Dear Frank,

to develope a new theory is always interesting, but to develop a goal-oriented theory make the real difference... I think, for you to read the TOGA meta-theory ( http://erg4146.casaccia.enea.it/index.html ) can be useful.

. Good luck!

Adam

Dear Adam, thanks for the input. I will go ahead a read about TOGA. Again, thanks!

I really can't contribute much, but I'd like to see the results of this research.

Dear Fabricio, I have added more in the question section to help you and others to gain a better idea of what Prelogic does or signifies for Set Theory, math, language and ultimately intelligence.

Please note, Prelogic is not an invention, but rather an understanding that has existed since the beginning. Personally, what I like most, it is the simplicity it offers and how it fuses separated fields into the language of Logic (or its beginning).

Hi Frank,

This is an intriguing topic that is close to my heart. In Cognitive Science there's been an ongoing debate for some 60 years as to whther Godel's Incompletness Theorems pose inherent limitations to Machine Intelligence. And as the Incompleteness Results would apply to both Naiv e Set-Theory as well as ZF Set Theory -- these results definitely relate to your topic.

You might want to Google my name for a couple of decent publications on this, and my references in these papers should prove helpful to you. I would be pleased to be of further assistance for your project if you like.

With Best Regards,

Albert E. Lyngzidetson

Well, assuming that all computational processes can be reduced to Turing machines, then it is certain that Gödel's incompleteness limits what can be computed, since it limits what a Turing machine can achieve. So far, the Church Turing thesis has been holding, but I don't know whether this will remain so.

Dear Albert,

YES, YES, YES and YES. The Incompleteness theorem is what I am working now and I really need help with it....

Let me share something with you. This SET forming system can also reason by creating what I call hyper Sets (basically, A intersect B), meaning that from a limited number of statements (axioms), the system can not only find the solutions (conclusions), but also create "new axioms" which, as long as they do not contradict the older axioms, everything works ok and furthers all conclusions. And hypothetically, this can go on and on as long as no contradictions are created. For example, the following statements say:

1) Mary is a girl

2) Lisa is a girl

3) Mary is a human

4) a human is a hominid

The only conclusion the system makes is: Mary is a hominid.

However, when the system imagines, it creates a new axiom (a girl is a human). The result is: a girl is a human (the added by imagination), Lisa is a human and Lisa is a hominid. These new conclusions do not contradict the initial statements but also serves to prove in the future if the initial axioms were correct. But what is really interesting, is that the axioms themselves are not what is important (whether they are true or not) but what is important is if the mechanisms that created the SETS (conclusions) were correctly formed (not compromised).

I do not know what this really means yet.... I am still struggling with the Incompleteness theorem trying to understand if some assumptions were made, meaning again, is not the axioms that matter, but rather the mechanisms that create conclusions what really has to be consistent.

Dear Fabricio, I am not sure this SET binding method of representing and grouping information actually applies to the Turing machine as far as I know the Turing machines means. This is a bit controversial I admit, but in this system, there are no algorithms per say, except for the mechanisms (Prelogic) for creating the sets among the entities representing concepts. I must confess, I am not very fond of computers.

Dear Albert, there is a lot of information about you...do you have a link to your papers please? -thanks

Hello Frank, you might want to check out some basic foundational work in set theory, and logic, perhaps even formal languages. Getting some of the basic naming conventions down will help you in your research. For example, the difference between a class and a set was formalized in NBG set theory, and properties of sets, are the class of such set in that system, so for example, the set of cars that has a class of red, which is the property red applied to cars, and it seem to me this is very close to what you mention. Russell's paradox has been dealt with by both Zermelo and Neumann through two different axioms, and is not really a pressing issue in foundations. It is, on the other hand, a curious introduction to naive set theory. Also, one or more axioms which produce other deductive truths from those axioms, is not usually referred to as another axiom, instead, it is often called a theorem, lemma or a proposition, depending on the circumstance.

By "imagination", it seems to me you are trying to formalize the conjecture, which is noble and necessary research. For this, I suggest you look into the research on learning algorithms, which, through the analysis of data, conjectures about the properties of such data can be made, and the algorithm can make a decision based on this conjecture. It is how we get self-driving cars.

In terms of Godel's incompleteness theorems, The simplest way to communicate what he did is that he proved that there are true statements that can be formed in a formal language which can not be proven as true in that language, he also proved that all complete systems must contain a contradiction or it can't be complete, and thus, any formal system that is consistent, must also be incomplete.

At a glance, based on the little I have seen here, it seems to me that you have created a string generator where the strings themselves generate based on the consistency of previous strings. What you have here, is not necessarily truth in the language of English necessarily, but you have included, on a more structural level, truth in how strings may be formed, which just happen to coincide with statements in English that may or may not be true to an observer. Such a system, if this is the case, is still under the rules of Godel's incompleteness, but you need to be clear about the system from which you are making your research claims. For example, in your program that you seemed to have worked out, "Mary is a girl" is a theorem in the program, despite the fact whether Mary is actually a girl or not to an English speaker/observer. In this sense, you should want to separate the formalism of your program which is not English that generates strings from a subset of English, which as a program, has a consistency and is subject to incompleteness, from all statements which can be made in English, which is a language that is complete (in the sense it contains at least all of predicate logic) and inconsistent.

Hi Frank,

I'm a bit confused when you say that your system creates 'new axioms' typically these are called 'theorems', as Axioms are primitive to a system and not reducible. Also, the term 'Hypet-Set' is usually used to refer to non-well founded sets as in the work of Peter Aczel. A good informal overview of Godel's results can be found at: http://www.scientificamerican.com/article.cfm?id=what-is-godels-theorem.

This is pretty technical stuff, so it might be pretty slow going at first.

You can find my papers Online from your Google-search results. I think the British Journal for the Philosophy of Science may also have them Online, but I haven't confirmed this.

Dear J Inman,

The so called Prelogic, is a mechanism (a physical event) that results when two or more entities (things, concept representing things, pixels, etc.) fused to create a group or Set. Accordingly, this mechanism, precedes Set Theory, and hypothetically, it equally serves as the foundation for the intelligent process (thinking). Therefore please excuse me from avoiding today's nomenclatures because I have to be careful of committing the system to any particular definition of a system that actually comes after Prelogic. As a matter of fact, this system diverges seriously from NBG because in this system classes are the equivalent to complex concepts and complex concepts are the result of fusing simpler concepts together (by which ever means is possible to fuse objects together). I know, this is controversial, but Prelogic contradicts NBG. Intelligence has to exist before Set theory can be understood, and before inteliigence emerges, this so called Pelogic must occur.

I absolutely agree, foundations are foundations and regardless of how we define them, nature continues doing what it does with or without us talking about it. However, the fact the Prelogic removes the Russell Paradox as a paradox is significant (again, only discussions of foundations) for Naïve Set Theory, making it not Naïve any more.

Personally -(this is only an opinion)- one issue I always dislike about Z and N Set theory versions is that they, almost politically, avoid or go around problems by saying.... this is not allow or that is not allowed. This is similar to saying: Jumping is safe with the exception rule of: not jump from cliff; therefore, jumping is safe. The reality is, jumping is "sometimes" safe; meaning that is not safe as a total rule. Including the exception as a rule is a bit a of a stretch.

I also find very interesting what you said about the Incompleteness theorem. It needs a contradiction to be consistent; therefore is not consistent. But part of my struggle is that the Incompleteness theorem mentions a true statement... according to what our prototype shows, true is actually irrelevant, that is my main pickle so far.

Essentially, until you formalize "Prelogic," I don't think anyone will be able to understand what you are saying by words like "fuse" or "foundation," which could have a various number of connotations attached to it which affects it's meaning. In fact, when one uses the word "foundation" in the context of logic and mathematics, we are usually referring to the axiom schemas that we choose to work with and any theorems that arise from those axioms which have universal consequences on the system or model. So either you are speaking in the language game of mathematics and logic, and abide by those rules, or you are speaking outside of it. So, it seems, you are speaking outside of logic by your use of "prelogic," In which case, by definition of naive, in the sense of the world of logic, which has a very specific connotation which is not negative as you perceive it, "prelogic" is still a naive theory despite the fact that somehow, the theory does not allow for Russell's paradox. Naive only means that the system is not formalized. A system is either formal, or it is naive. That's all the word means in this context. Philosophy of mathematics is a naive approach to mathematics, however, the philosopher of mathematics is at least aware of this distinction. Prelogic, to me, seems very metaphysical right now, but perhaps you are onto something. If you in fact have a working prototype of something, then that is something which can be formalized within the rules of logic, even if you can not describe it in terms of how it could be formalized.

To answer your question more directly here: Formal ZF set theory is one way to represent a certain logic, which includes, among some things, all of natural numbers and the operations we use with them. The Peano Postulates are another. Any artificial intelligence that may result from ZF set theory is illustrated by Church/Post/Turing and what has eventually become the study of machine learning. But ZF set theory is far from the only foundation to intelligence and logic. It is just one very powerful descriptive system that mathematicians use often and can describe easily many of the things common throughout mathematics.

But words like "group" and "set" are very loaded when in the context of a conversation about mathematics. And if you mean something else than what I might mean when I talk about an Abelian group or a the set of Natural numbers, or if your definition is more broad and also encompasses those types, you must explicitly define the words you are using so that other people can understand what you mean by them. That is all formalism is. You tell me what you mean by some word or symbol or process in language that does not contradict itself. You mis-paraphrased me. You said, I said about Godel's Incompleteness: "It needs a contradiction to be consistent; therefore is not consistent." But this is not true. It needs contradiction to be COMPLETE, that is, say all true statements. However, if there is a contradiction, it is inconsistent. Now here lies the rub: We don't care about contradictory theories, because they are essentially metaphysical and can prove anything. One contradiction leads to another, which can lead to unicorns in the forests of mars, as well as any and all truths. True is specifically what IS relevant to a logician. But not ALL truths, only those truths which are consistently verifiable. Since, without consistency, you can't get meaningful results.

To me, a mechanism and a physical event are two different things. An event, for instance, to me, conjures up the works of Heidegger, while a mechanism, is a sort of machine with an initialization process. But, here, it seems you use it to mean that Prelogic is a condition in the physical world, or maybe some apriori existing state of an indescribable logic. The first of which, is easily formalized, the second of which is metaphysical philosophy. You have to figure out what you are saying and to whom. But here is my adivce: don't get caught up in what you mean by such and such a word. Instead, figure out what other people mean by the words they use and then try to describe what you mean using that language. If what you mean doesn't exist, make it up and correctly define it, but don't confuse words that already exist without it being an explicit metaphor. And these rules of thumb should help you on your quest of formalizing your theory or presenting it philosophically, whichever is best suited for it.

Frank,

My questions arise from your example " the car is red and not yellow, while the submarine becomes conceptually yellow but not red." and your concept of meaningless entity.

1) How would your theory differ from the concept of interpretation in FOL?

2) Would your system be consistent and complete?

I hate to sound like the "nuance" of the room, but I believe we have something very similar to this from what you have described. It's called Automated Theorem Proving. Problems in this realm are what constructed modern Computer Science (the first hallmark to this was the Hilbert's Program being shown to be undecidable (i.e., via Godel, Turing, Church)). Some fun theorems in this realm are the decidability of Elementary Geometry (meaning you can write an algorithm that can test any theorem systematically by starting at axiomatic parts to construct those statements deductively (maybe not efficiently but it is possible in finite time). It is using axioms and using computational mechanisms to construct and verify if statements are true or not from their axioms. Some classes of problems can be decided upon such algorithms, but others are undecidable as they are too general as Fabricio pointed out. There may be a more applied twist from what you are describing, so I apologize if I am mistaken.

I would highly urge you to explore the literature related to this area. It was red hot in the 70s so there are a lot of results around (often is found in a lot of Computability Theory texts from around the 80s), but its purpose a lot of the time now is for Software Verification (i.e., building software verification tools that can test if instructions are correct or not based on the set of formal languages constructed). I am unsure (as I work mainly in Combinatorics, Algorithms, and Theory of Computation, not so much in Applied Computing) but I think maybe exploring what has been done in Machine Learning in the context of Verification Systems may be valuable for your venture which I do know some of these questions are being discussed about how to linguistically construct more complicated statements from simple ones (not so much atomic ones like we deal with in Automata Theory).

J. Inman, Great advice for everyone, especially one with a unifying "Prelogic" grounded in both metaphysics a priori physics. Thank you.

Dear Frank, I love the association you are hoping to make between Set Theory and the neural connections in our brains. I need to review all of the above comments. But, if you have an inspiration, don't let the details get in the way of your nascent idea. They can and should be worked out later.

Take the intuitive leap, my man!

Dear Inman,

The name Naive Set Theory does have a naive connotation in it, at least to some of us still, but I see how in younger students this will not be true.

I went and found one of my old dusty books and luckily I found the sentence that reads: Cantor himself, to be sure, did not for a moment lose faith in his theory in its full "naive" extent though he was unable to meet the challenge of Russell's antimony.

-Foundations of Set Theory A.A. Fraenkel, Page 4

The sets I talked about, I actually call them Cets to differentiate them from sets and thus avoid mixing with older, like you said, well defined or overly redefined homonyms. But then I face another researcher, demanding more general terms to explore the idea, at least, at the beginning (I can not win).

I have spent some decent amount of time in the linguistic, theory of language, Universal Grammar and Artificial Intelligence sections of research Gate (and the terrible Linkedin) and if I use Set as NBG (which this system already disagrees with) then the linguists and grammaticians naturally crash and the discussion is dead on arrival. So I have learned, I admit incorrectly, to talk in general terms to reach a broader audience, because unfortunately, Cets (sets of elements identifying concepts) have a broad consequence in many divergent, even never-associated fields of science, such as Natural Languages and Set Theory are based on the same: sets.

Nonetheless, there are several, minute but later significant differences that make difficult to use well defined terminologies such as some from logic. For example, I used conclusion and new axiom above. An axiom is a first level claim correct? However, is this prototype, as information (knowledge) begins to assemble into Cets, at any given time, an axiom can become a theorem (conclusion). For example, Mary is a girl is an Axiom on the above. If a new concept, say strissty is introduced (a strissty is a girl), and a new sentence arrives that says "Mary is a strissty" then "Mary is a girl" is no longer an axiom but a conclusion/theorem. In fact, "Mary is a human" went from axiom to theorem when "a girl is a human" (an abduction) was formulated. Knowledge is constantly changing, shifting and moving, meaning that I can not pin point a particular definition because the elements themselves are constantly changing their operations and therefore, what defines them (their definitions).

Also, this question is not only in Set Theory but in Linguistics (my old nemesis). And you are correct, the nomenclatures change, meaning that I should not share this topic simultaneously with other fields.

Dear Arturo, Good questions, but I am afraid I am not done with the research, meaning I can not answer them yet. I am still reading on the Incompleteness theorem, and is it not until I moved through the entire proof itself (is so freaking long) that I can only make comments so far to tickle your curiosity.

However, the prototype stands and proves some terrible or maybe good things about how is it that a set-group-collection can be used for representing information, and how this information, via more set-groups-collections can produce basically all the rational operations of intelligence.

This may be good (for all) as it can be really bad (for me at least); because it dictates that intelligence is made of Cets (Cets = sets of elements representing concepts) and anything beyond the ability for creating a Cet, can not be understood. This is why Set Theory is so important, and why the Russell Paradox serves not as a contradiction, but rather as a proof of the group-forming dependency of an intellect. In the Russell paradox, they way the statements configure it, make the set becomes an element which is return changes the set which again changes the elements and then the set again. This reverberation between set and elements never allows to finalized a set, and therefore, is conceived as contradictory (contradictory is the wrong word, but I have no other word for this, unless I make a new one.... maybe later)

Dear Antonio Lucero,

I kindly thank you for your support. I will keep on walking this path of self-destruction until finally, I can compare it, formalize it, present it, and go to the closest Pub and finish all the beer they have in stock.

I know I will stand for heavy criticism, but I am an idiot, so I will not stop.

Dear Albert,

Great link. In fact, it opens more questions than answers and deepens further my research.

So it appears that the incompleteness theorem essentially says that machine intelligence, as we know it today, will always find the limitation of a new missing algorithm? That there will always be some new task that the prior algorithms will never achieve?

Science is what we do when we don't know what we are doing (I don't know who said that but I love it)

@Daniel Page:

I didn't know that statements in Elementary Geometry were algorithmically decidable. Where can I find the complete proof?

There are many fundamental characteristics among the various topics discussed that need to be explored:

1) How the brain works: For example, in cognition, are we talking about innovation tasks or routine tasks? - big difference.

2) Logic: Traditional logic is linear - the brain's marvelous abilities are due primarily to its massively parallel processing.

3) The level of certainty that the brain accepts before reaching a conclusion depends on circumstances - it is not just true/false.

4) The brain processing involves many feedback loops.

5) Emotions and other non-logic factors can affect the processing performed by the brain.

Somehow, the scope of your analogy has to be explicitly delimited - start out small.

Hi Frank,

What Is for you the definition of "foundation of a theory" ? What is for you , more formaly speaking, the definition of ""foundation of intelligence.". Are you agree with the TOGA meta-theory basic axioms and definitions, see: http://erg4146.casaccia.enea.it/wwwerg26701/Gad-toga.htm ? Without the above anwers your question may have many different answers.

On the other hand: Accoeding to the Google, Your search - "Prelogic theory" - did not match any documents. (?)

Hi Adam,

Yes, you will not find anything on Prelogic because I have kept our research out of sight. I always believed that a prototype was an absolute need before any papers can be written.

Also, I must say, I am a little intrigued about TOGA, I don't think I understand it very well. Maybe we can talk directly? (phone call, email, etc.). Have you gone as far as developing a prototype? I know a prototype is not really needed for everything, but I feel a prototype gives you a shield that the opposition will have to prove it does not work before theories are discussed.

Dear Antonio,

Agreed. There are many things that can be challenged and talked about, and each has its own interpretation, nomenclature, field of science, even system of theories.

When it comes to theories, what ever we interpret is really up to us because nature continues with or without us, but this is still important to some of us... we need to know why things work the way they do.

However, this is what I think (only think) what our research and prototype may be able to prove (again, may be able to prove) is that: the basic creation of a set comprising entities that are representing concepts, can be trace back to how information is assembled so things can be understood. (this is so difficult to describe). In more horrific words.... why can Set theory be understood? because neurons are doing the same thing. Please, I am not saying Set Theory is the basis for intelligence. No. What I am saying is that the initial mechanism-action-procedure-task of forming a group (Prelogic) is what repeats over and over in the construction of math (set theory), language, reasoning, answering, abducting, and others.

There is doubt, this has immense implications if proved true, because it means that anything that doesn't follow these set forming task-procedure, can not be understood. An example of this may be the idea of infinite. Infinite has no end, the mind can not describe how big, or how it compares to others. The Russell Paradox is another. Also, even the very axioms are themselves "sentences" that follow these "set forming task" before they become a sentence that can be remembered and understood.

@Fabricio: Tarski proved it back in the day and can be found in the paper "What is Elementary Geometry?". You can locate it here http://www.corelab.ntua.gr/studygroup/Tarski_ElGeom.pdf

Usually when people refer to this result they are not looking for an efficient method of doing so (but then that becomes a complexity question, but that is independent of its computability). Now, in the paper Tarski formulates Euclid's elements in the form of axioms in logic then goes about it that way. Unfortunately some more advanced systems of geometry (often referred to as the AE fragment) of Euclidean Geometry (EG) is not known (as far as know) to be decidable. So once that is the case we could universally state Euclidean Geometry (as a whole, not just traditional EG) is decidable. Right now as far as I know it is still conjecture. Some logicians believe the AE fragment is decidable, but in pragmatic terms EG is decidable as this fragment is a set of rules that usually people don't look at as much in EG.

Hope this helps!

Thanks. Even if the problem is computationally hard, knowing that it is computable at all is quite impressive.

Yes. It indeed lends itself to the beauty of foundations where computer scientists and mathematicians in Logic can explore what we can or cannot prove or develop algorithms for.

Dear Mohammad,

Thank you kindly for your encouraging words. I still plenty of work and research ahead. This topic touches language, artificial intelligence and of course the math and science of set-collections.

Again, thanks

Dear Fabricio and Albert,

Central to the question if the Incompleteness theorem limits machine intelligence is also the question whether the incompleteness theorem ALSO limits human intelligence.

For biological intelligence repeats in the sense that there are procedures and their outcome which are common to all such intelligences. Empirically said, a kid from the Amazons can learn math in similar fashion than a kid from Tibet. They also speak, their languages which are made from the same basic elements (nouns, verbs, adjectives and adverbs) and they can both equally learn about the Incompleteness theorem.

The above children example shows that intelligence is indeed a standard procedure which as a procedure it will too be undermined by the Incompleteness theorem.

Controversially then, would human intelligence contradict the Incompleteness theorem?

Or, will the incompleteness Theorem also put the same kind of limitation on human intelligence?

This is actually a topic of vast debate among philosophers and mathematicians in Foundations but there is some consensus related to some matters. Some view the matter really as a matter related to context.

One lays down a formal system of rules and axioms, and can prove theorems based on the assumptions. Many agree the issue relates to the intuition that goes into developing new ideas because these ideas are out of the norm of the context the rules are established under (i.e., looking at the formal system then looking at what that system is capable of doing from a meta-level). To develop new ideas it relies heavily on knowing the context of ideas. If you come up with a new one a lot of the time it is not in the same box. Godel's Incompleteness Theorems are hallmarks of Mathematical Logic that have solid implications in all the formal sciences and what we can model (and prove) as mathematicians, computer scientists, statisticians, etc...

In a way if you were to give a method that did not fall under the same limitations as most mathematical concepts, you'd effectively be able to mechanize the procedure scientists develop new ideas with. This method would not be something that would fall under any universal Turing machine model.

The thing that confuses me about your question and the question you posed before is that you are modelling things with set theory, which will fall under the domain of Godel's work depending on the system you adopt (as any system will, and its implications to any undecidable system that can be reduced from a known one). Human intelligence relies on tangible things that can only be modeled empirically. Until you lay down the model, can you say one thing is the same as the other. The transitive properties of one system will come when a model is proposed for such.

If you want to ask a question about human intelligence versus formal systems, a more vague but one that is of debate is whether intuition can be modelled by a formal system. Godel had issues with this concept and it is very hard to pin down as we need more empirical results relating to this before we can formulate a solid theory that can be modelled and tested for comparing two things. This is a common mistake being done by cognitive science fellows, and applied fellows in my own field of Computing. Until that understanding is there, we cannot draw comparisons beyond what we know. We know our brains are not limited by the formal systems we create because it was our minds that intuited them in the first place while a computer cannot grasp such as it cannot fathom a test for decidability of any arbitrary program/algorithm fed to it (often a consequence of this is the validity of trying to simulate a brain on a computer because we know computation has limits).

Dear Daniel,

Please allow me to set the stage, so you and others can understand why this question appeared so soon and without its formal introduction. My research was privately funded and now it is threaten to end and keep me silent indefinitely. I wanted to let you know, via a few clues and examples, the existence of this set-based approach.

I am going take the freedom to repeat your argument:

"If you want to ask a question about human intelligence versus formal systems, a more vague but one that is of debate is whether intuition can be modelled by a formal system. Godel had issues with this concept and it is very hard to pin down as we need more empirical results relating to this before we can formulate a solid theory that can be modelled and tested for comparing two things."

So far, it appears that intuition may be modeled; but then again, I am only halfway there. I agree with all you are saying. Like Mark Inman said above, without a formal scientific paper and introduction, I am risking to loose many people in the way. But the basic idea is still the same.

If intelligence is indeed a process that naturally surfaces when entities begin to group or form sets with each other, then every intent to prove or disprove or understand any type of theory is intrinsically linked to the consistency of these "groups or sets.". This is similar to: We are going to use our own intelligence (which is based on groups-or-sets) to prove, discuss, argue and understand the very consistency of groups-or-set.

I am still studying the incompleteness theorem, for it too may be linked to the same recursive group-or-set dependency, and thus defeating its own purpose (the incompleteness of the incompleteness theorem??? ja, ja, ja).

Bests

Oh, by the way, I will too use my own intelligence, also dependent on the postulated group-or-set, to try to defeat or predicate the very course of my study (this is contradictory).

But interestingly, when the very consistency of the groups-or sets are challenged in our minds, it is then that the "illogical" or the "paradoxical" immediately surfaces.

For example, First order Logic, is intrinsically linked to the number of entities within a group-or-set. All its operations deal with the cardiniality of the entities within the sets. And because these entities (elements in the set) can not appear and disappear through any of the operations of FOL, it is then that the consistency of the system is respected.

@Frank said: "Or, will the incompleteness Theorem also put the same kind of limitation on human intelligence"?

My answer to this is Yes.

Godel proved incompleteness Theorem for mathematics,that is for human;then Turing proved that godel number is non-computable. So Godel First Incompleteness Theorem is accepted by most people for both human and computers.

Dear Yinsheng,

Yes, Godel proved incompleteness for a system that strictly makes usage of axioms which are foundation to the system, showing that new true statements will appears that although true (by external interpretation) cannot be deduced from the original small number of original axioms. However, I must emphasize, that the prove itself is a form a critic of the axiomatic based system and not others.

Just as significant then becomes the analysis of what makes a truth a truth.

If I may add this: the idea that intelligence is NOT a system (what some claim the incompleteness proves) also has tremendous negative implications. For example, how is it then that paradoxes are equally understood or viewed by people from completely different backgrounds? A child from the Amazons will see the Russell Paradox with the same logic framework that a child from England or Siberia.  

The incompleteness deals with a specific type of system, not all possible systems. 

This is an historical debate that originated in Turing's paper on machine intelligence in the 1950's, and culminated in the 1980s with intense debate in the Philosophy of Mind ans Cognitive Science.  I list a couple of my publications on this in my profile, but the contributions of Judson Webb, Chihara, Searle, and Dennet are equally noteworthy.

It seems clear to me that the difficulty of the Russell Paradox is that the logical system that Russell posited was a closed system. Whereas the conundrum comes when an unqualified statement is made that does not limit the universe of discourse to the closed system. When that happens, and we try to apply the statement outside of the closed universe of discourse, we encounter contradictions.

We just have to discipline our discourse to always be relative to a limited context. To do this we must be explicit as to the conditions and/or assumptions underlying the universe of discourse.

An another important area has been neglected, that is continious computation (for example, many simulating computers) which is related less to discrete variables, so less to set theory for that set theory is only for discrete variables. Consequetly, for some continious intelligent system, you can't say it is laid on the foundation of set theory.

Dear Zhang,

I just want to emphasize that the "foundations of Set Theory" is not "Set Theory." By "foundations" I am trying to describe the underpinning logic, within the intellect, that allows the human mind to comprehend (or intelligize) Set Theory itself.

Set theory is an outcome of the intelligent mind, it cannot be its foundation.

Hi Frank,

As Mark intonnates you're entering some pretty murky and deep waters here.The answers to your question are mitigated by some fundamental presuppositions you make regarding the nature of your inquiry.  For instance, are you proceeding on a Constructivivist, Logicist, or Formalist Paradigm?  

   As Cantor showed, sometimes our intuitions are less than reiiable in these abstruse matters.  Two sources that may benefit you in clarifying your approach may include Wittgenstein's remarks on the Philosophy of Mathematics and Peter Aczel's seminal work on hypersets. Finally, if you really want to dig deep, I would recommend Husserl's comments on the nature of mathematical phenomenology.  Anyway, best of luck to you... I look forward to reading what you come up with.

Frank, I think I got a gist of what you say. 

I do not think red, yellow, car and submarine are meaningless constructs, rather the contrary: All of us imagine each one of those in our heads as we say the sentence and we imagine each one of those in separate, as you seem to state later on.  

I think you are looking for what Dominic Hyde explained really well: Blurred sets. 

The book he used was called Introduction to Logic by Stephen Read: http://www.amazon.com/Thinking-About-Logic-Introduction-Philosophy/dp/019289238X          

Perhaps you could use the preferred theory of Hyde, Supervaluationism, to explain human convergence in reasoning, like all humans who are mentally normal could have a pattern for certain sentences, construction, and you then could use that.

I assume you are looking for a logical theory to explain what you see happening with the neurons, right? 

I would love to know the function of each neuron in this process. 

Am interested. 

Set theory and Boolean logic are related, but they are not the foundation of intelligence.