

I am going to write my point of view on Cantor’s approach in a very simple way that (I hope) can be understood by you.

Please read it, and open my eyes to important things that I omit, don’t understand, distorting and so on.

So here it is:

1) Let set C be a complete non-empty binary tree where complete non-empty binary tree exists iff both root AND all its leafs are in C.

2) Let RT be the root , let LF be the all leafs,
therefore RT AND LF are in C --> [RT , LF].

( In my point of view RT XOR LF are in C --> ( (RT… , …LF) OR (LT… , LF] OR
[RT ,…LF) ) AND NOT [RT , LF] where “…” means unreachable. )

Now, from my point of view I see these basic problems when RT AND LF are in C:

1) If both RT AND LF are in C, then C must be a finite set.

2) If C is a non-finite set (through my point of view C is forced to be a non-finite set) then the base value 2 (which is the fundamental structural property of the non-empty binary-tree) cannot exist. Also RT value is unknown.

We must realize that if RT AND LF are in C AND C is a non-finite set, then the structural property of our information (in this case we are talking about the binary tree structure) collapsed into itself (we have no infinitely many elements anymore) and cannot be used as an input by Math language .

Therefore the expression 2^aleph0 cannot exist and we cannot construct
the transfinite universes.

Shortly speaking, what is called uncountable is not uncountable but simply does not exist in any form of input that can be used by Math language.

Through my point of view base 2 can exit
iff (C is finite) OR (RT XOR LF are in C)

Please show me how C is a non-finite set where
(RT AND LF are in C) AND (base 2 exists --> binary tree exists).

Also please tell me what is the value of RT when RT AND LF are in C AND C is a non-finite set.
 
  • #130
I'm not talking about non-Euclidean geometry.

I'm talking about the idea of eliminating unnecessary assumptions.

Remember what SR was all about; it was proven that the speed of light is measured to be the same in all reference frames. Physicists of the time were trying to add new and mysterious things to physics to "save" their ideas of how the universe should work. Einstein said to heck with it and said that we might as well study what our measurements say.

My memory of timelines sucks, but I think mathematical formalism has several decades on Einstein. If not, it's in full swing now. *shrug* For instance, look at category theory; it cares nothing about the objects themselves, just the fact that they are objects, and that there are functions between objects.
 
  • #131
Einstein said to heck with it and said that we might as well study what our measurements say.
What measurements?

Again, what about QM and Bohr's complementary attitude?
 
  • #132
What complementary attitude, and what about it?
 
  • #133
Last edited:
  • #134
Hurkyl :


I am sorry, but what happend
to physics at 1905 by Einstein
Did not happened yet
to mathematics!

Moshek
 
  • #135
Originally posted by moshek
Hurkyl :


I am sorry, but what happend
to physics at 1905 by Einstein
Did not happened yet
to mathematics!

Moshek

I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?
 
  • #136
Originally posted by matt grime
I can't quite decide *exactly* what happened in 1905 that's so important, but, Moshek, and please don't take this the wrong way, on what basis are you qualified to say what has happened to mathematics at any point?

1905 is Einsteins "magic year": it is the year in which he published special relativity, the resolution of the photoelectric effect (for which he would get the Nobel prize later on), and the theoretical description of Brownian motion (that almost proved beyond doubt the existence of atoms). These three theoretical discoveries gave a profound paradigm shift in physics.

I have no idea if such coincedences have also been seen in the field of mathematics (I am not an expert), but I guess they probably have (Euler, Gauss, other giants?)...
 
  • #137
Dear Matt

Thank you for your nice question to me !

I will share with you today here very clear evidents that mathematics is standing today in front of changing paradigms like Einstein did in 1905 to Physics.

Dear Organic:

You may be interested in the following conference that stat today about the futher of Cardinals.

www.as.huji.ac.il/schools/math8/mathsprog.shtml[/URL]


Best
Moshek
 
Last edited by a moderator:
  • #138
Arguably these are partly mathematical discoveries, at least I was taught about two of the three of them in my mathematics degree.

I suppose Russell would have to qualify, as would the category theory 'paradigm shift' (notice that, organic, shift, not change, shift) and then there was the bourbaki school, and the current shift away from that line of thinking. The mathematics of today is vastly different from that in Gauss's day - a lot of the proofs of that period don't stand up to scrutiny now. Then science was seen as a branch of philosophy, natural philosophy, and had according standards of proof. The shift in mathematics to today's view was gradual, and didn't have this alleged sudden epiphany, but it is arguable that saying physics fundamentally changed in 1905 is missing the years of research that went on up to that point that allowed these new discoveries to be accepted.

And we haven't even mentioned Cantor, Godel, Zermelo-Frankel, Nash, chaos theory, Mandelbrot, and various others to a lesser degree (Brylinski's loop space, Tsygan's simultaneous and independent discovery of Cyclic homology with Connes, Keller and Rickard's independent and simultaneous work on derived equivalences, Brauer's ground breaking work on representation theory...). What standards do you want to use to compare them?
 
  • #139
Originally posted by moshek Dear Organic:

You may be interested in the following conference that stat today about the futher of Cardinals.

www.as.huji.ac.il/schools/math8/mathsprog.shtml[/URL]


Best
Moshek [/QUOTE]


I doubt it, Organic has not shown the lsightest interest in learning anything about mathematics to tihs point, why should he start now.
 
Last edited by a moderator:
  • #140
Matt :

Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?


Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.

That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.

Take care
Moshek
 
  • #141
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.
 
Last edited:
  • #142
Originally posted by moshek
Matt :

Did you ever read the 4 last lines
of Hilbert lecture at Paris 1900?


Organic already share with us that he dedicate almost 20 years from his life in trying the develop new perspective about mathematics and he share with us his discoveries very gently.

That fact that he is not famiyar with ordinary mathematics ( he admit that already ) is not necessarily relevant to the question if him material have the potential to make an Organic sift in mathematics.

Take care
Moshek


That'd be the Hilbert who inspired Organic's username? The one who also said we need never stray from the paradise Cantor created? Wonder why that hasn't sunk in?

You posted saying organic might be interested in some mathematics conference. What evidence do you have that he would even understand the simplest talk there, or care? I remember offering him a link to some papers of Keller explaining operads and A infinity algebras which give structures on trees that he said mathematics cannot describe. He dismissed them as worthless in less than 20 minutes.
 
  • #143
Matt:

I am sorry to hear that Organic dismissed significant mathematical paper. Maybe because he have not background to read it.
Well at list you can see that he was influence from Hilbert in the end of his famous lecture since he is trying to develop Organic type of a number with the duality of Quantity and structure.

As far as I can understand big change always started from to most simple thing.

Do you familiar with Wittgenstein attitude to mathematics?


Best
Moshek
 
  • #144
Matt,

Let me refresh your memory:

I wrote:
Matt, can you give me some address where i can read about A-infinity algebras?

Thank you.

Organic

Last edited by Organic on 01-22-2004 at 01:50 PM
Your response was:
And,no *I* won't tell you anything about A-infinity algebras. I don't beleve that will accomplish anything as you've yet to understand the basics of common mathematics, and these are very advanced concepts.
Then I have found myself these articles, and then I wrote to you:
Matt,

If you wrote this:
Your multiplication is not well defined then, because your 'numbers' are not well defined, that is there is no single element that represents it uniquely. For instance you give me several different answers for 2*3. To deduce anything from this you must acknowledge that '3' is not a number in your system and to call them numbers is misleading.

Then I have to say that you are much more closed system then I thought, that has 0 ability to understand new fundamental ideas about what numbers are.

You are in your closed system cannot understand how multiplication and addition are complement operations that do not changing the quantity of the number, but only its internal scructure, by breaking and unbreaking its internal self symmetry.

Now, I see you jump and saying: "here comes more undefined terms".

All the rigoruos terms are infront of your eyes but for you terms is written text, not for me.

Also I looked at:

http://arxiv.org/PS_cache/math/pdf/9910/9910179.pdf

http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf


They do not based on my fundamental ideas, therefore cannot express my mathematical system.
 
Last edited by a moderator:
  • #145
My apologies for my poor recollection. But you did dismiss out of hand these objects without thinking about them for anywhere near long enough to even begin to understand them.


I am familiar with Wittgenstein, and Gowers, who is currently the most vocal proponent of this position on mathematics.

This alleged multiplication on numbers. You do realize that it doesn't even take a pair of integers and then produce a third, which we can call their product. So to say it is the mulitplication that is 'complementary' to addition is disingenuous.


It is an operation on (certain) rooted trees, some subsets of which you label with elements of N. If you cannot even begin to see why you ought to learn about operads given this then it is you who is closed off to 'new' ideas. You've almost never accepted anyone else might have a valid point of view on this.
 
  • #146
Dear Matt !

Please sent me more information about Gowers work.

I am really glad that you know already Wittgenstein attitude to the possibility of creation new mathematic and the place of Klein bottle in this creation by adding one more dimension to mathematics as it appear in today paradigm which base on Logic in it center !

Have you notice already on the similarities of Organic work in mathematics to Prague work on creating first order logic?

Thank you
Moshek
 
  • #147
He's got a fields medal, I think you can locate him withtout too much trouble.

It is hard to notice any similarities with what organic writes and extant maths because he misuses terms all the time. My favourite at the moment being the insistence that the infinite rooted binary tree is a Cantor Set.
His abuse of the term enumerable or countable is now just annoying, as is his refusal to learn about any of the words he uses.

His use of trees is reminiscent of operads, and the operations are those of semi-groupoids or monads. He often asks us to offer current mathematical objects that does what his do, we have, with these, yet I doubt he's ever read about these to see. Nor does he state what his things 'do' (cf my repeated requests to find out how one decides which trees exhibit 'uncertainty') which as a Wittgenstein-follower you should admit is the most important thing.
 
  • #148
Matt,
This alleged multiplication on numbers. You do realize that it doesn't even take a pair of integers and then produce a third, which we can call their product. So to say it is the mulitplication that is 'complementary' to addition is disingenuous.
You can't understand it do you?

Any product in my number system is a structural/quantitative product.

Therefore multiplication and addition must be examined through this point of view, and from this point of view multiplication and addition are complementary operations.

This is a new approach, and partial treatment about it maybe can be found in A-oo algebras here:
http://arxiv.org/PS_cache/math/pdf/0108/0108027.pdf
in what is called "multiplication inside the inner-product-diagram" (page 26).
 
Last edited by a moderator:
  • #149
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
 
  • #150
I do get it, Organic, and because I get it I see that you're being imprecise:

you are calling it a multiplication on N, aren't you?

take two elements in N, say [2] and [3], you have shown what [2]*[3] isn't another natural number, [n].

[2] and [3] in your model are a sets of trees satifying certain properties, their product give you some of the trees in the set [6], doesn't it?

I don't think you can then be justified in calling it a multiplication on N.

I am a pure mathematician, and am happy with abstraction and objects that are apparently meaningless, but you do not provide enough information to say what's going on! Repeatedly I've said you're more than welcome to develop all this machinery, it's just that no one else understands what you're doing because you do not offer enough explanation or define anything in terms that allow other people to reconstruct your objects. If you forgot all the mumbo-jumbo about maths needing to remember the people participating in it, you'de get a lot further. Of course you can't do that because to you it's vitally important, though no one else can see why.
 

Similar threads

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