Russell's Paradox and the Excluded-Middle reasoning

[master_coda]

Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict itself identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?

Can you be master-coda which is not master-coda?

If you have A = "set of all sets that contain themselves" then all you know is that if a set is in A, then that set must contain itself. The definition does not say that A must also contain itself.

If B = "set of all even numbers" you cannot assume that B must itself be an even number. If C = "set of all sets that contain themselves" you cannot assume that C is a set that contains itself. If D = "set of all sets that do not contain themselves" then you cannot assume that D must not contain itself.


And, even if you were to add an axiom to your system that said "the set of all sets that do not contain themselves does not contain itself", Russel's paradox still applies. No amount of whining that the set must not contain itself will change that.

 

[Hurkyl]

In this case A cannot be in a state that do not satisfies the properties of its parts, and this it exactly what I say.

Then you have to prove it. You have to prove that A satisfies the condition required to be a member of A.

The set of all dogs is not a dog itself but it has a common property with each dog that included in it, which we can call it "Dogness".

Can you show one example of "dogness" that is shared by "the set of all dogs"?

1) Please show us how a set can get its name (identity) without any relation to the properties of its contents.

"The set of all dogs" gets is name precisely because:

(1) every dog is contained in the "set of all dogs"
(2) everything in the "set of all dogs" is a dog

There is no reason to think that the "set of all dogs" should have any properties of dogs. The relation here is:

A thing is a member of the "set of all dogs" if and only if that thing is a dog.


But, finally, "the set of all dogs" doesn't need to have a name; it can be discerned completely by the two properties I listed above.

In fact, pay attention to the fact that the name "the set of all dogs" is not really a name at all; it is a phrase stating what objects are in the set!


Speaking loosely, the "identity" of a set is entirely determined by the identities of its contents. This is stated in the axioms of ZFC by "A and B are the same set if and only they have the same elements".

By the way, to say that a set is not its members is a non-abstract and trivial thing to say.

You have this exactly backwards. A set is an abstract thing, whose "identity" is given entirely by the concept of "membership". The fact that you are unwilling (unable?) to separate the two ideas "properties of a set" and "properties of the elements of a set" is a very strong indicator of non-abstract thinking.

 

[Lama]

Master coda, think simple.

If by some operation or by avoiding some operation, a set contradict its own identity, then and only then we can conclude that a set loses its own identity.

And when a set loses its own identity, we cannot talk about its lost identity, as if it is still available for us.

 

[master_coda]

Master coda, think simple.

If by some operation or by avoiding some operation, a set contradict its own identity, then and only then we can conclude that a set loses its own identity.

And when a set loses its own identity, we cannot talk about its lost identity, as if it is still available for us.

Again, these are nothing more than rules you have just made up.

 

[Lama]

Hurkyl,

From your last post we can clearly see that you do not read all the posts that I write, and as a result most of what you write is based of luck of information.

But maybe this is the nature of your work, to read pieces of information and jump from thread to thread like a butterfly.

In short, please read ,for example, post #63.

 

[master_coda]

Hurkyl,

From your last post we can clearly see that you do not read all the posts that I write, and as a result most of what you write is based of luck of information.

But maybe this is the nature of your work, to read pieces of information and jump from thread to thread like a butterfly.

In short, please read ,for example, post #63.

Ha! You critizing people for lack of understanding is just funny. You don't even know the basic principles of logic, you don't know how to define things, and you can barely produce coherent English.

Perhaps you should consider the possibility that we all haven't fallen over ourselves worshipping your brilliance because you ideas make no sense, and not because we are stupid.

 

[Lama]

Again, these are nothing more than rules you have just made up.

No you use these simple rules to keep the identity of somthing.

These are the most simple rules that for example keeping you for not be me.

 

[Lama]

and you can barely produce coherent English.

My language is Hebrew.

..and not because we are stupid.

I think that Hurkyl, Matt Grime and you dear master coda are very intelegent and wise persons.

I simply have another point of view on the foundations of the language of Mathematics and to what directions it has to be developed.

 

[Hurkyl]

Self-identity says "a thing is itself"; nothing more, nothing less.

In the case where we're considering the thing "The set of all sets that contain themselves", self identity says:


"The set of all sets that contain themselves" is "the set of all sets that contain themselves".

Notice that this is a different statement than

"The set of all sets that contain themselves" is a "set that contains itself".


The other examples I mentioned are intended to show this. The following three statements are all false:

"The set of all dogs" is a "dog"
"The set of everything worth less than a dollar" is "worth less than a dollar"
The set of everything blue is blue.


The point is, "self-identity" cannot be used (by itself) to prove:

"The set of all T" is a "T".

 

[Hurkyl]

Perhaps you can demonstrate[with logical symbolism] how the composition fallacy is not a form of modus ponens error?

Any example would do.

Here's one: $\forall x \in A: P(x)$ therefore $P(A)$.

 

[Lama]

The point is, "self-identity" cannot be used (by itself) to prove:

Can a set which its identity is: "all_members_thet_do_contain_themselves" can contradict itself identity and include itself as a member of itself as if its identity is not important (or in other words: without losing its own identity)?

Can you be Hurkyl which is not Hurkyl?

"The set of all sets that contain themselves"

A set with no identity is only the green part.

Some identity (name) is the red part.

We need both green and red parts to make Math.

And the red part cannot contradict itself during Math operations.

 

[Hurkyl]

The problem is that you're not being consistent about when you use the green part.

Self identity says:

The set of all sets which contain themselves is The set of all sets which contain themselves.

Furthermore, we have the tautology

A set which contains itself is a set which contains itself.

However, you are saying

The set of all sets which contain themselves is a set which contains itself.

You can either have the green on both sides of "is", or on neither side; it is (usually) incorrect to mix and match.

 

[Lama]

Sorry Hurkyl,

But you have changed what I wrote.

The green part is: The set of...

The red part is: all_sets_that...

 

[PFanalog57]

Any example would do.

Here's one: $\forall x \in A: P(x)$ therefore $P(A)$.

If A then B


If each stick is breakable, then the whole bundle of sticks is breakable.

The fallacy of composition has a faulty premise.

 

[Lama]

If each stick is breakable, then the whole bundle of sticks is breakable.

It is like a self similarity of a fractal.

 

[Lama]

What I suggest is very simple:

1) The language of Mathematics is based on two systems: The relative and the absolute.

2) The absolute system is a finite or infinitely many elements with no unique identity.

3) The relative system is a finite or infinitely many possible unique names.

4) If a possible unique name is related to some absolute element, it determines its identity.

5) There are two basic types of operations on an element with a unique identity:

a) An operation that changes its identity.

b) An operation that does not change its identity.

6) In an excluded-middle reasoning, an absolute element can have simultaneously a one and only one unique name (identity).


If the absolute element is a set under an excluded-middle reasoning, then:

1) Its identity depends on the most abstract property of its content; therefore it cannot contradict the most abstract property of the content.

2) This identity cannot be changed under any operation, unless the most abstract property of the content is changed.

3) If the identity of a set is changed under an operation, its previous identity is not related to it anymore.

 

[master_coda]

Those rules you proposed are not any use when you still have a fundamentally flawed concept of "identity"; you still believe that any property that describes the contents of a set must describe the set itself.

 

[Lama]

Those rules you proposed are not any use when you still have a fundamentally flawed concept of "identity"; you still believe that any property that describes the contents of a set must describe the set itself.

An identity of a set cannot be but with a relation with the most abstract proprty of the content.

It is very fundamental.

By "most abstract proprty" I mean thet when this proprty is omited, the set losing its identity, and return to the state of a non-unique absolute element.

The idea of the relative/absolute relations can be used as a fundamental idea in more then one aspect of the Langauge of Mathematics.

For eample, please read this: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

 

[master_coda]

An identity of a set cannot be but with a relation with the most abstract proprty of the content.

It is very fundamental.

By "most abstract proprty" I mean thet when this proprty is omited, the set losing its identity, and return to the state of a non-unique absolute element.

The idea of the relative/absolute relations can be used as a fundamental idea in more then one aspect of the Langauge of Mathematics.

For eample, please read this: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf

You seem to be treating abstact ideas as if they are phsyical objects that can be changed or destroyed.

For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.

A set which exists cannot be somehow made to not exist. You can construct a new set that has weaker properties (the equivalent of "removing" the properties or "identity" of the original set) but that does not make the original set no longer exist.

 

[matt grime]

none of the things you suggest mathematics must be, or do, is informed, lama/doron/shmesh/etc, and just demonstrates your complete lack of understanding, and you total ignorance of the world you claim to talk about. Are you even aware of topoi where the 'excluded middle' fails to be true? No, you aren't. Mathematics is far richer than you can even begin to understand, and the repeated demonstrations of your ignorance of it are not particulalry endearining.

You are also inconsitent in the extreme. One need only look at you belief in dichotomic options to see that.

 

[Lama]

none of the things you suggest mathematics must be, or do, is informed, lama/doron/shmesh/etc, and just demonstrates your complete lack of understanding, and you total ignorance of the world you claim to talk about. Are you even aware of topoi where the 'excluded middle' fails to be true? No, you aren't. Mathematics is far richer than you can even begin to understand, and the repeated demonstrations of your ignorance of it are not particulalry endearining.

You are also inconsitent in the extreme. One need only look at you belief in dichotomic options to see that.

There is a little problem here dear Matt.

You did not show us that you understand my ideas about Math.

Please also read my answer to master coda.

For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.

So, you do not understand the idea of the relative/absolute system.

In an excluded-middle reasoning an absolute element (set, point, ...) can have simultaneously a one and only one unique name (identity).

And I do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.

The identity of an absolute element is its literal name like: point named 'pi', point named 'e',
point named '1', point named '0', the set of 'all_sets_that_do_not_contain_themselves' ... and so on.

So nothing disappears here.

A set which its literal name is 'all_sets_that_do_not_contain_themselves' cannot contain itself (in an excluded middle reasoning) exactly as some absolute point cannot have more than one literal name (in an excluded-middle reasoning).

Conclusion: Russel's Paradox cannot be defined in an excluded-middle reasoning.

Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.

I am waiting for your detailed remarks of my papers.

Thank you,

Lama

 

[PFanalog57]

What I suggest is very simple:

The language of Mathematics...

A description is an abstract representation of a concrete physical instantiation.



Mathematics is a meta-language, highly abstract. A description contains the concrete physical instantiation in the abstract sense and the concrete object contains the description in the physical sense.


Here is the definition of "algorithm":

http://en.wikipedia.org/wiki/Algorithm

"Algorithm

From Wikipedia, the free encyclopedia.

Broadly-defined, an algorithm is an interpretable, finite set of instructions for dealing with contingencies and accomplishing some task which can be anything that has a recognizable end-state, end-point, or result for all inputs. (contrast with heuristic). Algorithms often have steps that repeat (iterate) or require decisions (logic and comparison) until the task is completed."

DNA is an algorithm, a finite set of instructions, which can construct a carbon based life form.

The life form physically contains the DNA and the DNA contains the life form in an "abstract" sense.

At a fundamental level of existence, it is postulated that "nature" could be constructed of tiny strings, and those strings, loops, or branes, could even be constructed of string "bits".

These bits could encode information, analogous to the universe's "DNA"? A set of instructions built into the fabric of space/time and mass/energy?

"If, then, it is true that the axiomatic basis of theoretical physics cannot be extracted from experience but must be freely invented, can we ever hope to find the right way? I answer without hesitation that there is, in my opinion, a right way, and that we are capable of finding it. I hold it true that pure thought can grasp reality, as the ancients dreamed." (Albert Einstein, 1954)

At the most fundamental length scales, the fundamental paticles, called "strings", could be constructed of even more basic units i.e. bits? analogous to a computer code?

1010100010...etc.

Universal algorithms?


Some interesting ideas on "string bits":

http://xxx.lanl.gov/PS_cache/hep-th/pdf/9607/9607183.pdf

http://xxx.lanl.gov/PS_cache/hep-th/pdf/9707/9707048.pdf

Introduction

In string-bit models, string is viewed as a polymer molecule, a bound system of point-like constituents which enjoy a Galilei invariant dynamics. This can be consistent with Poincar´e invariant string, because the Galilei invariance of string-bit dynamics is precisely that of the transverse space of light-cone quantization. If the string-bit description of string is correct, ordinary nonrelativistic many-body quantum mechanics is the appropriate framework for string dynamics. Of course, for superstring-bits, this quantum mechanics must be made supersymmetric.

According to string theory, the uncertainty in position is given by:

Dx < h/Dp + C*Dp

Which points towards a type of "discrete" spacetime?


A metric space has distance function r(x,y), characterized by involvement with the real numbers, R, such that the metric space and R are embedded simultaneously in the full structure of manifold M. A topological space consists of sets of points which are defined[in this case] to be the intersections of cotangent bundles.


If space is *quantized* yet also continuous, then it too, has the property called "wave-particle" duality. If space consists of indivisible units, then a measurement of space means that Fermat's last theorem holds, for it.

According to the Pythagorean theorem:

x^2 + y^2 = z^2

All possible integer solutions are then rerpresented as:

[a^2 - b^2]^2 + [2ab]^2 = [a^2 + b^2]^2

a^4 -2(ab)^2 + b^4 + 4(ab)^2 =

a^4 + 2(ab)^2 + b^4 = [a^2 + b^2]^2




all odd numbers can be represented as:

[a^2 - b^2] or Z^p - Y^p

if Y is an "even" natural n and Z is odd, same for a and b .

Fermat's last theorem, for integers a,b,Z,Y,p:

[a^2 - b^2]^p + Y^p = Z^p

[a^2 - b^2]^p = Z^p - Y^p

a^2 - b^2 = [Z^p - Y^p]^[1/p]

When Z^p - Y^p is a prime number, it cannot have an integer root.

a^2 - b^2 is not an integer, for [Z^p - Y^p]^[1/p] , for a,b,Z,Y,p, unless p = 2.


To every set A and every condition S(x) there corresponds a set B whose elements are exactly those elements x of A for which S(x) holds. This is the axiom that leads to Russell's paradox. For if we let the condition S(x) be: not(x element of x), then B = {x in A such that x is not in x}. Is B a member of B? If it is, then it isn't; and if it isn't, then it is. Therefore B cannot be in A, meaning that nothing contains everything.

This means that relativity holds in the "topological" sense and T-duality is correct.

Quantum entities are described as probability distributions, which are attributes of an underlying phase space, where the properties-attributes such as "spin" and "charge" are not the attributes of individual particles, but they are universally distributive entities, being the attributes of a "coherent wave function". It is this wave-distribution property that then "decoheres" into the ostensible "wave function collapse", as waves become localized particles that are "in phase" creating standing-spherical-wave resonances, which are condensations of space itself. The continual collapse-condensation of space into matter-energy is the continual "change", i.e. the property called "time". The spherical waves, or probability distributions are represented by the Schrodinger wave function, "psi".


The information density of the universal system must be increasing. The increase of information density is analogous to a pressure gradient.

[density 1]--->[density 2]--->[density 3]---> ... --->[density n]


[<-[->[<-[-><-]->]<-]->]

Intersecting wavefronts = increasing density of spacelike slices

As the wavefronts intersect, it becomes a mathematical computation:

2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ...2^n

 

[Lama]

For example you talk about a set as if it is something that can be made to go away; as if by changing the properties of the set, you somehow make the original abstraction disappear.

So, you do not understand the idea of the relative/absolute system.

In an excluded-middle reasoning an absolute element (set, point, ...) can have simultaneously a one and only one unique name (identity).

And I do not mean to some variable symbols like 'a', 'A', 'b' , 'B' ... and so on.

The identity of an absolute element is its literal name like: point named 'pi', point named 'e',
point named '1', point named '0', the set of 'all_sets_that_do_not_contain_themselves' ... and so on.

So nothing disappears here.

A set which its literal name is 'all_sets_that_do_not_contain_themselves' cannot contain itself (in an excluded middle reasoning) exactly as some absolute point that cannot have more than one literal name (in an excluded-middle reasoning).

Conclusion: Russel's Paradox cannot be defined in an excluded-middle reasoning.

Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.

I am waiting for your detailed remarks of my papers.

Thank you,

Lama

 

[Lama]

Dear Russell E. Rierson,

Can you simply say how you connect my idea of relative/absolute system to your interesting ideas?

 

[PFanalog57]

Dear Russell E. Rierson,

Can you simply say how you connect my idea of relative/absolute system to your interesting ideas?

In a word, "duality".

 

[Lama]

In a word, "duality".

What is "duality" for you?

 

[PFanalog57]

What is "duality" for you?

The laws of physics become the laws of geometry. Certain invariants hold, which are analogous to the "absolutes". There are also analogous non-absolutes, or relational perspectives, on the surface of the geometry.

 

[PFanalog57]

What is "duality" for you?

Here is an article on symmetry and duality. It appears to be very interesting:

http://1omega.port5.com/articles/Sym_dual/SYM_DUAL.HTM

 

[Lama]

Dear Russell E. Rierson,

It is about the time to build our community, which is based on the "Duality" principle.

I have learned during the last 20 years that without a community support, no fundamentals can be changed in science.

Do you have any ideas?

 

[PFanalog57]

Dear Russell E. Rierson,

It is about the time to build our community, which is based on the "Duality" principle.

I have learned during the last 20 years that without a community support, no fundamentals can be changed in science.

Do you have any ideas?

Duality, or possibly, a three valued logic as phoenix explains, solves russell's paradox

 

[Lama]

Russell E. Rierson,

"Duality" principle is not necessarily a 2-valuad logic, when you examine it by an Included-Middle Logic.

Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.

I am waiting for your detailed remarks of my papers.

Thank you,

Lama

 

[moshek]

The simple principle of Duality is an Archimedean point
That will make a shift in the whole mathematics.

Moshek

 

[PFanalog57]

Russell E. Rierson,

"Duality" principle is not necessarily a 2-valuad logic, when you examine it by an Included-Middle Logic.

Please read http://www.geocities.com/complementarytheory/No-Naive-Math.pdf (and all of its links) for better understanding.

Lama

The number of fractions from zero to one, is equal to the number of "natural numbers", from zero to infinity.

 

[Lama]

The number of fractions from zero to one, is equal to the number of "natural numbers", from zero to infinity.

Please give a detailed explanation.

 

[PFanalog57]

Please give a detailed explanation.

In your paper, you wrote that any segment, has the same magnitude as the entire real line.

I was trying to understand