Deeper

[Guest Doron Shadmi]

In order to define a framework that is researched and manipulated by a formal language, we first have to define the limitations of the formal language itself.

 

Since any language cannot work without information, we first define what are the minimal and maximal conditions that cannot give us any information, when we try to research them.

 

The minimal concept that cannot be researched is Emptiness, because no information can be found in it.

 

The maximal concept that cannot be researched is Fullness, because it is beyond measurement or manipulation of information.

 

So, formal language can work, if and only if we can measure or manipulate information by our rigorous and consistent logical/gremial rules of our framework.

 

It is obvious that this information can be found only within the middle domain that exists between Emptiness and Fullness, where Emptiness and Fullness concepts, are included in the middle domain only as its non-reached (permanent unexplored) limitations.

 

By this basic approach, which comes before any formal method, we achieve 3 fundamental conditions that must exist in the basis of any consistent framework:

 

1) We clearly define its operational domain.

 

2) Any measureable element that can be found in the middle-domain, cannot be but a consistent element of the middle-domain, because it is based on a non-destructive associations between the limits of the middle-domain.

 

3) Our framework is symmetrical by default, because it is based on constructive associations between opposites.

 

These initial conditions are so comprehensive until they can use concepts like Redundancy and Uncertainty as first-order conditions of our framework.

 

It means that we not just use information, but also first define the full range of the information concept itself, before it is used by some logical/gremial rules.

 

This approach is deeper than any formal logical/gremial rules, and exposes new abilities of the language of Mathematics, which are beyond the Standard formal method.

 

(Urelement: http://mathworld.wolfram.com/Urelement.html)

 

Empty set is {}.

 

Full set is {__}.

 

Emptiness and Fullness concepts are used as the weakest and strongest information forms that cannot be manipulated by any mathematical tool.

 

It means that thay are the weakest and the strongest non-composed information forms, which are used as the limits of any formal or informal language.

 

By using the set concept we can distinguish between emptiness and fullness, because:

 

|{}| = 0

 

where |{__}| = The one, where "The One" is a non-composed element, which is different from "one of many".

 

By using these limitations, we first clearly distinguish between actual infinity (Emptiness and Fullness) and potential infinity, which is any collection of infinitely many elements.

 

In short, potential infinity is the composed form of infinity that cannot reach the state of the non-composed actual infinity.

 

Please look at this diagram, in order to understand better my idea:

 

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

 

As for symmetry:

 

(If {} then {__}) AND (if {__} then {}) = 1

 

By this proposition we get the true value, which is based on a symmetry between opposite concepts.

 

The logical basis of Emptiness is: oo …E nor E nor E … oo = 1

 

The logical basis of Fullness is: oo …F and F and F … oo = 1

 

For more detailes please see pages 4,5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf

 

 

Point is {.}.

 

Segment is {._.}

 

The difference between Sgment's edge and a point, can be found in:

 

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

 

a) There is no urelement between {} and {.}.

 

b) There is no urelement between {.} and {._.}.

 

c) There is no urelement between {._.} and {__}.

 

By {} <--x(={.}) we mean that {.} is a potential {}.

 

By x(={._.}--> {__}} we mean that {._.} is a potential {__}.

 

The least useful input cannot be anything but a combination of {.} AND {._.} forms; therefore x is at least both {.} AND {._.} information form.

 

Now we can write ({},{__}):={x|{} <--x(={.}) AND x(={._.})--> {__}}

 

 

Some examples that are based on this approach:

 

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

A proof that cannot be accomplished by using standard N members:

 

Theorem: 1*5 not= 1+1+1+1+1

 

Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1

 

To understand this proof, please read at least page 13 of http://www.geocities.com/complementarytheory/ONN2.pdf

 

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

A test that shows the advantage of - and + operations in an included-middle logical reasoning framework, can be found in pages 22-29 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf

 

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

Complementary relations between Multiplication and Addition binary operations can be found in pages 7-8 of http://www.geocities.com/complementarytheory/ONN1.pdf

 

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

A fundamental new approach about the Natural numbers can be found in:

 

http://www.geocities.com/complementarytheory/ONN1.pdf

 

http://www.geocities.com/complementarytheory/ONN2.pdf

 

http://www.geocities.com/complementarytheory/ONN3.pdf

 

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

A new approach about 0.9999... = 1 can be found here:

 

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

 

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

A new approach about the Limit concept can be found here:

 

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

 

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

A new approach about Russell's first paradox, can be found here:

 

http://www.geocities.com/complementarytheory/Russell1.pdf

 

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

A new approach about Cantor's diagonal methods can be found here:

 

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

 

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

 

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

A new approach about Collatz' problem can be found here:

 

http://www.geocities.com/complementarytheory/3n1proof.pdf

 

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

A new approach about the Real numbers can be found here:

 

http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

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

A new approach about the Infinity concept can be found here:

 

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

 

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

A new approach about the Function concept can be found here:

 

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

 

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

A new approach about the Logic concept can be found here:

 

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

 

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

[Guest Doron Shadmi]

In order to define a framework that is researched and manipulated by a formal language, we first have to define the limitations of the formal language itself.

 

Since any language cannot work without information, we first define what are the minimal and maximal conditions that cannot give us any information, when we try to research them.

 

The minimal concept that cannot be researched is Emptiness, because no information can be found in it.

 

The maximal concept that cannot be researched is Fullness, because it is beyond measurement or manipulation of information.

 

So, formal language can work, if and only if we can measure or manipulate information by our rigorous and consistent logical/gremial rules of our framework.

 

It is obvious that this information can be found only within the middle domain that exists between Emptiness and Fullness, where Emptiness and Fullness concepts, are included in the middle domain only as its non-reached (permanent unexplored) limitations.

 

By this basic approach, which comes before any formal method, we achieve 3 fundamental conditions that must exist in the basis of any consistent framework:

 

1) We clearly define its operational domain.

 

2) Any measureable element that can be found in the middle-domain, cannot be but a consistent element of the middle-domain, because it is based on a non-destructive associations between the limits of the middle-domain.

 

3) Our framework is symmetrical by default, because it is based on constructive associations between opposites.

 

These initial conditions are so comprehensive until they can use concepts like Redundancy and Uncertainty as first-order conditions of our framework.

 

It means that we not just use information, but also first define the full range of the information concept itself, before it is used by some logical/gremial rules.

 

This approach is deeper than any formal logical/gremial rules, and exposes new abilities of the language of Mathematics, which are beyond the Standard formal method.

 

(Urelement: http://mathworld.wolfram.com/Urelement.html)

 

Empty set is {}.

 

Full set is {__}.

 

Emptiness and Fullness concepts are used as the weakest and strongest information forms that cannot be manipulated by any mathematical tool.

 

It means that thay are the weakest and the strongest non-composed information forms, which are used as the limits of any formal or informal language.

 

By using the set concept we can distinguish between emptiness and fullness, because:

 

|{}| = 0

 

where |{__}| = The one, where "The One" is a non-composed element, which is different from "one of many".

 

By using these limitations, we first clearly distinguish between actual infinity (Emptiness and Fullness) and potential infinity, which is any collection of infinitely many elements.

 

In short, potential infinity is the composed form of infinity that cannot reach the state of the non-composed actual infinity.

 

Please look at this diagram, in order to understand better my idea:

 

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

 

As for symmetry:

 

(If {} then {__}) AND (if {__} then {}) = 1

 

By this proposition we get the true value, which is based on a symmetry between opposite concepts.

 

The logical basis of Emptiness is: oo …E nor E nor E … oo = 1

 

The logical basis of Fullness is: oo …F and F and F … oo = 1

 

For more detailes please see pages 4,5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf

 

 

Point is {.}.

 

Segment is {._.}

 

The difference between Sgment's edge and a point, can be found in:

 

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

 

a) There is no urelement between {} and {.}.

 

b) There is no urelement between {.} and {._.}.

 

c) There is no urelement between {._.} and {__}.

 

By {} <--x(={.}) we mean that {.} is a potential {}.

 

By x(={._.}--> {__}} we mean that {._.} is a potential {__}.

 

The least useful input cannot be anything but a combination of {.} AND {._.} forms; therefore x is at least both {.} AND {._.} information form.

 

Now we can write ({},{__}):={x|{} <--x(={.}) AND x(={._.})--> {__}}

 

 

Some examples that are based on this approach:

 

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

A proof that cannot be accomplished by using standard N members:

 

Theorem: 1*5 not= 1+1+1+1+1

 

Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1

 

To understand this proof, please read at least page 13 of http://www.geocities.com/complementarytheory/ONN2.pdf

 

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

A test that shows the advantage of - and + operations in an included-middle logical reasoning framework, can be found in pages 22-29 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf

 

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

Complementary relations between Multiplication and Addition binary operations can be found in pages 7-8 of http://www.geocities.com/complementarytheory/ONN1.pdf

 

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

A fundamental new approach about the Natural numbers can be found in:

 

http://www.geocities.com/complementarytheory/ONN1.pdf

 

http://www.geocities.com/complementarytheory/ONN2.pdf

 

http://www.geocities.com/complementarytheory/ONN3.pdf

 

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

A new approach about 0.9999... = 1 can be found here:

 

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

 

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

A new approach about the Limit concept can be found here:

 

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

 

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

A new approach about Russell's first paradox, can be found here:

 

http://www.geocities.com/complementarytheory/Russell1.pdf

 

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

A new approach about Cantor's diagonal methods can be found here:

 

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

 

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

 

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

A new approach about Collatz' problem can be found here:

 

http://www.geocities.com/complementarytheory/3n1proof.pdf

 

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

A new approach about the Real numbers can be found here:

 

http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

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

A new approach about the Infinity concept can be found here:

 

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

 

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

A new approach about the Function concept can be found here:

 

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

 

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

A new approach about the Logic concept can be found here:

 

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

 

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

[Sayonara]

Do you ever stop to think that the people who want to know this are more than capable of finding your web site?

 

We aren't your pimp.

[Guest Doron Shadmi]

people who want to know this are more than capable of finding your web site

Then please tell me what do you thing about my point of view?

 

Thank you.

[premjan]

I think your approach is more amenable to a computer algorithm than to proofs (which is what mathematicians traditionally do more of). Maybe you are posting in the wrong section?

[Sayonara]

Then please tell me what do you thing about my point of view?

Thank you.

I'm not really interested in it, personally. That doesn't have anything to do with the question I asked though.

[Guest Doron Shadmi]

I think your approach is more amenable to a computer algorithm than to proofs (which is what mathematicians traditionally do more of). Maybe you are posting in the wrong section?

Dear premjam, proofs are based on logical deductions of consistent axiomatic systems.

 

But cocsistent axiomatic systems cannot be consistent if their first-order level has an asymmtric proprty, and ZF set theory is not symmteric because it uses {} and ignore {__}.

 

conclusion: All proofs that are based on ZF does not hold.

 

Mathematicians will do the best they can do in order to eliminate this simple fact, but "the king is still naked".

[premjan]

the problem is that proofs are generally designed for people (mathematicians) to follow, not for computers. Your approach might reduce the number of steps for a proof, especially a mechanical proof, which might make it useful in computer theorem provers.

[premjan]

as for the incorrectness of ZF proofs, you have to show this by an actual counterexample, just saying that the system has a flaw is not likely to be good enough.

[Guest Doron Shadmi]

the problem is that proofs are generally designed for people (mathematicians) to follow' date=' not for computers. Your approach might reduce the number of steps for a proof, especially a mechanical proof, which might make it useful in computer theorem provers.

[/quote']

On the contrary, computers will be used only just as a tool to reach any mthematical object and the Proof concept will be replaced by the CuRe concept.

 

Please read http://www.geocities.com/complementarytheory/CuRe.pdf in order to understand my point of view.

 

Thank you.

[Guest Doron Shadmi]

as for the incorrectness of ZF proofs' date=' you have to show this by an actual counterexample, just saying that the system has a flaw is not likely to be good enough.

[/quote']

I clearly show here that the Cantorian transfinite system, which is based on ZF, SIMPLY DOES NOT HOLD, we do not need more then that.

 

Please see for yourself: http://www.geocities.com/complementarytheory/TRANSFINITES.pdf

 

In this case mathematicians will also do their best in order to eliminate this fact.

 

But facts which are based on symmetry cannot be eliminated, unless you break the symmetry but then you get an inconsistent pure framework.

[premjan]

simply does not hold because of asymmetry, but I hardly consider that a law in the same vein as e=mc^2.

[Guest Doron Shadmi]

'=' is based on symmetry, where '>' or '<' are asymmerty.

[premjan]

your "proof" that Cantor is wrong depends on his use of 2^aleph0 for the magnitude of R. This notation is not compulsory, and in hindsight perhaps ought to be discarded.

 

What Cantor ought to have said was "power set" instead of 2^aleph0. Unfortunately Cantor is not here to explain himself.

[Guest Doron Shadmi]

The difference between power sets can be found only between collections of inifinitely many elements, but Cantor's transfinite cardinality goes beyond any member in any collection, therefore it cannot be but an actual infinity, and actual infinity cannot be manipulated by the language of Mathematics.

[premjan]

is the difference between power sets used in the diagonal argument?

[Guest Doron Shadmi]

Yes, aleph0 < 2^aleph0 < 2^(2^aleph0) thay are enumerable by my approach.

 

The Cantorian aleph0 cannot be a Natural number if aleph0+n = aleph0, and this is exactly some result of a Cantorian transfinite cardinals arithmetic.

 

So the Cantorian transfinite system is based on self contradiction, if aleph0 is a Natural number, therefore aleph0 must be beyond any Natural number, and in this case, it does not belong anymore to any model, which is based on infinitely many elements.

 

The only Cantorian alternative is that aleph0 is Fullness (an infinitely long pointless solid element) and then it cannot be manipulated by the language of Mathematics.

 

Even the most primitive compression, shows the triviality of the Cantorian aleph0, for example:

 

By Cantor: aleph0+1=aleph0, aleph0-2^aleph0 has no meaning, aleph0 < 2^aleph0, 3^aleph0=2^aleph0, etc...

 

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...

 

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

[Sayonara]

Doron, you have still not answered my question.

[premjan]

I guess transfinite cardinals are a bit like infinitesimals (dx,dy) used by Newton which are also nonsensical, strictly speaking, or like renormalization in QED which is also mathematically nonsensical. However, some limited manipualation with these entities does work. In your model aleph0 is basically unmanipulable?? In your system, how does one arrive at the fact that aleph0 is the cardinality of the naturals? Is it possible to say this?

[Guest Doron Shadmi]

In my system, aleph0+1 > aleph0, so aleph0 is a general magnitude for any collection of infinitely many elements.

[Guest Doron Shadmi]

Doron' date=' you have still not answered my question.

[/quote']

Please ask questions, which are related to the subject of the tread, thank you.

[atinymonkey]

Ooooh, catfight.

 

 

No hair pulling O_o

[premjan]

so your notation has nothing to do with set cardinality at all? I guess it does not strictly need to. Then again, {__} is apparently not related to the cardinality of {. in N}?

[Sayonara]

Please ask questions, which are related to the subject of the tread, thank you.

How about if you answer my question now.

[Guest Doron Shadmi]

so your notation has nothing to do with set cardinality at all? I guess it does not strictly need to. Then again' date=' {__} is apparently not related to the cardinality of {. in N}?

[/quote']

In my system I have two kinds of cardinals and two kinds of ordinals, for details please start from page 3 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf

 

Thank you.