Russell's Paradox and the Excluded-Middle reasoning

[kaiser soze]

Lama,

You overlook several important aspects of maths. Maths is not just a language, it is also a way of thinking. Thinking mathematically is an acquired skill - no one is born with it, it is developed and refined through doing maths (and in this case maths is what mathematicians are doing!). Another important aspect of maths is abstraction - it is in fact one of the cornerstones of maths. Abstract thinking is not trivial and not "natural" but is essential to understanding mathematic concepts, constructs and realms.

I have noticed that you avoid abstraction, and try to address things graphically - while this may help in some cases, it is inappropriate in others, and deminishes understanding and grasping of abstract notions. Most of the issues you are referring to are very abstract by definition - I think that when you master (traditional) mathematical thought and abstraction your views on your current ideas will change.

Kaiser.

 

[Lama]

Dear kaiser soze,

First, thank you for your kind post.

I agree with you about what can maybe called "the art of abstraction".

My gateway to this art is triggered by visual imagination, which means that the graphic forms are only tools exactly as linear lines of symbols are tools.

When you understand something deeply, it is always in the abstract level, where tools like pictures, symbols, sounds, and so on, do not exist anymore.

In short, each person can use his favorite trigger to develop his abstraction skills, and no method is better then the other.

The problem arises when some community of people does not distinguish between its tools and the internal cognitive abstract states of mind.

Maths is not just a language

A Langauge is many things in many levels of abstraction, which can be expressed and developed by dance, picture, writing, smell, music, movement, touch, sound, silence, colors, taste, light, sport ... and infinitely many other ways (and combinations of them).

And no community of people can put this natural power in boxes and tell us what is the right way of thinking and what are the right tools that we have to use, if we want to develop abstract cognitive skills.

Only one thing has to be developed: the art of internal/external dialog, and the rest is based on it.

 

[terrabyte]

Each consistent system includes within it the seeds of its paradigm shift, and in my opinion, this is the essence of the Langauge of Mathematics.

mostly this is only so because we choose to define it that way. For every created system we have elements that we define having values. this is all well and good until we come to the point where we start comparing and contrasting these values against each other. suddently we have to define not only what elements "Are" but also what they "Are not". in doing so we create these "seeds" for paradigm shift that you speak of. they're not necessarily inherant in the system, they're just a result from applying the system to define something.

indeed you'd have a hard time labelling "an honest person" if you first did not define "a lie". you could say "this person always tells the truth" but then the logical path that comes to your mind is "is there anything but 'the truth'?". there has to be or everyone would be an "honest person" and there would be no reason to define that term at all.

 

[Lama]

Hi terrabyte,

they're not necessarily inherant in the system,

I agree with you, because seeds do not strat to grew without an external trigger, like water for example.

So a paradigm shift needs both "seeds", and us as "water".

 

[Chronos]

Langauge is inefficient. It naturally tends to be self-contained and paradoxical. Math is better, but, not entirely. It too contains self contradictions. What Godel said, in essence, is that no theory can prove 'a priori' assumptions. In effect, Godel says his theory of incompleteness, is incomplete. Ironic. Logic and math are first cousins.

 

[Lama]

By Godel we can understand that any consistent(and therefore limited) system cannot be complete(and therefore without limits), and any complete system cannot be consistent.

In short, the concepts consistent and complete are preventing/defining each other.

For example, please see this picture: http://www.geocities.com/complementarytheory/comp.jpg

As you see the two black profiles and the white vase are clearly preventing/defining each other.

Please also see http://www.geocities.com/complementarytheory/CompLogic.pdf , which is a short paper of mine on included-middle reasoning.

This is, by the way, the deep reason why universal quantification cannot be related to a collection of infinitely many things.

It means that the word 'all' can be meaningful only if it is related to a collection of finitely many things.

 

[CrankFan]

"This is, by the way, the deep reason why universal quantification cannot be related to a collection of infinitely many things." -lama

Really? Why is that? specifically?

"It means that the word 'all' can be meaningful only if it is related to a collection of finitely many things." -lama

Ok, that's your opinion but you've not provided any real explanation (just some hand-waving about Goedel incompleteness) why we can't use universal quantification to say meaningful things about (infinite) sets, like for example the set of all positive integers:

Ax(not(x = 0) -> (not Ay (not (x = Sy))))

Any, non-zero natural number x is the successor of some natural number y.

These sentences, both the informal and almost-completely-formal version seem perfectly meaningful to me.

Previously you speculated that I was a professional mathematician. I'm not. In fact I'm not even remotely close to being a professional mathematician. I'm a guy who likes to learn mathematics when I have some free time. So I guess I'm an enthusiast, or something like that. Anyway, getting back to the topic... People aren't correcting you because they're bodyguards of math, whatever that is supposed to mean. People are correcting you because you make statements about mathematics which are either known to be false or are unsubstantiated.

 

[matt grime]

"By Godel we can understand that any consistent(and therefore limited) system cannot be complete(and therefore without limits), and any complete system cannot be consistent."

that is not an accurate or succinct interpretation of what Goedel's incompleteness theorem actually states, and you have its negation correspondingly incorrect. You may wish to look it up.

 

[matt grime]

"It means that the word 'all' can be meaningful only if it is related to a collection of finitely many things."

you may wish to explain to the uninitiated what your definition of "all" is mathematically, since it does not accord with its usage in predicate calculus.

 

[Lama]

Matt and Carnkfan,

Thank you for your posts.

Please this time read all of http://www.geocities.com/complementarytheory/ed.pdf

In this short paper I clearly show another possible and non-standard point of view, which explains why, for example, a universal quantification cannot be related to R collection.

 

[Chronos]

Mathematically, the term 'all' only has meaning in set theory.

 

[Chronos]

Lama, one thing I am curious about. I am a bit naive when it comes to logical operators and you seem quite comfortable with them. To me, the mathematical equivalent of the 'Liar Paradox' [this sentence is false], is 'x is not equal to x'. The math that follows completely breaks down in my mind. 1 multiplied by 'x' and 2 multipied by 'x' can easily be reduced to the expression '1 = 2'.

 

[Lama]

Matt,

that is not an accurate or succinct interpretation of what Goedel's incompleteness theorem actually states

Ok, let us put it this way:

By inconsistent system we can "prove" what ever we want with no limitations
but then our "proofs" are inconsistent.

A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proved by the current axioms of this system, and we need to add more axioms in order to prove these statements.

So any consistent system is limited by definition and any inconsistent system is not limited by definition.

Now, a complete system that suppose to prove everything is inconsistent by definition, therefore no consistent system (and therefore incomplete) can prove anything, and because of this we can find within any consistent system statements that are well-defined within the framework of the consistent system, but they cannot be proved within this framework, unless we add more axioms to the system, and so on, and so on.

So if by using the word 'complete' we mean 'inconsistent', then no 'complete' system is consistent.

Let us return to the universal quantification 'all'.

As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements ,even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements, is an inconsistent idea.

For example:

Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.

In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.

Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf

Form this point of view a universal quantification can be related only to a collection of finitely many elements.

An example: LIM X---> 0, X*[1/X] = 1

In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.

'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.

 

[Lama]

Hi Chronos

1 multiplied by 'x' and 2 multipied by 'x' can easily be reduced to the expression '1 = 2'.

Sorry, please explain this.

 

[Russell E. Rierson]

Sorry, please explain this.

A = B

A^2 = A*B

A^2 - B^2 = A*B - B^2

A^2 - B^2 = B*[A - B]

[A + B]*[A - B] = B*[A - B]

A + B = B

and

A = B

2*B = B

2 = 1

If one forgets that A - B = 0, because we cannot divide by 0. Unless you have an amazing discovery ...Lama?


A/0 is undefined.

0/0 is indeterminate.

2*X cannot equal 1*X

The interval from zero to one has an infinite number of fractions and it is a finite unit.

A = not-A ?


LIM X---> 0, X*[1/X] = 1

Some may argue with the epsilon-delta formalism of Carl Wierstrass but there is also Abraham Robinson's non-standard analysis, which puts infinitesimals on a rigorous footing.

 

[matt grime]

Your second description of godel's theorem is even less succinct and still inaccurate. Perhaps you'd like to see what i can recall of it?

If S is any (set) theory in which there is a model of the natural numbers (you've never managed to include that), then there exist a proposition, P, such that both P and not P are consistent with S.

Your definition of 'all' is garbage.

 

[Lama]

Your definition of 'all' is garbage.

I told it to ahrkron and also I tell you, my work is like a mirror at the first stage.

If you understand it you can see beyond your own reflection, but in your case you see nothing but your face in it.

If S is any (set) theory in which there is a model of the natural numbers (you've never managed to include that), then there exist a proposition, P, such that both P and not P are consistent with S.

Since you cannot go beyond the standard formal definitions, you cannot understand that I am talking about x_AND_not_x which are members of set S of 'any undefined element of some (set) theory'
where S is well defined in this (set) theory.

What we get in this case is a well-defined set S as a 'Trojan horse' which includes in it elements that cannot be defined in the framework of the examined theory.

So as you see i do not stay in the original formal definition of Godel's theorem, but I use the deep principle of it to develop another point of view, which is much more interesting (in my opinion) then Godel's theorem.

In short, Godel's theorem is often used to show the limitations of a system, but my point of view is to show that any limited system actually lead us to search beyond its domain, which is a positive approach of the same idea.

Since you do not aware to the power of the philosophical thinking as a profound tool for deep mathematical fundamental ideas, you cannot see but the reflection of your formal face in my philosophical mirror.

Matt, if you think that a good mathematician is a walking encyclopedia of formal mathematical knowledge, which is used to produce results within a particular brunch of the standard framework, and he never think beyond his own domain, then your way is not my way.

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

Your definition of 'all' is garbage.

Theorem: Matt's response is a garbage.


Proof:

By inconsistent system we can "prove" what ever we want with no limitations, but then our "proofs" are inconsistent.

A consistent system is based on a finite quantity of well-defined axioms, but then we can find in it statements which are well-defined by the consistent system but they cannot be proved by the current axioms of this system, and we need to add more axioms in order to prove these statements.

So any consistent system is limited by definition and any inconsistent system is not limited by definition.

Let us re-examine the universal quantification 'all'.

As I see it, when we use 'all' it means that everything is inside our domain and if our domain is infinitely many elements ,even if they are limited by some common property, the whole idea of "well-defined" domain of infinitely many elements, is an inconsistent idea.

For example:

Someone can say that [0,1] is an example of a well-defined domain, which is also a collection of infinitely many elements, but any examined transition from the internal collection of the infinitely many elements to 0 or 1, cannot be anything but a phase transition that terminates the state of infinitely many smeller states of the collection of the infinitely many elements, and we have in our hand a finite collection of different scales and 0 or 1.

In short, the well-defined ‘[0’ or ‘1]’ values and a collection of infinitely many elements that existing between them, has a XOR-like relations that prevents from us to keep the property of the internal collection as a collection of infinitely many elements, in an excluded-middle reasoning.

Again, it is clearly shown in: http://www.geocities.com/complementarytheory/ed.pdf

Form this point of view a universal quantification can be related only to a collection of finitely many elements.

An example: LIM X---> 0, X*[1/X] = 1

In that case we have to distinguish between the word 'any' which is not equivalent here to the word 'all'.

'any' is an inductive point of view on a collection of infinitely many elements, that does not try to capture everything by forcing a deductive 'all' point of view on a collection of infinitely many X values that cannot reach 0.

If my definition of 'all' is garbage, then 0*(1/0)=1

Since 0*(1/0) not= 1, Matt's response is a garbage. QED.

 

[Chronos]

hi lama.

i think i understand your point. but, i don't think you can logically prove mathematics anymore than you can mathematically prove logic. i prefer math to explain science and logic to explain philosophy. The essence of Godel was to say that all systems are self-referential [they necessarily include 'a priori' assumptions] and assumptions cannot be proven or disproven within any such system. Relativity, in this sense, may be the most profound statement about reality we are capable of grasping.

 

[Russell E. Rierson]

Back to the "Fallacy of Composition" for the educational benefit of Hurkyl and co.

The properties of the whole distribute over the individual parts OF the ...whole

[WHOLE]--->[PARTS]

If W then P

W

therefore P



Properties of the parts don't necessarily become properties of the whole.

If W then P

P

therefore W

is false logic, also known as the fallacy of composition via modus ponens error.

QED

 

[matt grime]

"So as you see i do not stay in the original formal definition of Godel's theorem, but I use the deep principle of it to develop another point of view, which is much more interesting (in my opinion) then Godel's theorem."

but have you proved it is true?



in any case you are demonstrating explicitly again that you are using a different definition and usage from the rest of the world in your use of "all" amongst other things, so don't correct us or tell us mathematics is wrong when an "all" is used (ie your problems with Cantor's Theorem) when you are refusing to use the words correctly, as they were intended.

tell you what: 2 > x for all x in (0,1), wow, i used the word all for a collection of infinitely many things...


you're also scoring high on the crackpot index again with your combined insistence that what you have is amazing and so much better than crappy mathematics as practised by religious dogmatic fanatic and admitting that your theory is allowed not to be very good because it's new and you aren't a raligiously fanatical adherent of limited mathematics.

 

[Lama]

in any case you are demonstrating explicitly again that you are using a different definition and usage from the rest of the world in your use of "all" amongst other things

So, do you support the idea that Mathematics is no more then a rigorous agreement between people?

"Math, in my opinion, is first of all a rigorous agreement that based on language.

Symmetry is maybe the best tool that can be used to measure simplicity, where simplicity
is the best platform for stable agreement.

Any agreement must be aware to the fact that no model of simplicity is simplicity itself.

This awareness to the difference between x-model and x-itself is the first condition for
any stable agreement, because it gives it the ability to be changed."

http://www.geocities.com/complementarytheory/CATpage.html

but have you proved it is true?

It is trivially understood that in any consistent (set) theoretical system, there is at least one well-defined set that includes any of the undefined elements of this consistent (set) theoretical system.

Actually it is an axiom, which may be called "The axiom of the paradigm-shift".

crackpot

"crackpot" is a reflexive response of a 'limited system' when it is forced to look beyond its limited domain.

In that case I am proud to be called a "crackpot".

 

[matt grime]

So it's an axiom in your system?

Do you recall that you once claimed that you'd successfully answered the Collatz conjecture because you'd "proven" that it was equivalent to the axiom of infinity, and was thus not provable? Have you changed you mind about axioms being true then?

 

[Lama]

Do you recall that you once claimed that you'd successfully answered the Collatz conjecture because you'd "proven" that it was equivalent to the axiom of infinity, and was thus not provable? Have you changed you mind about axioms being true then?

No Axiom can be proven within its own framework, because it is an arbitrary true, that is based on our intuitions, therefore a statement which is equivalent to an axiom, cannot be proven within its own framework, as I show in http://www.geocities.com/complementarytheory/3n1proof.pdf

Only statements that are not equivalent to an axiom can be proven within their framework.

In short, only statements that are based on axiom(s) but do not equivalent to anyone of them, can be proven within their framework.

For example:

The Natural numbers are axiom's products and not the axioms themselves, so we can prove things which are based on N members' properties only if we do not need to go to the level of the axioms themselves.


We can use here an analogy, which is based on chemistry:

In order to prove something in the level of the muscles tissue, we do not need quarks level.

 

[Chronos]

So it's an axiom in your system?

Even a domesticated bunny with pink eyes could see that trap... hilarious matt. But forgive lama for being logically inconsistent, he means well.

 

[Lama]

But forgive lama for being logically inconsistent

It depends on the interpretation of "what is a proof?"

My interpretation excludes the axioms of some given consistent system, because they are arbitrary true within the framework of their own system.

Their 'true' validity can be examined and proven only within some Meta framework, where they cannot be considered as axioms anymore.

In that sense, any consistent system has two sides:

a) The internal side is the consistent and therefore limited system, which is based on its own arbitrary true (self evidenced) axioms.

b) The external side which is not limited but then not necessarily consistent with this system.

c) In that sense, no axiom can be proven within its own framework, and on the same time, this axiom must be proven within some Meta-framework.

So, as we can see, an axiom is an element that always exists between internal and external frameworks.

 

[kaiser soze]

Occam's Razor:

"one should not increase, beyond what is necessary, the number of entities required to explain anything"

http://pespmc1.vub.ac.be/OCCAMRAZ.html

 

[Lama]

Hi kaiser soze,

So can we ignore the butterfly wings when we construct the model of a hurricane? (http://mathworld.wolfram.com/ButterflyEffect.html)

 

[arildno]

Hi kaiser soze,

So can we ignore the butterfly wings when we construct the model of a hurricane? (http://mathworld.wolfram.com/ButterflyEffect.html)

Certainly, because that effect would be included in the initial conditions, not the system of differential equations which is THE MODEL

 

[Lama]

Certainly, because that effect would be included in the initial conditions, not the system of differential equations which is THE MODEL

Differential equations are nothing but a tool here.

The Model in this case cannot be less then a combination of initial conditions AND the system of differential equations, otherwise our nonlinear phenomena cannot be understood.

In short, we can never be 100% sure what is necessary and what can be omitted, when we deal with nonlinear complexity, for example.

 

[arildno]

Blather Again, Lama!

 

[kaiser soze]

Lama,

You missed the point. A common interpretation for Occam's Razor is: In science, the simplest theory that fits the facts of a problem is the one that should be selected.

Occam's Razor is a rule of the thumb for selecting theories to explain phenomena.

Kaiser.

 

[Lama]

Blather Again, Lama!

This is a very poor response, I think that you can do better than that.

 

[Lama]

You missed the point. A common interpretation for Occam's Razor is: In science, the simplest theory that fits the facts of a problem is the one that should be selected.

Occam's Razor is a rule of the thumb for selecting theories to explain phenomena.

Do you really think that a non-trivial problem or phenomena are some clear objects that are waiting for us to explain them?

When we cut out things we change our initial conditions, and if our explored system is sensitive to initial conditions then your "simplest" theory misses the point.

In short, the quality of a theory is much more important than the quantity of its components, when we deal with non-trivial systems.

Any way, you did not explain in details, why do I have to use Occams' Razor on my arguments, so please do that, thank you.

 

[Russell E. Rierson]

Q1:

If the barber shaves those, and only those men who do not shave themselves, then does the barber shave himself?


Q2:

If an assertion A, is true and its negation, ~A is also true, it becomes a form of the "liars paradox".

Suppose a person called X, stands up and says, "This assertion is false."

Let S denote the statement uttered; let p be the proposition the person makes by uttering S. Then the utterance of the phrase "This assertion" refers to the claim p. It follows that, in uttering the words "This assertion is false," X is making the claim "p is false". Thus , p and "p is false" are one and the same:

p = [p is false]

By making the claim, X is implicitly referring to the context in which the claim is stated. Let c symbolically represent the context for which the sentence refers.

X's uttering of the words "This assertion" refers to the context, c, which entails p.

[c entails p]


That is to say, p must be the same as [c entails p] due to the fact that X is referring to both p and [c entails p] via the utterance of the phrase "This assertion."

If X's assertion is true then [c entails p] is true

p = [p is false]

[c entails p is false] is true


This creates a contradiction, ergo X's claim that [p is false] is false.

[c entails p is false] is false


This appears to be the same contradictory state of affairs as in the previous cases of the Liars Paradox.

Conclusion?:

c cannot be the appropriate context.

Consequently, the paradox becomes a theorem/demonstration.
When X utters the Liar sentence, X is uttering a falsehood, and the context in which the claim of falsehood is made cannot be the same as the context in which the Liar sentence S, was uttered...



Thus context c, becomes a subjective/qualia operator.

 

[kaiser soze]

Lama,

Tell me what phenomenom you are trying to explain using your theory(ies) and I will explain how Occam's razor rule of the thumb should be applied.

Kaiser.