Does anyone have any expertise or insight to Godel's Incompleteness Theorem?
This is what it is:
http://mathworld.wolfram.com/GoedelsIncompletenessTheorem.html
Google it on Mises.org. Hoppe has some brief commentary on it in his Economic Science and the Austrian Method. As an aside, it's interesting with regard to its applicability in the case of ontological dualism.
-Jon
Freedom of markets is positively correlated with the degree of evolution in any society...
I don't know the proof, but the theory simply says that any axiomatic number system is incapable of proving that it can either prove or disprove any proposed expression in this number system.
In other words, no system can prove its own completeness while remaining consistent.
Check my blog, if you're a loser
G. theorem shows that science is always inconclusive and there will be always certain amount of subjective belief in scientific truth.
To prove that a theorem is “true” you verify that theorem is derivable from axioms or from other true theorems derived from axioms, if so you assert that theorem is true. To prove that theorem is “false” you show that asserting it to be true you would contradict axioms or any other true theorem derived from axioms. Notice that also “false” theorems are derivable from axioms, you arrive at false theorems from denial of axioms or true theorems. Thus suppose theorem A, if there exist at least one true theorem which A contradicts the A is false otherwise (if none of true theorems or axioms are contradicted) A is true. It is obvious that theorem cannot be both true and false.
Suppose you want to describe world in its entirety, you want to be able always to tell what happens next. It was believed that world observes the laws of LOGIC. Logic is no more no less then finding the true or false theorems in the sense described in second paragraph above. First you state your axioms and then you develop your theorems, development of theorems is very primitive thing which can be done by little kid or monkey or computer it is just applying certain transformation (or rule of inference) on base axioms, first you have just axioms, then you apply transformation first time, you get your first set of theorems, then you apply same transformation on axioms and your first developed set of theorems, you get then another set of theorems, you can continue that way ad infinitum ….., this way you will be able describe all the happening in the world. Notice that suddenly the notion of “infinity”, of “every-thingness” has pop up here, and this were the problem starts.
So we naively think that every theorem is derivable from axioms and it cannot be both true and false. Goedel proved that it is not so.
There is a branch of LOGIC called formal logic. In formal logic you use instead of words logical symbols, e.g. instead “word” implication you use symbol “->” instead the world “relation” you use symbol “R”, instead of word assertion you use symbol “|-“, a theorem expressed formally is just ugly juxtaposition of formal symbols having no meaning for layman not knowing meaning of particular symbols.
To be able to prove something about formal logic (and hence about logic itself), Goedel would define precise mapping of logical symbols to numbers, e.g. instead of “->” he would use “1”, instead of “R” he would use “2”, instead of “|-“ he would use “3”, having that mapping defined he was able to prove certain things about any logical formal system just by “playing” with numbers which stand for the symbols. He proved that there does not exist logical formal system dealing with numbers which can be both complete and consistent. In every such a system there will be theorems derivable from its axioms which are both true and false, so you are left with 2 choices either reject such theorems in which case your system will be incomplete (it will not contain some theorems derivable from axioms), or you can accept such theorems in which case your system will be inconsistent (it will contain theorems which are both true and false).
Numbers are in their nature infinite. Instead using the words of natural language or symbols of symbolic language you can use numbers, think that for every word or symbol you will use certain precise number. Goedel proved that no matter how hard you will try to avoid contradict yourself there will always be possibility of contradictory statements.
Hence it follows ,that no scientific or economic theory cannot be completely showed to be true, it can only be BELIEVED to be true.
Yes, I know a little about Godel's incompleteness theorems (there are two).
The easiest way for a lay-person (like myself, my interest in the subject is purely amateur) to understand his theorem is to start with Russell's paradox. Russell's paradox is the following:
Consider the set V = { x : x is not an element of x } (Read: "V is the set of all sets which are not members of themselves").
Is V an element of itself? If we say yes, it is, then it is not an element of itself since that is the definition of V. If we say no, it is not, then it must be an element of itself because it is the set of all such sets. Either way, we arrive at a contradiction. Therefore, there is no set V. This is a set theoretic form of the liar's paradox, "This statement is not true."
Godel was contemplating issues of provability, particularly the attributes of consistency and completeness. A formal system is consistent if no contradiction can be proved within it. It is complete if every true theorem in the formal system is provable. It is obvious why consistency is a desirable property of a formal system. Completeness is a desirable property because a logically complete formal system would be mathematically universal - every true math theorem could be proved without a need to invoke new axioms, such as the axiom of choice.
Stripping away the formal arguments, Godel's theorems rest on what could be called "Godel's paradox" (though he nowhere lays out such a paradox in plain language), namely, "This statement is not provable" instead of the liar's paradox, "This statement is not true." Godel first established a method to be able to say "this statement" in formal logic (to create self-reference) and then found a method to assert non-provability. Godel reasoned that such a statement is in fact true. "This statement is not provable" is true if the system in which it is asserted is consistent. However, since it is true, it is therefore not provable. And since it is not provable, there is at least one true statement in every consistent formal system which is not provable. Therefore, every consistent formal system is incomplete (cannot prove every true statement).
Godel's first incompleteness theorem states that a formal system (of "sufficient" power) is consistent if and only if it is incomplete. You can't have both. From Wikipedia, "for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory." The second incompleteness theorem states that a formal system (again, of "sufficient" power) can prove its own consistency if and only if it is inconsistent.
Contemporary mathematician Gregory Chaitin has significantly extended Godel's results to show that almost all statements in any formal system (of sufficient power) are not provable! He says it this way, "Most mathematical theorems are true for no reason." In fact, everything I know about Godel I've learned through Chaitin's writings. You can read the argument I've presented here in Chaitin's words here and you can watch a video on his groundbreaking mathematical constant (the "Halting Probability" or Omega) here (you can skip the first 4 minutes of gratuitous self-aggrandizement by the introducer).
Clayton -
This is a set theoretic form of the liar's paradox, "This statement is not true."
February 17 - 1600 - Giordano Bruno is burnt alive by the catholic church. Aquinas : "much more reason is there for heretics, as soon as they are convicted of heresy, to be not only excommunicated but even put to death."
Juan:Russell's 'paradox' is nonsense. This is a set theoretic form of the liar's paradox, "This statement is not true." That's not a paradox - just cheap self-reference.
Take it up with Russell, Godel, Turing and Chaitin. :)
I don't know any real paradox. All paradoxes are not paradoxes anymore once you take a deeper look into them, but they're used to prove a point, no matter how stupid they look.
Juan:Take it up with Russell, Godel, Turing and Chaitin. :) It's not worth it =] As a side note, a statement like "pigs fly" can be true or false. But a statement like "this statement is true" can't be true or false. A bunch of words can't say anything about themselves. The statement is effectively meaningless.
Well, I've read and understand Godel's proof. What specific flaw do you find in his proof that has escaped the mathematical establishment for 77 years?
I think some clarification is in order.
1. There is nothing wrong with Gödel's Incompleteness Theorem.
2. There is nothing wrong with Mises' Action Theorem.
3. There is nothing wrong with Praxeology (The Action Theorem applied)
There is a reason the Action Theorem is not analytical a priori, but synthetic a priori.
Mises, Rothbard, Hoppe would all agree that the validity of the Action Theorem does not rest on pure formal logic. Once validated, however, it becomes possible to deduce praxeological laws logically.
ClaytonB:Take it up with Russell, Godel, Turing and Chaitin. :)
Perhaps we should add Alonzo Church just for good measure? Entscheidungsproblem and all that.
Juan:Maybe the mathematical establishment has the same love for truth as the economic establishment does ? The thing is, you can't analyze logic using logic. It's just self-recursive nonsense. Just like asking, " What's the meaning of the word meaning ? "
Godel's Incompleteness Theorems are about the limits of formal systems, not an attempt to bootstrap logic.Clayton -
ClaytonB: Contemporary mathematician Gregory Chaitin has significantly extended Godel's results to show that almost all statements in any formal system (of sufficient power) are not provable! He says it this way, "Most mathematical theorems are true for no reason." In fact, everything I know about Godel I've learned through Chaitin's writings. You can read the argument I've presented here in Chaitin's words here and you can watch a video on his groundbreaking mathematical constant (the "Halting Probability" or Omega) here (you can skip the first 4 minutes of gratuitous self-aggrandizement by the introducer). Clayton -
Thank you for the YouTube reference Clayton. Chaitin, in part 7, mentions something very exciting to me. The lower bound complexity of his flip-a-coin Turing machine is maximally unknowable. Please correct me here because I'm not a computer scientist, but what this means is that you cannot compress the bit stream output from this program into less bits, e.g. knowing the first 100 bits cannot tell you anything about the 101st bit. Thus it is, by his own definition of randomness, random; correct?
So, let us for the sake of argument disregard John Searl's idea of the mind not being able to be modeled as a Turing machine, and assert the opposit: The human mind/brain is a Turing machine. Next, let us imagine someone being truly able to model this machine. Lastly we monitor the bit stream output. The question then becomes: Would the lower bound complexity of this modeled mind be maximally unknowable?" If so, would it be random as Chaitin would define it? And, if random, would we not have found a foundation of free undetermined thought, free undetermined action. That is, anyone trying to predict human action with claimed full certainty would be unscientific?
I know I have moved into counterfactuals here, but if you could give an affirmation of my thoughts above, Clayton, I would appreciate it very much. In a sense my thoughts could be formulated as such: "Assume we created Artificial Intelligence, would there not be a possibility to use AI to disprove someone trying to negate the idea of free thought and free action?"
Nothing cheap about self-reference, at all. The central lesson of Turing's and Church's research is that a sufficiently powerful formal system is capable of self-simulation* (Turing's universal machine or Church's fixed-point operator can self-simulate) and this is what enables us to say "this statement" in formal mathematics.
As Chaitin discusses in the YouTube lecture I linked, Turing's argument is the most straightforward (compared to Godel's and Church's). In essence, Turing proposes what could be a physically realizable device (in every respect except having infinite tape) and asks what is essentially a question of physical science: Is there a method to decide whether such a device halts or not by inspecting its tape, short of running the machine for an indeterminate amount of time and waiting to see whether it halts or not? The answer - which is ultimately equivalent to Godel's Theorem - is no. Turing shows that there are unsolvable problems (Godel had showed there were unprovable truths). If you accept the Church-Turing thesis, then these unsolvable problems are completely unsolvable in a classically physical world (it turns out they are unsolvable in a quantum physical world, too). By reasoning about an imaginary physical device, Turing avoids the problem of self-reference completely. The culprit (as was suspected from the time of Cantor) is not self-reference. The culprit is randomness, but that would be an entirely new thread. :)
*Please do not confuse self-simulation with semiotics and other blatantly circular nonsense. There is a world of difference between self-simulation and self-interpretation.
It is amusing that people sometimes use Goedel's proof as a proof against apriorism in general, though. Perhaps because they're positivists (even if not consciously) and have a messed up notion of knowledge. It does have some interesting applications in the philosophy of the mind, in favour of dualism.
corpus delicti: ClaytonB: Contemporary mathematician Gregory Chaitin has significantly extended Godel's results to show that almost all statements in any formal system (of sufficient power) are not provable! He says it this way, "Most mathematical theorems are true for no reason." In fact, everything I know about Godel I've learned through Chaitin's writings. You can read the argument I've presented here in Chaitin's words here and you can watch a video on his groundbreaking mathematical constant (the "Halting Probability" or Omega) here (you can skip the first 4 minutes of gratuitous self-aggrandizement by the introducer). Clayton - Thank you for the YouTube reference Clayton. Chaitin, in part 7, mentions something very exciting to me. The lower bound complexity of his flip-a-coin Turing machine is maximally unknowable. Please correct me here because I'm not a computer scientist, but what this means is that you cannot compress the bit stream output from this program into less bits, e.g. knowing the first 100 bits cannot tell you anything about the 101st bit.
Thank you for the YouTube reference Clayton. Chaitin, in part 7, mentions something very exciting to me. The lower bound complexity of his flip-a-coin Turing machine is maximally unknowable. Please correct me here because I'm not a computer scientist, but what this means is that you cannot compress the bit stream output from this program into less bits, e.g. knowing the first 100 bits cannot tell you anything about the 101st bit.
Hmmm, I'm not sure he says anything about the output of individual Turing machines on the basis of their input. A highly random Turing machine could have an incredibly dull and predictable output, e.g. just printing '1' indefinitely.
Thus it is, by his own definition of randomness, random; correct? So, let us for the sake of argument disregard John Searl's idea of the mind not being able to be modeled as a Turing machine, and assert the opposit: The human mind/brain is a Turing machine.
So, let us for the sake of argument disregard John Searl's idea of the mind not being able to be modeled as a Turing machine, and assert the opposit: The human mind/brain is a Turing machine.
I think of the mind as a Turing machine, though not in the usual, corny AI sense.
Next, let us imagine someone being truly able to model this machine.
Well, be careful here. The machine is the model. We cannot model it in terms very much simpler than itself if it is very complex, and the human brain is the most complex object we know of in the physical universe. This is why Calculus or DiffEq isn't much help in predicting human behavior.
Lastly we monitor the bit stream output. The question then becomes: Would the lower bound complexity of this modeled mind be maximally unknowable?"
No. But Chaitin says something far more interesting later (I think end of pt. 7, beginning of pt. 8) that is what I think you're driving towards (and which I personally find very exciting): you cannot prove that an N-bit program is elegant (incompressible, random) with less than N-bits of "axioms" or "assumptions." Another way to say this is that an N-bit compression program can only compress, in computable time, strings whose Kolmogorov-Chaitin complexity (KC complexity) is less than N-bits.
Now, induction and compression are integrally linked - to "comprehend" data in the Kolmogorov-Chaitin sense is to compress it to its shortest form, that is, to discover the underlying law or theory that explains the data (by compressing out all the redundancies and leaving only the structure). If we imagine the human brain (which is obviously an induction machine) as a compression program with N bits of KC-complexity, then the human brain cannot compress strings (data) whose KC complexity is N-bits or greater. Since the human brain is itself N bits of KC complexity, this means that the human brain cannot self-predict (non-determinism, free choice) and that speaking of an ultimate "theory" of the human brain is nonsense even in a deterministic universe. Any detailed model of the brain could not be much smaller (in bits of complexity) than the brain itself. Determinism, in itself, is no help to prediction when you are at the limits of computability. In other words, the future behavior of deterministic systems can be strongly unknowable to non-omniscient observers.
This does not exclude the possibility of attaining speedups through simulating the operation of the human brain (e.g. in silico), or of partial "compression" (i.e. explanation) of the operation of the human mind. But the age-old fantasy of commoditizing thought is just that, in my opinion: fantasy. The human brain has none of the features of highly compressible objects, and there's no reason to believe there's very much hidden redundancy within it (evolution is rarely wasteful), so I don't see humans improving upon the brain in any other than simple speed-up or fixing obvious evolutionary mistakes.
Another way of saying it is this: evolution gave us N-bit brains, so we live in an N-bit world whether we like it or not. :)
(NOTE: The above paras are 100% my own metaphysical speculations spun off from Chaitin's work... I have found no such metaphysical speculations affirmed by any other more reputable sources... read at your own risk.)
If so, would it be random as Chaitin would define it? And, if random, would we not have found a foundation of free undetermined thought, free undetermined action. That is, anyone trying to predict human action with claimed full certainty would be unscientific?
Again, you have to be careful, because strings of complexity less than N-bits may be predictable or 'patternful' to the N-bit complexity human brain. It's not because the brain itself is maximally incompressible that we cannot model its future behavior. Rather, it is because it is of no more complexity than itself.
I don't think we have to invent AI to know that. Unless there's some god-like humans out there with significantly more complex brains than the rest of us in the unwashed masses, human behavior is ultimately unpredictable (therefore gives the illusion of "free choice" even in a deterministic universe) by humans.
BTW, there could be some synergy between computability theory and the Misesian calculation problem. ;)
Thank you Clayton. A very helpful reply. I'm on unfamiliar ground but it made sense. Just to be clear, I agree we do not have to invent AI to know about free choice. I was just thinking along the lines of: "Even though you are certain about a theory, and thus about the outcome of an experiment designed to test it, it could still be exciting to observe the outcome" (if that even makes sense, lol).
oops
Juan:Maybe the mathematical establishment has the same love for truth as the economic establishment does ? The thing is, you can't analyze logic using logic. It's just self-recursive nonsense. Just like asking, " What's the meaning of the word meaning ? " Feel free to consider me part of the ignorant and unwashed masses, of course =]
Are you a Gauge Institue guy, Juan? lol
ClaytonB:By reasoning about an imaginary physical device, Turing avoids the problem of self-reference completely. The culprit (as was suspected from the time of Cantor) is not self-reference. The culprit is randomness
I have some problems with this. If you speak of the halting problem, that is about self-reference as far as I can see. How does randomness enters the picture and becomes the main culprit?
scineram: Juan:Maybe the mathematical establishment has the same love for truth as the economic establishment does ? The thing is, you can't analyze logic using logic. It's just self-recursive nonsense. Just like asking, " What's the meaning of the word meaning ? "Feel free to consider me part of the ignorant and unwashed masses, of course =] Are you a Gauge Institue guy, Juan? lol
Juan:Maybe the mathematical establishment has the same love for truth as the economic establishment does ? The thing is, you can't analyze logic using logic. It's just self-recursive nonsense. Just like asking, " What's the meaning of the word meaning ? "Feel free to consider me part of the ignorant and unwashed masses, of course =]
Ooh! Ooh! Perpetual motion really is possible but there's a government conspiracy lead by the reptilian Atlanteans to cover it up!
scineram: ClaytonB:By reasoning about an imaginary physical device, Turing avoids the problem of self-reference completely. The culprit (as was suspected from the time of Cantor) is not self-reference. The culprit is randomness I have some problems with this. If you speak of the halting problem, that is about self-reference as far as I can see.
I have some problems with this. If you speak of the halting problem, that is about self-reference as far as I can see.
How is the halting problem any more self-referential than the gravitational problem, except that we are physical beings asking a question about the behavior of a (imaginary) physical object? The proof that the halting problem is unsolvable relies on giving HALT(x) itself as input, i.e. HALT(HALT) but even this is not self-reference per se, it's just providing a bit pattern to a function that happens to be the same as the bit pattern comprising the function itself. Its purpose is not self-reference but diagonalization.
How does randomness enters the picture and becomes the main culprit?
As I said, that is a separate thread unto itself. Basically, randomness enters the picture because most mathematical facts are true for no reason. This statement (Chaitin's) is not an attempt to be avant-garde or postmodern, it's a statement about the statistical behavior of an exhaustive enumeration of mathematical theorems (or programs, as the case may be). Now, this statement is (unintentionally) confusing because when Chaitin says "most mathematical theorems" he does not mean most mathematical theorems which we find interesting, but most mathematical theorems in an objective, exhaustive enumeration of all possible mathematical theorems. We humans just happen to find the theorems which are true for a reason (compressible) interesting exactly because they're true for a reason (our brains look for things which are "patternful" and ignore things which are apparently patternless or try to impose patterns on them).
Read about Chaitin's constant for a concrete example of a mathematical object which is defined "almost constructively" (to use Chaitin's words), yet whose value is as random as the flipping of a metaphysical coin. Chaitin has even used his constant to demonstrate that number theory contains pure randomness. This is a surprising result because we would expect that structure should be all-pervasive in mathematical systems which contain no apparently chaotic or disordered elements. What Chaitin has found is that order is infinitesimally rare in the universe of all mathematical truths and this is why there are always facts which are true, but not provable (or "true for no reason" where "for no reason" means "without possibility of being proven true").
I was not giving the link for serious considerations, but to show that math cranks are actually out there.
I think Goedel is pretty smart, and I think many mathmeticians would agree. I would assume the vast majority of those who discredit him are unfamiliar with the details of his theorems and mathematics in general.
When you are dealing with a discrete, deterministic system, theorems inside this system can be proven true or false. Goedel, to my knowledge, has proven his theorem inside such an environment. There is no necessity for measurement or other such flexible observations. You "test" the theorem by determining if it is consistent with the system's axioms (required basis for any such system). There are no real world discrepancies with empirical data or the truthfulness of theories upon which a new theory is based.
Attempting to apply the system to the real world would be required to introduce the problems I believe you are describing.
I think Goedel is pretty smart, and I think many mathmeticians would agree.
Yeah, that's why I came here - because the establishment told me that this was the place for sound economic theory. Give me a break.
Do you have an argument against Goedel's theorem, or do you just enjoy being the last skeptic in the room? Do you have an argument or evidence to reinforce your viewpoint? Do you believe that theory inside discrete, deterministic systems based upon axioms have the same obstacles as theory based upon the behavior of the real world, such as physics or economics?
I would assume the vast majority of those who discredit him are unfamiliar with the details of his theorems and mathematics in general.
Chaitin: When I was a small child I was fascinated by magic stories, because they postulate a hidden reality behind the world of everyday appearances. Later I switched to relativity, quantum mechanics, astronomy and cosmology, which also seemed quite magical and transcend everyday life. And I learned that physics says that the ultimate nature of reality is mathematical, that math is more real than the world of everyday appearances. But then I was surprised to learn of an amazing, mysterious piece of work by Kurt Gödel that pulled the rug out from under mathematical reality!
Chaitin:"Applying mathematical methods to study the power of mathematics is called meta-mathematics, and this field was created by David Hilbert about a century ago."
To say "Kurt Godel was smart" is an understatement of cosmic proportions. Godel possibly thought more deeply about mathematics than any man ever has.
Juan:http://www.cs.umaine.edu/~chaitin/summer.html Chaitin: When I was a small child I was fascinated by magic stories, because they postulate a hidden reality behind the world of everyday appearances. Later I switched to relativity, quantum mechanics, astronomy and cosmology, which also seemed quite magical and transcend everyday life. And I learned that physics says that the ultimate nature of reality is mathematical, that math is more real than the world of everyday appearances. But then I was surprised to learn of an amazing, mysterious piece of work by Kurt Gödel that pulled the rug out from under mathematical reality! That's not mathematics - that's metaphysics.
It's also not Godel's 1931 paper On Formally Undecidable Propositions in Principia Mathematica and Related Systems. Please read the paper - the introduction is not inaccessible even to a layperson - and any one of its many online (or hardcopy) expositions and tell me what specific error Godel makes in his reasoning. :)
I can safely laugh at claims such as " And I learned that physics says that the ultimate nature of reality is mathematical"...
Most of Chaitin's online articles are metaphysical discussions - he makes no claims to the contrary. But he is discussing what he sees as the philosophical implications of his mathematical work (most of which was done during the late 60's and early 70's). I fail to see the problem.
By the way, with Chaitin, I hold that the physical universe is fundamentally mathematical. It's a kind of Pythagoreanism called "digital physics." Look it up.
In the meantime, I'd like to see a specific objection to Godel's reasoning.Clayton -
I would agree that popularity is not necessarily correlated to correctness, although in academic communities this is more often more likley the case. (Note - there is a huge difference between completely independent cranks and conflicting schools.) What I meant about Goedel and the establishment is that I have not been made aware of any dissenting opinions that have been able to convince me Goedel is wrong. In 70+ years since this widely known theorem has been around, you'd think you'd at least come into contact with movements that present a strong case. Maybe I'm wrong. Do you know of any strong rebuttals?
I read the paper. I'm not arguing about mathematics, but axiomatic systems. That is, the resulting system when certain things are accepted as rather than proven to be true. I don't believe anyone claimed Goedel's theory definitely applied to general reality, only systems that were based upon axioms derived from real life observations.
Chaitin says that the theorem can be applied to things that are undefined. I don't think this is exactly true. Without any axioms, nothing really means anything. And he seems to define randomness. His examples still seem to rely on axioms. As far as randomness goes, I find the work of Wolfram interesting on this one (note: I know it is not completely his work, and that he is a huge prick about that).
Chaitin is saying that he believes reality is an axiomatic system. He's not going to be able to prove this ever, I don't think. Goedel's incompleteness theory, nonetheless, does not simply invalidate all theorems inside such a system, only saying that it is impossible to know if all theorems inside this system are true or false, or whether the axioms themselves can be proven consistent. For most people's purposes, Godel's work changes nothing.
If reality is an axiomatic system and Goedel's incompleteness theory does apply, this probably won't have any fundamental effect on science in general anyway. I believe most people already assume that we can't know if all theories can be verifiably proven true or false.
It is important to realize that Godel's incompleteness theorem may or may not apply to nature, because there are no means to determine if it is an axiomatic system. Currently, M-theory's failure to incorporate the 4 forces into a set of compatible and consistent equations shows that if nature is axiomatic, these axioms are unknown. Further, things like fluctuating cosmic constants and such would lead us to believe that if there are axioms, we are a long way away from knowing them.
Juan:I think Goedel is pretty smart, and I think many mathmeticians would agree. Ah yes. If the majority of the establishment claims 'X' is true, it logically follows 'X' is true...
By the way, as Chaitin discusses in the video lecture I linked, Godel's work on incompleteness has been largely ignored by the mathematical establishment. Chaitin's work, which is even worse for the establishment, is - as he says in the video - considered "pornographic" in the mathematical community... it's something most people know about but nobody discusses.
Nevertheless, the work of both scholars has not been refuted (there is one guy who rejects some of Chaitin's arguments, but I think his reasoning is incorrect). In that sense, they are "mainstream" that is, they are not cranks like the "Guage Society" which cineram linked to. That in itself doesn't make it true and you're welcome to be the first to find an error in their reasoning. Call me credulous, but I'm persuaded. :-)
Most of the Incompleteness theorem deals with the problem of how certain sorts of closed logical arguments are flawed in as much as they require external referents to make them 'normal' (or complete). There are exceptions, which I believe Godel recognized too, such as the Pythagorean Theorem of Right Triangles, and I think one other mathematical relationship (I can't remember the name of it right now...Sorry). Anyways, if you try to apply the Incompleteness Theorem to a given logical relationship, you're trying to show that its attempt to be 'closed' it simply setting itself up to be unproven or more exacting to state unprovable (as it will just go around and around in deductive breakdowns where it will reference itself forever). Oddly, if you look at the work of Topos Theory, it hints at the same situation, but in a more obvious manner than the Incompleteness Theorem itself (that some sets of logic have to remain 'open' to be consistent).
"The power of liberty going forward is in decentralization. Not in leaders, but in decentralized activism. In a market process." -- liberty student
meambobbo:Chaitin says that the theorem can be applied to things that are undefined.
You'll need to explain this statement - Godel's theorems apply to formal systems, there are no "undefineds" in formal systems (their intent is to avoid ambiguity).
I don't think this is exactly true. Without any axioms, nothing really means anything.
None of Chaitin's or Godel's work has anything to do with having no axioms. :-)
And he seems to define randomness.
Kind of... randomness in Chaitin's view is the complete exhaustion of order and structure and, by virtue of this, is undefinable. His definition is not a "definition" in the sense of making randomness well-defined, which would be an obvious inconsistency.
His examples still seem to rely on axioms.
Of course.
As far as randomness goes, I find the work of Wolfram interesting on this one (note: I know it is not completely his work, and that he is a huge prick about that).
Note that Wolfram is an intellectual debtor to Chaitin. :-) His ANKOS cites Chaitin many times.
Chaitin is saying that he believes reality is an axiomatic system.
More specifically, he believes that the physical world is a computer. This is not so extreme, a mainstream theoretical physicist (Seth Lloyd) has written a small book on the subject, Programming the Universe.
He's not going to be able to prove this ever, I don't think.
It's not about proof. It's about constructing a new paragdigm from which to do mathematics (and physics) to, hopefully, gain new insights into the structure of the physical world.
Goedel's incompleteness theory, nonetheless, does not simply invalidate all theorems inside such a system, only saying that it is impossible to know if all theorems inside this system are true or false, or whether the axioms themselves can be proven consistent.
The axioms can only be proven consistent if the system itself is inconsistent (2nd incompleteness theorem).
For most people's purposes, Godel's work changes nothing.
Watch the video - Chaitin work refutes this widely held view. Incompleteness is pervasive in mathematics. The vast, vast majority of mathematical facts are true "for no reason."
Chaitin sees the primary impact of his work to be on mathematics itself. He advocates doing math in a "physics-like" manner, i.e. creating hypotheses, performing experiments and then drawing (tentative) conclusions on the basis of the results.
Clayton,
I was referring to his section on "some things are true by accident". Obviously, there are still axioms, even if they are not expressed as mathematical equations. I believe we are in complete agreement, although you seem to have a much more fundamental grip on why you believe you are correct, whereas I am just blabbering.
Yes, random is definable as being undefinable, or something like that. It's a tough idea to take on face value. You kind of figure that randomaeity might be a human thought phenomenon rather than a mathematical concept. One dimensional cellular automata based upon 8 simple rules produces "randomness," but this obviously isn't random. Is random really unpredictable, or just to human perception?
Wolfram - Chaitin connection! I now know why Chaitin's name sounds so familliar.
Very interesting on the rest of your post. I will look more into this. Wolfram seems to have argued in the opposite direction for experimentation -> he sought to bring his principles based in the math world into the real world, while Chaitin wants to bring the scientific method into mathematics.
What I meant by Godel's work changing nothing, I was referring to applying the scientific method to nature. Godel's work didn't suggest that scientific reasoning upon nature was a waste of time or should be conducted differently. Yes, it SHOULD have quite an impact on the math world. Correct me if I'm wrong here.
meambobbo:Yes, random is definable as being undefinable, or something like that. It's a tough idea to take on face value.
Randomness is what is left when there is no more order or structure to be found. Program-length gives us an objective, but uncomputable way to "define" randomness. It's the fact that the definition is uncomputable (no program or algorithm can be written to recognize random vs. non-random numbers) that saves it from inconsistency.
You kind of figure that randomaeity might be a human thought phenomenon rather than a mathematical concept.
Well, what mathematics is not a human thought phenomenon. Think about it.
One dimensional cellular automata based upon 8 simple rules produces "randomness," but this obviously isn't random.
Well, you have to distinguish between randomness and non-determinism. It's deterministic, but that doesn't mean that it's non-random. That's the key result of Godel's, Turing's and Chaitin's work... deterministic systems generate true randomness.
Is random really unpredictable, or just to human perception?
Predictability and randomness are related but not synonymous. There's another name which Chaitin mentions in the video, that of Ray Solomonoff. Solomonoff applied the ideas of computability theory to induction and developed a theory of algorithmic prediction. Essentially, he formalized what we mean when we talk about induction. In my opinion, Solomonoff has resolved the problem of induction which is the source of much hand-wringing in philosophical circles. But that's just my opinion.
Wolfram - Chaitin connection! I now know why Chaitin's name sounds so familliar. Very interesting on the rest of your post. I will look more into this. Wolfram seems to have argued in the opposite direction for experimentation -> he sought to bring his principles based in the math world into the real world, while Chaitin wants to bring the scientific method into mathematics.
You could say that.
Here's an interesting paper by Cristian Calude, From Heisenberg to Godel via Chaitin. He argues that the Heisenberg Uncertainty Principle and logical incompleteness are directly related (you'll find the story in the opening paragraph interesting...) Mind-blowing stuff.
For anyone intersted, Peter Smith, logician and senior lecturer at Cambridge, posted in a blog the other day his notes (PDF) for his four lectures on Gödel’s theorems this term. He also says to watch this space for supplemental things such as exercises, handouts and what not. If you haven't visted his blog before, there is a wealth of resources there.
Just scanned Godel Without (Too Many) Tears... this looks like a very solid technical (but not too technical) introduction.