Russell's Paradox and the Excluded-Middle reasoning

[matt grime]

If Russell's paradox cannot be defined, how are you able to reason about it and, erm, define it? Your position is itself paradoxical.

You are not adding anything new to the arguments of this paradox, just inventing new terms and seemingly misapplying existing ones, however the poor presentation means it is hard to decide what you are trying to say some times.

Why do you insist on saying 'excluded middle logic' is limited as if this is somehow disturbing news? Every mathematical system is limited by and to its axioms. There are systems where excluded middle isn't used (see Martin Hyland's examples of topoi). Note, we are not giving credence to any of your attempts ot defining logical systems since you have not produced anything that we can consider to be consistent, or readable.

 

[Lama]

Hi Matt,

Long time no see.

If Russell's paradox cannot be defined, how are you able to reason about it

In my paper I show that what is considered as a paradox in naive set-theory, is no more then a false statement.

Please show me where can I find similer interpretation to this "paradox".

But the main point of my paper is about Godel's incompleteness theorem.

You can find it at post #120.

 

[Alkatran]

I don't know if this was covered yet, but you guys seemed to be confused about what "not"ing a number does in programming.

not X = -X -1

The value of true is -1 and false is 0:
not false = true = -0-1 = -1
not true = false = --1 -1 = 0

The operation that would switch all the bits of a number is XOR.

X XOR True = (X with all 0s switched to 1s and 1s switch to 0s)
X XOR False = X

 

[Lama]

For example:

If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all



If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all



Some conclusions:

a) The self identity of the Barber of Seville is based on the false statement all = not_all.

b) Self identity, which is based on a false statement, is no more then a false statement.

c) No false statement is a paradox in excluded-middle reasoning.

d) Therefore Russell's paradox is not defined in excluded-middle reasoning.



In general we can conclude the above about any self-referenced definition, which includes in it all condition.

If an all condition is omitted form a self-referenced definition, then the possibility of self identity as a false statement, is avoided in an excluded-middle reasoning.

 

[Lama]

If the statement contradiction = not-contradiction is a contradiction is false, the statement contradiction = not-contradiction is not a contradiction is true?

In logic we can say that our true result is a false statemant.

This is the reason why some false reuslt can be found in our logical system.

Only the true stands behind any result.

Also in excluded-middle reasoning any examined concept cannot have more than one unique identity,
so a = not_a cannot be but a false statemant (which is the true reuslt) in this case.

 

[CrankFan]

"A new point of view on Russell's first paradox..." -Lama

Maybe you missed the part in the text you linked which said that:

"In Cantor's system, M is a well-defined set."

And that's pretty much all you need to know, that given the same assumptions that Cantor was making (in particular the abstraction principle) the Russell Set is perfectly valid. That you might want to add new rules, which further instruct what is and what isn't a set in naive set theory is immaterial, because once you've added those additional rules you would no longer be talking about naive set theory.

http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Naive_set_theory#Specifying_sets

When you say things like:

"In excluded-middle reasoning, each element must have a one and only one unique identity. An element without a unique identity cannot be a participator in the excluded-middle "game". -Lama

Not only is mumbo-jumbo like this nearly incomprehensible, not only is it not anything close to a proof that the Russell set isn't well defined in terms of naive set theory, but it's also completely irrelevant because phrases like "a participator in the excluded middle game" and "unique self identity" aren't known well enough for you to use them before defining precisely what they're supposed to mean in terms that everyone can understand.

I can guess your intended meaning, but if that's what you expect us to do, then you shouldn't believe that what you've provided is rigorous or a proof of anything. It's more like a guessing game.

Naive set theory is not foundational in any other sense than historically. Modern set theories don't have classical antinomies, like the Russell set. However, most introductory set theory texts will discuss them for the sake of describing key historical developments and motivating discussion about how we can properly axiomatize a set theory so that it retains the richness of naive set theory and also avoids classical problems.

Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.

http://en.wikipedia.org/wiki/Russell's_paradox#Set-theoretic_responses_to_the_Russell_Paradox

[edited for typos]

 

[Lama]

No proposition can make a statement about itself...

If we look at this propositoin, we can say that within an excluded-middle reasoning, if a self reference of a proposition changes the propositon, then and only then it cannot be referred to itsef, because in an excluded-middle reasoning, each element has exactly one and only one uniqe identity.

By tautology x = x means: x is itself, otherwise we cannot talk about x.

Now we can ask if a teotology is also recursive, for example: x = x = x = ...

If we do not get any new information by this recursion, then x = x is enough, which is like a one_step_recursion.

So, Russel's paradox is like if by teotology we examine if x is not_x or x = not_x , which is no more then a false statement from an exluded-middle point of view.

In an excluded-middle reasoning no false statement is a paradox.

Again:

The element x_AND_not_x cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.

Therefore Russell's Antinomy is nothing but a false statemant and not a paradox in excluded-middle framework.

note the correct use of iff, sometimes denoted <=>, and not =, since 'equals' is not an operator in boolean logic

'=' is used here for the tautology of a = a.

a = not_a is no more than a false statement in excluded-middle reasoning.

A and not_A

As usual, you miss the point.

A and not_A cannot be defined in excluded-middle reasoning, because any examined concept cannot have more than a one unique identity.

Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.

Our true result in this case is no more then a false statement, and all the big affords that professional mathematicians like you put in their theories to avoid this "paradox", are no more than a full gas in neutral.

 

[arildno]

Lama:
Are you willing to let yourself be tested by members on this forum in order to establish whether or not you have understood anything in standard maths?

So far, you have given no indication that you possesses any such understanding.
Your own ideas would be looked at more closely if it could be established beyond any doubt that you understand what standard math is.

 

[Lama]

Hi arildno:

If you do not want to understand that x_AND_not_x is beyond (cannot be well-defined, and it means that the "paradox" cannot be defined) the domain of x_XOR_not_x (which is the basis of an excluded-middle reasoning), then your basic attitude, in my opinion, is to be no more then a full time job bodyguard of The language of Math, and (as I see it) you do not give yourself any chance to see fundamental things from a different point of view.

Maybe you missed the part in the text you linked which said that:

"In Cantor's system, M is a well-defined set."

So what if Cantor thought that M is well-defined in his system.

The Langauge of Mathematics is not based on gurus, but on fundamental concepts that are never beyond re-examination.

I look at this "paradox" from an included-middle reasoning, and the affect is similar as if I look on 2-D system from n>2-D system.

Someone who looks on some system from a first-order higher level of reasoning system (where 2-D reasoning system is only a proper sub-system of it, and I clearly show it in my papers) can easily show new interpretations to fundamental concepts of the Langauge of Mathematics.

If you stick to the standard 2-D reasoning, you will never understand my work.

I made my move to new points of view that re-examine the most fundamental concepts of this beautiful language.

Take for example persons like Matt Grime, which in my opinion make here a very good job as the bodyguard of Math.

It took me some time (almost 2 years) to understand that I am talking to a full time job bodyguard, so now I take what I take and I do not care anymore that full time job bodyguards do not want to or can’t understand my work.

You, Matt Grime, CrankFan, Master Coda, Hurkyl, kaiser soze, Ahrkron , and more full time job bodyguards of Math, did not show even a little step to see things from new points of view on the most fundamental concepts of the language of Math.

Form my side, I clearly an simply show why Standard-Math approach does not hold in these most fundamental concept.

I learn my mistakes, and I am trying to improve the basis of the reasoning of my work, but because all you do is to be full time job bodyguards, you do not distinguish that.

For example, please look at the attitude of Ahrkron to my work:
https://www.physicsforums.com/showpost.php?p=243538&postcount=37

In my opinion this is nothing but a poor, non-detailed and limited approach to someone's ideas.

More examples:

Please look at the attitude of Matt Grime to my work:

I wrote to him:

https://www.physicsforums.com/showpost.php?p=242567&postcount=28

As an answer I got:

https://www.physicsforums.com/showpost.php?p=242639&postcount=29

Another example of Matt Grime's attitude:

https://www.physicsforums.com/showpost.php?p=250748&postcount=41

And my non-friendly reply to him:

https://www.physicsforums.com/showpost.php?p=250819&postcount=42

that he chose to ignore.

Also be aware to the name 'CrankFan' that can say a lot about his limited attitude to the possibility if new interpretations of fundamental concepts of the Langauge of Math.

All I asked in 'Theory development forum' is a little more flexible approach that can examine "well-defined" terms from (time to time) a new point of view.

What I have found is a community of hard minds that do not want anyone to change the fundamentals of their religion.

In short, I do not accept Cantor's M definition, and I clearly and rigorously show why I do not accept it.

No full time job bodyguard can understand it.

 

[geistkiesel]

"A new point of view on Russell's first paradox..." -Lama

Maybe you missed the part in the text you linked which said that:

"In Cantor's system, M is a well-defined set."

And that's pretty much all you need to know, that given the same assumptions that Cantor was making (in particular the abstraction principle) the Russell Set is perfectly valid. That you might want to add new rules, which further instruct what is and what isn't a set in naive set theory is immaterial, because once you've added those additional rules you would no longer be talking about naive set theory.

http://en.wikipedia.org/wiki/Naive_set_theory
http://en.wikipedia.org/wiki/Naive_set_theory#Specifying_sets

When you say things like:

"In excluded-middle reasoning, each element must have a one and only one unique identity. An element without a unique identity cannot be a participator in the excluded-middle "game". -Lama

Not only is mumbo-jumbo like this nearly incomprehensible, not only is it not anything close to a proof that the Russell set isn't well defined in terms of naive set theory, but it's also completely irrelevant because phrases like "a participator in the excluded middle game" and "unique self identity" aren't known well enough for you to use them before defining precisely what they're supposed to mean in terms that everyone can understand.

I can guess your intended meaning, but if that's what you expect us to do, then you shouldn't believe that what you've provided is rigorous or a proof of anything. It's more like a guessing game.

Naive set theory is not foundational in any other sense than historically. Modern set theories don't have classical antinomies, like the Russell set. However, most introductory set theory texts will discuss them for the sake of describing key historical developments and motivating discussion about how we can properly axiomatize a set theory so that it retains the richness of naive set theory and also avoids classical problems.

Discussing this kind of stuff as if you're working on or attempting to resolve a 100 year old problem is a joke.

http://en.wikipedia.org/wiki/Russell's_paradox#Set-theoretic_responses_to_the_Russell_Paradox

[edited for typos]

i avoied this thread for a lot of reasons, but when i styarted slowly to difest the mater what Lama has to say, that was once gobble-gobble as was everybody elses's post, I started to get the drift. It isn't listening to th Barber if seville, but it also isn't what you described the matter to be, at least not to me and I am a slow learner.

quote Lama:
If the Barber of Seville does not shave himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = less_than_all or in other words: all = not_all

If the Barber of Seville shaves himself, then he does not fit to his own self identity, which is:

To shave all of the people in Seville, only if they do not shave themselves, and in this case we can conclude that all = more_than_all or in other words: all = not_all

This isn't difficult to understand, and it isn't gobbledy gobble. Nullify a unique aspect of something and it isn't unique any more. Adios paradox, right? I had only heard of the paradox peripherally, and could barely grasp the ssence of understanding the "paradox" and now I understand the negated paradox. A learned piece of history: Russel wasn't "everything in logic and reason" was he? And another really nice thing about it is that 2 + 2 = 4, most of the time.

 

[Lama]

This isn't difficult to understand, and it isn't gobbledy gobble

Thank you dear geistkiesel for allowed yourself to be opened to another point of view of this "paradox".

I can understand why professional mathematicians do not want to look on fundamental mathematical concepts from a different point of view.

In My opinion, the most basic important thing of any formal or informal language, is the ability of an open dialog on any concept of any formal or informal language.

If this basic and important thing is omitted from some language, then (in my opinion) this language is a dieing language.

 

[matt grime]

No, Doron, we don't want to look at basic objects from different points of view, which is why there is only one set theory... oh no, look at that there isn't just one set theory, there are many, damn, wrong again. If you don't believe me try looking up some stuff in journals of computational mathematics. The last one I looked through had about 5 different set theories mentioned in the first 3 papers. Once more your lack of knowledge and unwilllingness to open yourself to others' ideas has led you to believe in something false. There are also many difference logic types too, not that you seem to accept this.

Would you care to offer any mathematical arguments or examples to show where your opinion is true? And where the refusal to accept it has led to problems?

 

[Lama]

contiune...

On Godel's incompleteness theorem:

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

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

The identity of an element is its literal name like: a number named 'pi', a number named 'e', a number named '1', a number named '0', a set named 'not_all_sets_that_do_not_contain_themselves' ... and so on.

Strictly speaking, a well-defined element in an excluded-middle logical reasoning system, cannot be but an element that has a one and only one unique literal name.

The set that includes 'all of the elements that do not have well-defined and unique identity' has a unique self identity.

Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well-defined set are false and true, neither true nor false, contingently true or false etc.)

This is a positive approach of Godel's incompleteness theorem, which says:

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.

In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:

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.

 

[Lama]

Hi Matt,

Once more your lack of knowledge and unwilllingness to open yourself to others' ideas has led you to believe in something false.

Do you understand my ideas? Prove it!

Would you care to offer any mathematical arguments or examples to show where your opinion is true? And where the refusal to accept it has led to problems?

The problem is never in some system and always in the limited minds of the person how refuses to look at fundamental things from different points of view.

Since you never showed any motivation to understand my work about fundamental concepts of the language of Mathematics, you cannot understand, for example this paper: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf
which answers to your last question.

The last one I looked through had about 5 different set theories mentioned in the first 3 papers

Another typical response of you.

Instead of compare between the contents of these theories, to my theory, you count how many of them can be found, which is a very deep approach.

Also read: https://www.physicsforums.com/showpost.php?p=253586&postcount=129

 

[matt grime]

Ahem, the reason I, and no one else it appears, can understand your "theories" is because they do not make any sense for many reasons. How can anyone demonstrate understanding of that which cannot be understood. You don't even appear to be able to understand the fact that your allegations about mathematics are without foundation, so why should we listen to the arguments based upon faluty premises?

 

[Lama]

Ahem, the reason I, and no one else it appears, can understand your "theories" is because they do not make any sense for many reasons. How can anyone demonstrate understanding of that which cannot be understood. You don't even appear to be able to understand the fact that your allegations about mathematics are without foundation, so why should we listen to the arguments based upon faluty premises?

This superficial and non-detailed monolog was infront Matt Grime's mirror.

And because he thinks that to talk to a mirror is a dialog, he cannot understand what is a dialog.

An example of a dialog between Muddler an Lama can be shown here:
https://www.physicsforums.com/showthread.php?t=33687

I think that you have something smurt to say about https://www.physicsforums.com/showpost.php?p=254672&postcount=133

Do you now why i am so nice to you?

because in this year you are going to get your phd in Mathematics, and then you are going to be a teacher for a new generation of mathematicians that most of them will not survive your limited doctrine and become your duplicates.

If there are too many mathematician that have a limited doctrine like you, then (in my opinion) there is a real danger that in couple of generations this beautiful language will die because of luck of creative and opened minds, which understand that no language (formal or informal) can survive without an opened dialog, that allowed itself to re-examine any of its fundamental concepts.

No full time job bodyguard of the Langauge of Mathamatics, can understand it, and in my opinion, you are one of the best.

 

[Chronos]

I'm a little fuzzy about the logic here. Sounds a lot like Eubulides 'Liars Paradox', which has been around for about 2400 years. For amusement see

http://www.utm.edu/research/iep/p/par-liar.htm

 

[Lama]

Hi Chronos,

Yes Russell's first paradox is based on the principle of the liar's paradox.

For example:


X = Liar

Y = Honest


X cannot have any property of Y, and Y cannot have any property of X, In an excluded-middle reasoning or in other words:

A true Liar cannot say directly (by using the word "I") or indirectly (by using the word "ALL") that he is a Liar, because he can’t say the truth.

A true Honest cannot say directly (by using the word "I") or indirectly (by using the word "ALL") that he is a Liar, because he can’t lie.


X has a one and only one unique self-identity.

Y has a one and only one unique self-identity.

The logical condition between X and Y in an excluded-middle reasoning is : X_XOR_Y.

The paradox is based on X_AND_Y but because any well-defined element in an excluded-middle reasoning cannot have more than one unique self-identity, then no element, which its identity is based on X_AND_Y is well-defined in the domain of excluded-middle reasoning.

It means that the Liar's paradox (and also Russell's first paradox) is not well-defined concept within excluded-middle reasoning.

The set that includes 'all of the elements that do not have well-defined and unique identity' has a unique self identity.

Therefore it is a well-defined set in the framework of excluded-middle logical reasoning, but no one of its members can be considered as a well-defined element within the framework of excluded-middle reasoning (the best that can be done is to say that the members of this well-defined set are false and true, neither true nor false, contingently true or false etc.)

This is a positive approach of Godel's incompleteness theorem, which says:

Within any consistent system, there is at least one well-defined set, which its content cannot be well-defined within the framework of the current logical system.

In short, in any consistent system we can find pointers, which lead us beyond the domain of the current system, or in another words:

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.

 

[matt grime]

But, Doron, you don't want to learn about other views of mathematics, as you've made abundantly clear. Your idea of a dialogue is expounding your views, not in learning about mathematics. As you are the one who wants the dialogue, you'd think you'd at least be prepared to listen to others?

 

[Lama]

Matt,

I am listening to you and I write my non-standard ideas, which sometimes refer directly to what you say, sometimes refer indirectly, and sometimes take me to a new place where I can find ideas which are beyond the scope of the original starting point.

Then I write my non-standard Ideas in non-standard methods, because the standard methods most of the time cannot express correctly my non-standard ideas.

You have in your mind the standard concept of how the Langauge of Mathematics should be expressed and developed, but I have both new fundamental ways of how this Langauge can be developed and new fundamental points of view on the most fundamental concepts of this language.

In this most fundamental level when the subject is a paradigm shift of some concept, and someone inviting you to a dialog in these conditions, you have to take your first steps when you are armless.

No full time job bodyguard of Math that using his standard point of view as a weapon, and come to save the holy Math from crackpots like me, will be able to understand what I have to say.

 

[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.