Transfinites?

[matt grime]

Firstly, this is your interpretation of what symmetry means (with respect to sets and natural numbers), and if it contradicts the usual model of ZF then it is inconsistent with ZF. This would make ZF inconsistent and would be important if you could demonstrate how to deduce your interpretation of symmetry from the axioms of ZF or in some claimed model of ZF. Can you do that?

[Guest Doron Shadmi]

You miss the point dear Matt.

 

ZF is an asymmetric pure method because the Full Set is not defined within its framework.

 

Simple as that, and we do not need more then that in order to clearly concluded that ZF is not a consistent pure Mathematical framework.

 

In other words, any pure method that is not symmetrical in its first order level, cannot define but non-artificial products.

[matt grime]

and what's the definition of non-artificial, and hence artificial?

[Guest Doron Shadmi]

and what's the definition of non-artificial' date=' and hence artificial?

[/quote']

 

When the fundamental building-blocks of a formal system, clearly defined by a symmetrical domain (in my case the symmetrical domain exists by constructive associations between opposite concepts like Emptiness and Fullness), then any product of this domain is considered as a non-artificial product.

 

In the case of ZF, because Fullness is not defined, the result is the artificial products of the transfinite universe that cannot distinguish between Actual-Infinity and Potential Infinity.

 

In short, the before any definition, we must care about the symmetrical condition of our domain.

[haggy]

No dear, the Language of Mathematics cannot be understood, if concepts like Infinity, Limit, Number are not understood first, before any technical definition.

 

Why are you so averse to having what you call technical definitions. The whole point of definitions is to ensure absolute clarity concerning the issue at hand. If one doesn't maintain absolute clarity there is room for confusion. An example of this would be the following:

Say we were to try to prove some theorem about triangles and it started...

Let P be the centre of the triangle.

Now, to the person initially writing it this might seem absolutely clear in meaning. But does this person mean the incentre or the circumcentre etc?

 

Doron, dealing with the concepts of Infinity, Number and Limit in a formal manner isn't a cop-out. Rather it is a means for all involved to know exactly what is being considered at a certain point. This is where much of your explanations don't have any formal definitions that would ensure absolute clarity.

 

In short' date=' for the past 100 years, pure mathematicians play with fancy notations without understanding the deep meaning of them.

 

Furthermore, they are doing their best in order to avoid any exploration of these fundamental concepts (like, Infinity, Number, Limit, Logic, etc…), and if someone tries to touch them, then immediately he can hear the same old broken record: ”It is not Mathematics”, “It is a philosophy”, “You do not understand Mathematics”, etc…[/quote']

There is also another broken record, that of your condescending: "You are not capable of understanding my included middle reasoning" or "You are confined by your 0_XOR_1 reasoning"

[matt grime]

So, once more, by putting conditions on set theory that no one else does you make ZF inconsistent? That is nonsense since consistency is a purely an internal attribute.

[Guest Doron Shadmi]

So' date=' once more, by putting conditions on set theory that no one else does you make ZF inconsistent? That is nonsense since consistency is a purely an internal attribute.

[/quote']

By Godel, no consistent system can be totally sure if it is consistent or not, therefore if we ignore fundamental concept like SYMMETRY, we for sure put our pure system in non-consistent state.

[Guest Doron Shadmi]

There is also another broken record' date=' that of your condescending: "You are not capable of understanding my included middle reasoning" or "You are confined by your 0_XOR_1 reasoning"

[/quote']

You have no case here, because I show clearly and simply why I say it, and I do not see anything in your post that clearly shows problems in my climes.

This is where much of your explanations don't have any formal definitions that would ensure absolute clarity.

Please give me a detailed example.

Why are you so averse to having what you call technical definitions?

Technical definitions are a very important tool to create a common agreement between a group of people, but first the ideas behind the technical "written/spoken" agreement must be deeply understood.

 

In my work I show that Cantor did not understand the Infinity concept, because he ignored SYMMETRY when he created his framework.

 

And without SYMMETRY no pure framework can be consistant, simple as that!!

[haggy]

*bangs head against wall*

[Guest Doron Shadmi]

*bangs head against wall*

Is this your way to give some details about my work?

[premjan]

what is this symmetry principle that you keep talking about Doron? Can you give an example of it?

 

In my work I show that Cantor did not understand the Infinity concept, because he ignored SYMMETRY when he created his framework.

 

And without SYMMETRY no pure framework can be consistant, simple as that!!

[Guest Doron Shadmi]

Yes,

 

If someone uses a concept like "The Empty Set" he cannot ignore its oppsite symmetrical concept, which is "The Full Set".

 

And Cantor ignored The Full Set.

[matt grime]

And what's the full set doron? the set of all sets? so what about russell's paradox? is it because we are restricted by or 0 xor 1 thinking?

[premjan]

Yes' date='

If someone uses a concept like "The Empty Set" he cannot ignore its oppsite symmetrical concept, which is "The Full Set".

And Cantor ignored The Full Set.[/quote']

 

I have been down this route before, and it strikes me as a typical "theistic" or "philosophical" fallacy. Collections are not known for infinite elements but are known only for finite ones (e.g. a basket of fruits is a collection of entities, you can decide whether something is in the basket or outside of it. it is possible to imagine a basket without fruit but a little harder to imagine a basket containing all possible fruit).

 

The basket containing no fruit is physically realizable.

The basket containing all possible fruit (even an infinite number) is a purely imaginary concept. Hence it makes sense that Cantor would treat the infinite as something that had to be demonstrated through construction whereas the empty was relatively easier to demonstrate using a physical analogy.

 

I don't consider symmetry to be an infallible principle, merely a guide of some sort.

 

What about man? What is the opposite of man? Is it God? Do you believe that God exists?

 

Have you heard that the universe was created by "symmetry breaking"? This is one of the modern physics concepts. Scientists believe in the efficacy of symmetry as an organizational tool but do not worship it.

 

Do you worship at the altar of symmetry?

[Guest Doron Shadmi]

And what's the full set doron? the set of all sets?

Matt Grime,

 

Do not play it close again, you know very well that the full set is a non-composed infinitely long element, which is beyond any mathematical manipulation, exactly as Emptiness is beyond any mathematical manipulation.

 

So please tell us, why do you ask this kind of questions?

[Guest Doron Shadmi]

Do you worship at the altar of symmetry?

Hold your horses dear premjan, I am not talking at this stage (in this thread) about the physical existence, but on the minimal conditions of any pure consistent framework.

 

No pure framework can be considered as a consistent method if Symmetry is ignored.

 

(If you want to understand the goal of my work, then please look at http://www.geocities.com/complementarytheory/CATpage.html ).

[premjan]

yeah, I guess I went overboard with the altar and worship remark. I agree that it would be more consistent not to talk of any infinitely long logical expressions. If your system is really better than the competition (Cantor in this case) it ought to be easier to implement as an algorithm (perhaps you could create a Mathematica module or something like that). That would be a real coup for your system. That's probably a good way to get some coverage into the mathematical (or at least the computing) community.

[Guest Doron Shadmi]

If you look at http://cyborg2000.xpert.com/ctheory/ you will find some applet that can represent ONNs.

 

It is based on Cartesian product algorithm and because of this it draws same Left-Right ONNs that can be ignored.

 

Current PCs cannot draw beyond ONN7 or ONN8, and we need an ONN Turing-Like machine in order to draw ONNs in "no-time".

 

This technology does not exist yet, and it is definitely not a one man's work.

[premjan]

do you have an example of a problem that works well in ONN that is not tractable in dyadic logic? that will help me to get a real hold on what you are trying to get at.

[premjan]

OK I found it at that site you posted earlier: SAT is apparently decidable in Monadic logic (I still need to get my brain around how an undecidable problem turns decidable just by changing the logic system).

[Guest Doron Shadmi]

A question to Matt Grime,

 

When we use a Symmetrical framework, we immediately understand that if we go beyond a model which is based on infinitely many elements, we find ourselves in a model that cannot be manipulated by any language, including the language of Mathematics and its reasoning.

 

I call this model Actual infinity or Fullness, which is described by me as an infinitely long non-composed (indivisible) solid element.

 

This kind of a model is too strong to be used as an input by any language (including Math), exactly as Emptiness is too weak to be used as an input by any language (including Math).

 

Strictly speaking, 2*Emptiness or 2^Emptiness is meaningless (be aware that it is not the same as {} and Power({})={{}} that in this case manipulates the set and not its “content”, which is Emptiness) and also 2*Fullness or 2^Fullness is meaningless.

 

Because Cantor ignored Fullness he made a fatal conceptual mistake and invented Aleph0 in such a way that cannot be but a model of Fullness.

 

And the reason is very simple, because if we take a model which is based on infinitely many elements and force on it a Universal Quantification, then we are no longer in the state of Potential infinity (which is described as infinitely many elements that cannot be completed to a state of an infinitely long indivisible and non-composed solid element) but in a state of Actual infinity, which cannot be manipulated by the language of Mathematics.

 

In short, Cantor created the Transfinite universe by comparing between aleph0 and 2^aleph0, which from a symmetrical point of view (where Fullness is the opposite of Emptiness) is equivalent to Fullness and 2^Fullness comparison, which is a meaningless mathematical operation.

 

 

http://www.dpmms.cam.ac.uk/~wtg10/ is perfectly written and exactly because of this reason, it is a good example of the failure of the Standard approach about the language of Mathematics.

 

The Standard approach basically takes care about the technical methods that can help us to communicate according some agreed common rules, but this approach generally ignores any research of the insights and the deep meaning of the concepts, that stand in the basis of these agreed common rules.

 

Fancy mathods to create precise propositions according to rigorus gremial rouls, are definatly not enoph.

 

As a result of this approach, Cantor ignored Symmetry and missed the full understanding of the Infinity concept, which led him to use a universal quantification on a collection of infinitely many elements.

 

This combination clearly cannot be done if by using Symmetry, you first care about the consistency of your framework (and in this case: if {} then {__}).

 

So all those fancy technical methods and their clear explanations that can be found in http://www.dpmms.cam.ac.uk/~wtg10/ did not help Cantor to avoid his fatal conceptual mistake about the Infinity concept (he used Emptiness and ignored Fullness, and by this he missed the possibility to distinguish between Actual Infinity and Potential Infinity).

 

This lack of educational methods to research the insights and the meaning of the concepts behind them, simply blocks Modern Mathematics from being exposed to real re-examination of its most fundamental concepts, which lead it step by step to become a scholastic closed and technical-only framework that cannot be changed by fundamental new ideas.

 

Strictly speaking, fancy methods to create precise propositions according to rigorous gremial rules, are defiantly not enough to create good mathematicians, if most of the time they play with rigorous gremial rules, without any educational methods that give them the opportunity to develop their understanding of the concepts that can be found in the basis of these rigorous gremial rules.

 

In short, the academic system is a very good place to become a very good technician, which is no more than a walking encyclopedia of rigorous gremial rules, without any original creativity behind them.

 

Here is a typical example of the academic approach about new ideas:

 

Please pay attention to the last two posts of these thread, before the Super Moderator of this forum closed it:

 

Some explanation:

 

In the first post you can find a typical response of a member that rewrites what he was thought in the university, instead of using his own abilities to research the subjects of this thread:

 

http://www.physicsforums.com/showpost.php?p=338863&postcount=177

 

 

And now pay attention to the basic attitude of Tom, the Super Mentor of this forum:

 

http://www.physicsforums.com/showpost.php?p=339299&postcount=178

 

The whole thread can be found in:

 

http://www.physicsforums.com/showthread.php?t=16133&page=1&pp=15

 

The big problem that I want to show in this post, is not related to any specific parson that was mentioned in this post.

 

The big problem is this:

 

How can it be that after more then 100 years of mathematical activity, we do not find any professional mathematician that discovered Cantor's fatal conceptual mistake about the Infinity concept, and do you think that there is any connection between this fact and the educational methods of the academic system, along these 100 years?

 

My solution to Aleph0 concept

 

My concept of aleph0 is based on "cloud-like" magnitude of any collection of infinitely many elements.

 

For example:

 

aleph0+1 > aleph0

 

If A = aleph0 and B = aleph0 - 2^aleph0, then A > B by 2^aleph0, where both A and B are collections of infinitely many elements.

 

Also 3^aleph0 > 2^aleph0 > aleph0 > aleph0 - 1, etc...

 

So, as you can see my aleph0 is much more flexible and rich concept than the standard Cantorian approach.

 

Fore more details please look at: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf

 

Strictly speaking, Actual infinity is too strong to be used as an input.

 

Potential infinity (which never reaches Actual infinity, and therefore cannot be completed) is the name of the game.

 

For further information please look at:

 

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

 

http://www.geocities.com/complementarytheory/9999.pdf

 

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

 

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

 

Also Cantor's proof, which is not based on the second diagonal method http:// http://en.wikipedia.org/wiki/Cantor's_first_uncountability_proof (The first is 1-1 and onto between N and Q), is actually failed because of a very simple conceptual mistake, which is:

 

If A set, c point, and B set are clearly distinguished from each other, then there cannot be a gapless state between them, simple as that!!

 

 

In short, Cantor uses simultaneously two different models (3_distingueshed_states_AND_a_solid_line) that are clearly contradicting each other.

 

Therefore this proof does not hold.

[Guest Doron Shadmi]

Doron' date='

 

People use set theory because it has given years of fruitful mathematics. It also is just fine and dandy consistent. If you were serious about set theory, you'd learn something developed after 1930. There are plenty of great constructive set theories that fit your limitations on the concept of actual infinity, yet they do not deny the consistency of normal set theory. There are many constructive theorems about the limits of the applicabilty of ordinary set theory and its usefulness in generating concepts for discussion.

 

Try Bishop and Bridges book on constructive mathematics. They are able to succesfully introduce all the concepts in this thread constructively.[/quote']

 

 

1) I am not talking about any specific version of Set Theory.

 

 

2) I am talking about a fatal conceptual mistake that Cantor did when he used the idea of sets.

 

 

 

My argument about the Cantorian transfinite system (where aleph0 is not in N), is based on the most simple things that, in my opinion, we should care about when we define a consistent framework, and these things are:

 

 

Simplicity and Symmetry.

 

 

In this case, we do not think about quantity (fewest possible elements) but about simplicity.

 

Only the simplest thing can be considered as a building-block of some pure framework.

 

In the case of Emptiness, it is the lowest concept that cannot be manipulated by any framework that is based on information, and the language of Mathematics is first of all an information system, like any language, formal or informal.

 

When we have the lowest concept that cannot be manipulated by any framework, then if we want to save the simplicity of our framework, we cannot ignore anymore its internal symmetry.

 

So, if we use Emptiness, then in order to save the simplicity of our framework, we use symmetry, and define Fullness as the highest concept that cannot be manipulated by any framework.

 

By saving the simplicity and symmetry of our framework, we actually define its operational domain, where we can work and do interesting Math.

 

Cantor missed this important insight and created an asymmetrical framework that do not aware to the highest concept that cannot be manipulated by any framework, which is Fullness

 

And the result is the transfinite universe that ignore Actual infinity (which is both Fullness and Emptiness concepts).

 

When we look at this diagram http://www.geocities.com/complementarytheory/CompLogic.pdf we can clearly and simply see that Actual infinity is an inevitable fundamental concept of any set theory, which uses Emptiness as one of its concepts.

 

Strictly speaking, any set theory that uses Emptiness, cannot ignore Fullness, in order to be consistent.

 

Since the Cantorian approach uses Emptiness but ignore Fullness, it cannot be a consistent framework, and this conclusion is stronger then any proof which is based on some axiomatic system.

[Guest Doron Shadmi]

Doron' date='

 

People use set theory because it has given years of fruitful mathematics. It also is just fine and dandy consistent. If you were serious about set theory, you'd learn something developed after 1930. There are plenty of great constructive set theories that fit your limitations on the concept of actual infinity, yet they do not deny the consistency of normal set theory. There are many constructive theorems about the limits of the applicabilty of ordinary set theory and its usefulness in generating concepts for discussion.

 

Try Bishop and Bridges book on constructive mathematics. They are able to succesfully introduce all the concepts in this thread constructively.[/quote']

 

 

1) I am not talking about any specific version of Set Theory.

 

 

2) I am talking about a fatal conceptual mistake that Cantor did when he used the idea of sets.

 

 

 

My argument about the Cantorian transfinite system (where aleph0 is not in N), is based on the most simple things that, in my opinion, we should care about when we define a consistent framework, and these things are:

 

 

Simplicity and Symmetry.

 

 

In this case, we do not think about quantity (fewest possible elements) but about simplicity.

 

Only the simplest thing can be considered as a building-block of some pure framework.

 

In the case of Emptiness, it is the lowest concept that cannot be manipulated by any framework that is based on information, and the language of Mathematics is first of all an information system, like any language, formal or informal.

 

When we have the lowest concept that cannot be manipulated by any framework, then if we want to save the simplicity of our framework, we cannot ignore anymore its internal symmetry.

 

So, if we use Emptiness, then in order to save the simplicity of our framework, we use symmetry, and define Fullness as the highest concept that cannot be manipulated by any framework.

 

By saving the simplicity and symmetry of our framework, we actually define its operational domain, where we can work and do interesting Math.

 

Cantor missed this important insight and created an asymmetrical framework that do not aware to the highest concept that cannot be manipulated by any framework, which is Fullness

 

And the result is the transfinite universe that ignore Actual infinity (which is both Fullness and Emptiness concepts).

 

When we look at this diagram http://www.geocities.com/complementarytheory/CompLogic.pdf we can clearly and simply see that Actual infinity is an inevitable fundamental concept of any set theory, which uses Emptiness as one of its concepts.

 

Strictly speaking, any set theory that uses Emptiness, cannot ignore Fullness, in order to be consistent.

 

Since the Cantorian approach uses Emptiness but ignore Fullness, it cannot be a consistent framework, and this conclusion is stronger then any proof which is based on some axiomatic system.

[zaphod]

truly disheartening

[Guest zeroatx]

In this case, we do not think about quantity (fewest possible elements) but about simplicity.

 

Define "simplicity".

 

Only the simplest thing can be considered as a building-block of some pure framework.

 

Define "pure". Define "framework".

 

When we have the lowest concept that cannot be manipulated by any framework, then if we want to save the simplicity of our framework, we cannot ignore anymore its internal symmetry.

 

Define "symmetry".

 

So, if we use Emptiness, then in order to save the simplicity of our framework, we use symmetry, and define Fullness as the highest concept that cannot be manipulated by any framework.

 

I call this model Actual infinity or Fullness, which is described by me as an infinitely long non-composed (indivisible) solid element.

 

Please define "emptiness". Please redefine "fullness". Your definitions ("the highest concept that cannot be manipulated by any framework" and "an infinitely long non-composed (indivisible) solid element") are nonsensical. The first is semantically flawed to such a degree that it is meaningless. The second definition is simple fantasy; i.e. not relevant to the physical world. What do you mean by "highest"? What do you mean by "concept"? What do you mean by "manipulated"? "Any"? What do you mean by "framework"?

 

By saving the simplicity and symmetry of our framework, we actually define its operational domain, where we can work and do interesting Math.

 

Again, what do you mean by framework? What do you mean by "actually"?Define "interesting".

 

Cantor missed this important insight and created an asymmetrical framework that do not aware to the highest concept that cannot be manipulated by any framework' date=' which is Fullness

 

And the result is the transfinite universe that ignore Actual infinity (which is both Fullness and Emptiness concepts).[/quote']

 

What? Your grammar is so faulted that it renders the above quote meaningless.

 

When we look at this diagram http://www.geocities.com/complementarytheory/CompLogic.pdf[/url'] we can clearly and simply see that Actual infinity is an inevitable fundamental concept of any set theory, which uses Emptiness as one of its concepts.

 

Your math has the same flaw as your language: it is semantically hollow.