Goedels Incompleteness Consequences
Goedels Incompleteness ConsequencesGoedelsIncompletenessTheorem is maybe the misused theorem in the history of mathematics.
GoedelsIncompletenessTheorem has important consequences with regard to the nature of truth and the possibilities of artificial intelligence.
Or perhaps it doesn't.
[extracted from GoedelsIncompletenessTheorem]
Any axiomatic system sufficiently complex (where sufficient is along the lines of 'can contain a theory about its own theories') will contain true theories which cannot be proved using the rules of the system.
If they can't be proved or disproved then in what sense are these "theories" true? Goedel's theorem only shows that the notion of a single absolute "truth" is nonsense (actually, the notion that "truth" as commonly conceived exists at all is easily disproved). Goedel's theorem isn't some bizarre epistemological statement about a Truth Beyond Truth, or Truth Beyond Human Reckoning, as many pseudo-intellectual pseudo-philosophers claim, and as the above statement implies. 19th century philosophy of mathematics was centered around Truth. 20th century mathematics has rejected the notion entirely, but you still find people (especially pseudo-intellectuals) who have never made the paradigm shift. It doesn't help that logicians still use the word "true" while meaning something completely different by the term from what the general population means. Logicians misuse "true" instead of using 'correct'. In a very serious sense, the paradigm shift which started at the beginning of the century in mathematics still hasn't completed. The same occurs in QuantumPhysics where classical notions like the HeisenbergUncertaintyPrinciple and other nonsense still plague the field. -- RichardKulisz
Goedel's theorem does not show that the notion of a single absolute "truth" is nonsense (though that might be so anyway). The idea that it does is every bit as much a "bizarre epistemological statement", and just as untrue, as the idea that it says anything about a "Truth beyond Truth". It is also not true that 20th century mathematics has entirely rejected the notion of truth. You may argue that it should have, but as a matter of fact it has not. -- GarethMcCaughan
Can you provide examples? Especially ones where whoever is involved isn't misusing the label "truth" to refer to the notion 'correct'? And you're right; I don't distinguish between "Truth" and "truth"; I think the concept is inherently mystical even when people don't religiously capitalize it. -- rk
I don't know what you want examples of. And I'd like to know what you mean by misusing the label "truth" to refer to the notion "correct", and why you think it's a misuse, because otherwise I don't understand your restriction well enough to be confident of knowing whether I'm obeying it or not. -- GarethMcCaughan
The terms "true" and 'correct' are not the same. When a logician says that a sentence is "true", he means that it bears a particular relationship to a given, usually implicit, axiomatic system; something along the lines of "_ accurately represents part of _". To a logician, "true" is a property of the context of a sentence, an external property imposed upon the sentence, and not a property of the sentence itself (as is implied by "truth value"). Of course, this is not at all what "true" means and there is already a perfectly good word that expresses this meaning, 'correct'. The difference between the two words (that one is an internal property whereas the other is an external one) is important and dictates such things as the fact that "truth" exists while "correctness" is massively awkward (if being correct is a relation then it's more accurate to say that one is subject to it instead of possessing it) and also the fact that "true" is nonsense. Because when it comes down to it, sentences do not have any inherent property called "truth value"; it's an artifice invented by people who can't use the correct word. If "truth" were an inherent property of sentences, then a formal definition of the term 'sentence' would have to include it. -- RichardKulisz
I am rather baffled by the paragraph above. I agree that there's a useful distinction between "true within a model" or "provable by a system", and "actually true", and that the last of these feels fuzzier than the others. I have no idea why you think "correct" is better than "true" for any of these. But I think you exaggerate the difference; the variety of truth you call "internal" is always relative to something, even if only the meaning of the terms in the sentence under consideration, and everyone knows that. As for the statement that "sentences do not have any inherent property called `truth value'": whether that's, er, true, depends on what you mean by "inherent". If you mean "independent of the language the sentences are interpreted in, and the state of the bits of the world they refer to", then of course sentences don't have "inherent" truth value, and no one thinks they do, and the fact isn't very interesting. If you mean "independent of arbitrary human conventions" or something, then you may be right or wrong but something more than ProofByIntimidation would probably help your case. Finally, I don't understand at all why you're so sure that (1) if "truth" were an inherent property of sentences then a formal definition of the term "sentence" would have to include it, and that (2) it doesn't. -- GarethMcCaughan
By an inherent property I mean a property of the object itself. A car has a certain mass, shape and the ability to carry passengers. Ownership of the car is not a property tied directly to the car; it's an arbitrary concept imposed by social convention. Colour is an example of a property of physical objects that is generally believed to be inherent in them, but it isn't since it's imposed on these objects by the human neurovisual system.
A mathematical statement has predicates, logical operators and free or bound variables. Each instance of a statement has specific predicates which are part of the nature of that statement. Statements do not inherently have any "truth value" and whether a statement is "true" or not has nothing to do with the nature of the statement itself. A statement's "truth value" comes about from its relationship to an axiomatic system. Statements may be "true" in reference to one axiomatic system and "false" in another. Goedel's theorem proves that statements may not have any "truth value" at all! In particular, if I interpret statements in relation to the empty axiomatic set then no statement has any "truth value". "truth value" is an artifice invented by logicians, it has nothing to do with the nature of statements, logic or mathematics.
In English, if I say "Billy is a sleazeball." that sentence has no specific "truth value". If I say "Klaatu niktau." the concept of its "truth value" isn't even meaningful. The subject-verb-object trinity is part of sentences in all languages (though with different orders) but it doesn't provide sufficient meaning (sufficient context) to a sentence in order to assign it a "truth value".
And the reason 'correct' is so important (aside from the fact that this is what logicians mean by "true") is because it provides a simple algorithm to weed out sense from nonsense. If you can't replace "true" with 'correct' in a sentence then that's because it is used in a mystical manner. The word 'correct' isn't open to such abuse. -- RichardKulisz
There's an ambiguity here. If by "a mathematical statement" you mean a closed sentence in some formal theory, then I think all mathematicians would agree that to say that such a thing is or isn't "true" is at best sloppy (but convenient) usage. Formulae are just strings of symbols. But consider the English statement "The number 7 is prime". Do you think it's "bullshit nonsense" to say that that is "true" (or, if you insist, "correct")? What about the statement "There is no way to arrange 7 pebbles in a rectangular array with more than one pebble along each side"? (That might want a little tightening up; please assume it done.) What about the statement "WIBBLE is a theorem of system FOO", where FOO is a formal system generally regarded as modelling number theory and WIBBLE is a sentence in that system corresponding to "7 is prime"? -- GarethMcCaughan
I think the statement "The number 7 is prime." is interpreted within a specific (implicit) axiomatic system and that when you add this system to the sentence (and tighten it) then you get a closed sentence in a formal theory. And that is what makes it a correct statement to me. Now, if your statements are about theorems and the systems that contain them, then these statements are still merely correct, incorrect or neither (as opposed to "true") since they are again interpreted within a specific (implicit) axiomatic system. Saying that the statement "WIBBLE is a theorem of system FOO" is a meta-statement of system FOO is just a sloppy way of saying that it is not a statement of system FOO but a statement of a system that strictly contains FOO. If the system FOO did not exist as part of the context of the discussion, then the statement would be incorrect. Statements are always part of some system even if people leave the system implicit (to save on time and sanity) or even deny the system entirely. -- rk
It would have been nice if you'd answered the question about pebbles; the point there is that mathematics is not just a matter of pushing symbols about in formal systems, but also relates in some (hard to describe precisely) manner to the world around us. Does your last statement mean that you think every statement ever made - even ones that have nothing at all to do with mathematics - ought to be considered as parts of strictly formalized systems, and that the only reason for not thinking about them that way is economy of thought?
It's an oversimplification to say that colour is something we impose on objects, just as it's an oversimplification to say that colour is an inherent property of objects. An object's absorption and reflection of light of different wavelengths is as objective a property (or, if you prefer, set of properties) of that object as one could ask for. What we all "colour" is a particular sample, so to speak, of that property. If you accept the principle "If something is an intrinsic property of an object, then so is anything that is completely determined by that something" then that means that "colour" is intrinsic.
If one accepts that principle then colour is not an intrinsic property of physical objects. The source of light determines colour at least as much as the optical properties of the object. There's a reason I said 'mass' and not 'weight' above.
The source of light determines how the object appears to us. The colour of the object isn't the same thing as how it appears.
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Once again, all this is an oversimplification because different people have slightly different colour senses; but (e.g.) the distinction between "bright red" and "bright green" isn't so very subjective, even if it mightn't be an interesting distinction to creatures with different visual systems. -- GarethMcCaughan
If you're looking at variability (and it is a decent first approximation) then colour varies more than weight and less than "truth". -- rk
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LOL. I gave up on chronological sequencing a long time ago and topical sequencing will need a refactoring.
One other thing. You originally claimed that 20th-century mathematics has rejected the notion of truth. I still haven't seen any justification of that claim. 20th-century mathematicians continue (on the whole) to regard theorems as true, whether or not they use that particular word. One has only to listen to the conversations of mathematicians to realize this. (I oversimplify a little. Many mathematicians would regard a theorem that depends on something like the Axiom of Choice or the Axiom of Dependency as something less than simply "true". But if you can find me more mathematicians who don't regard the Prime Number Theorem as true than I can find who do so regard it, I'll be very surprised indeed. -- GarethMcCaughan
Platonism is dead nowadays and truth is the quintessential Platonic concept. It doesn't matter whether or not mathematicians still think in Platonic terms during their day to day work as long as they never publish anything that would betray their beliefs on the subject, just like it doesn't matter whether or not scientists personally believe in some invisible omnipotent alien as long as they never publish anything on the subject. You seem to want to just listen to what mathematicians say; well, that standard sucks. There's a sharp difference between what people believe and the Party Line which they assiduously obey. As an insider, it would be difficult to see the difference.
Lots of mathematicians nowadays abuse the word "true" to mean 'correct' because they don't realize, they don't want to realize, and ultimately they just don't care, how different the philosophy of mathematics was a mere century ago. To most of them, there can't have been anything fundamentally different in the beliefs of 19th century math and since mathematicians used the word "true" then, they now blindly follow the tradition. This in no way implies that they mean the same thing by the word.
From a formal point of view, there is absolutely nothing wrong with a contradictory axiomatic system. Indeed, it's impossible to create a "meta-axiom of mathematics" which rules out contradictions from consideration as "true". It is impossible to say that contradictory systems are "less true" a priori (as you imply mathematicians believe). Rather, it's possible to say that the unique contradictory system is uninteresting. It is even possible to say that it is not a part of physical reality. When mathematicians say that the prime number theorem is "true", they mean only that it is part of an axiomatic system that is part of reality. But this all gets into the privileged position of physical reality, which is metaphysics, which is philosophy, which is definitely not something your average mathematician gives a damn about.
What's important is that whatever system is the basis for what mathematicians mean by something being "more true" than something else, that system is axiomatic and thus subject to Goedel's theorem. So in order to maintain the opening statement of this page (that Goedel's theorem proves axiomatic systems contain unprovable "true" statements), you have to buy in to Penrose's idiot rantings about the human mind transcending the limitations of "mere machines".
-- RichardKuliszI don't think Platonism is dead, however much you would like it to be. (Some varieties are certainly dead; I don't think anyone believes in Plato's version any more.) I think it would be easy to see the difference between believing something and "assiduously obeying" a party line from "inside", and I find it odd that you apparently believe that doing mathematics is a disqualification for thinking about these things. ("As an insider, it would be difficult to see the difference.")
Ideally, I'd want someone with qualifications both in philosophy and mathematics to handle philosophy of mathematics. But if I have to choose then I'll pick philosophy, just like I'd pick history as more important if we were talking about history of mathematics. Of course, I don't much care for qualifications (not least because I don't have them) but I think they indicate the frame of mind to bring to the subject. As a group, mathematicians disqualify themselves by not caring about the philosophy of their subject.
Well, OK. I happened to see, the other day, a book consisting of the proceedings of a conference held in 1995 or thereabouts entitled something like "Truth in Mathematics". I think it's clear that involvement in such a conference indicates that one cares about both mathematics and its philosophy. The first sentence of the first chapter of the book (after the purely prefatory matter) begins: "The concept of truth occupies a central place in mathematics and its philosophy". The book was published by a major scientific publisher, which at least suggests that the conference wasn't organized by some group of offbeat weirdos. I think this suggests that the notion of truth is not dead in either mathematics or philosophy of mathematics.
Additionally, people like to fool themselves and the more their own beliefs diverge from the Party Line, the more they will fool themselves. But in any case, I concede that all kinds of (should be) obsolete ideas reach the public.
Regarding both Platonism in specific and philosophy of math in general, what do mathematicians believe now anyways?
I think it varies a lot from mathematician to mathematician. Some are (more or less) outright platonists. Some are outright formalists. Most practicing mathematicians don't think at all about the metaphysics of mathematics, but still think of mathematical statements being "true" or "false".
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There is no such thing as "the unique contradictory system", and it is hard to take pronouncements about the implications of mathematical logic seriously when they are interspersed with this sort of blunder.
All contradictory axiomatic systems are identical; you can prove and disprove anything within any of them. The fact that it is unique also explains why nobody bothers investigating contradictory systems; as soon as you reach a contradiction, you know that the properties of the system are well-known. Ahhh, of course; do you mean that the contradictory system can be characterized by many different sets of axioms? But if you have two systems from which you can derive the exact same set of statements then I think that whether A or B is an axiom is ultimately arbitrary; a social nicety among mathematicians. -- rk
Different systems can have different languages. Inconsistent systems with different languages are not identical; they need not even be isomorphic. Some systems in which it's possible to do a lot of mathematics don't have the property that "ex falso quodlibet", so it's not true that in all contradictory systems one can prove everything. I was not referring only to the fact that one can choose different axiomatizations for systems with the same sets of theorems.
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When mathematicians say that PNT is "true" they do not mean that any particular axiomatic system is "part of reality". They might mean that the statements provable in that system are "part of reality", but that isn't the same. (The point is that there's no attempt to claim a privileged position for the axioms.)
That depends on your conception of 'reality'. If you conceive of reality as an axiomatic system (possibly because that's the only definition of the term that doesn't lead to deciding properties of reality a priori and thus into Big Trouble epistemologically) then the axioms that prove PNT might literally be part of reality. -- rk
I don't think it's true that viewing "reality" as an axiomatic system is the only approach that doesn't lead to big trouble. It's clear that without considerable extension it isn't a sufficient approach; there are relations between statements other than purely logical ones, after all. (On the whole I prefer to avoid the term "reality" altogether, though I'm not strict about this.)
Even if you try to handle all-of-reality axiomatically, you're going to find it necessary to have different axioms with very different statuses. Consider the "laws" of (1) logic and (2) physics, and also the axioms describing particular contingent facts about the world that will be necessary if the axioms are going to be sufficient to model everything. It seems unfortunate to treat all three in exactly the same way. At least, it seems so to me.
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I don't think it's at all clear that when mathematicians say that mathematical statements are "true" they are thinking of some underlying axiomatic system. (They might be thinking with one, but that's entirely different.)
How is it different?
By "thinking with one" I mean: perhaps their brains are working according to some underlying axiomatic system, which is probably appallingly complicated and whose theorems certainly don't correspond in any simple way to the theorems accepted by the mathematicians. By "thinking of one" I mean: believing that ultimately the world should be thought of as an axiomatic system, and seeing their work as discovering portions of that system.
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Your alleged paraphrase of my "contention" distorts it severely, as I am certain you realize. Please do not do this if you want the discussion to continue. (I feel that I ought to add that I do not agree with Penrose's arguments about non-computability.) -- GarethMcCaughan
Kurt Goedel's theorem also proves (indirectly) that algorithms don't (not just can't) simulate intelligence. [No it doesn't, see below for rebuttal]
One nifty implication of this theorem is to dispel any illusion that the purpose of science is to find *the truth*, meaning a formal, self-consistent, accurate model of everything.
Why is that? Even ignoring the dubious inference that GoedelsIncompletenessTheorem really maps so directly to science, why can't science still have the purpose of finding *the truth*? Microsoft's purpose is to have 100% market share in every market for every product and service everywhere on Earth, even though they probably won't achieve that.
Another way of thinking about it is that science can always attempt to find better and better ways of explaining 'how things work' even though the Radical Skeptic can still dismiss these claims as unprovable. Some epistemologists (forgot who, fix later) even claim we may have moral obligation to do so, but that's a side issue.
Heck - science never claimed to be able to prove anything, just to be able to disprove things. GoedelsIncompletenessTheorem doesn't change a thing.
I always believed that Mathematics was a human construct that closely modeled the real world, in which case we can make it anything we want, including consistent as it is invented and mutable. You seem to believe that it is a property of the universe and therefore is both discoverable and immutable. I'm not a mathematician, but the last time I checked this debate was still raging, so your last 'Clear' statement doesn't appear to be too clear. -- BryanDollery (lurking nit-picker)
Well, if we can redefine "consistent" to mean "like mathematics" or redefine "mathematics" to have no content beyond "0=0", then of course we can "make mathematics consistent". I don't think even the most thoroughgoing conventionalist would find either of those options acceptable, though.
It seems to me that for something (whether a human construct or anything else) to deserve the name "mathematics", it needs to have certain properties - I don't feel competent to decide exactly what they are! - and this restricts our freedom to "make it consistent". Similarly, we aren't at liberty to define "consistent" just any way we please. From where I'm sitting, I think these restrictions tie us down too much for it to be likely that we can choose whether mathematics is consistent or not.
It may well be in our power to choose whether certain areas of mathematics are consistent (suppose someone discovered an inconsistency in the most commonly used varieties of set theory; then we'd probably just change the axioms and try again), but that doesn't seem like the same thing as choosing whether mathematics itself is consistent. Possibly this is because actually asking whether mathematics is consistent is meaningless, and we should really be asking about particular well-defined mathematical theories instead of asking vaguely about mathematics in general, as if there were universal agreement on what mathematics is! -- GarethMcCaughan
What exactly do you mean by a "mathematical theory"? Mathematics is a problem domain (a bunch of questions) against which mathematicians create axiomatic systems. Goedel's theorem talks about the relationship between these systems and the problem domain. It says that no finite system of axioms can resolve (answer 'yes' or 'no') all the questions in the extremely ill-defined problem domain called "mathematics". This is because solutions to questions in mathematics are used to create more questions, including self-referential and circular questions. -- RichardKulisz
Goedel's theorem is not really about the relationship between the formal systems and the problem domain, and your summary of what it says is misleading. Goedel's theorem says that any axiomatic system with such-and-such properties can resolve all questions within a particular very well-defined problem domain. (What problem domain depends on what axiomatic system you take.)
I didn't have any very precise definition of "mathematical theory" in mind (there is a formal notion of "theory" used by mathematical logicians, but I wasn't intending anything quite so rigid). I was just pointing out that if you're going to ask "is X consistent?" then you need to be sure that you know what the scope of X is, and "mathematics" is too fuzzy a concept for that to be true. "Euclidean geometry" or "Zermelo-Fraenkel set theory" are nearer the mark. -- GarethMcCaughan
That doesn't surprise me. My own judgement is that the Incompleteness theorem is a very technical and/or trivial statement. I know little about the actual statement of Goedel's theorem, just that people like to use it to bolster the most ridiculous nonsense;
Most people don't even distinguish between the problem domain of mathematics and the axiomatic systems of mathematics, and that dooms them to incomprehension of Goedel's theorem. Could you explain what properties of axiomatic systems are important and how the problem domain is defined?
As for set theory, I know that everyone calls it a theory, and I do too, but it doesn't seem to be consistent with the everyday notion of 'theory'. A scientific theory always involves some supposition about reality, usually "this model represents reality", but mathematical structures don't possess any such thing. -- RichardKulisz
I'm not sure what's going on in this discussion. You're making a lot of accusations, but I'm not sure whom you're accusing (me? some other person or persons writing here? "the public"?), and I'm not sure how to respond to someone saying (more or less) "I think this is trivial but I don't know anything much about it".
My fault (and again ;-) I'm just doing exposition so you could say I'm accusing the public). A lot of people think that Goedel's theorem has something deep to say about the nature and human attainability of "truth". They also assume that "truth" is meaningful as a concept and that it exists. Insofar as the theorem says anything about "truth", it says "revolutionary" and "earth-shattering" things about it. But since I don't accept that "truth" exists, or is even very meaningful, that makes the theorem comparatively trivial.
I don't know why you think it makes the theorem trivial. It may perhaps make it uninteresting to you, but that's a different matter entirely. The statement "There is no formal system that enables us to assign truth values mechanically to all statements of elementary number theory in a consistent way" seems to me to be highly non-trivial and very interesting, whether or not it's interpreted in terms of "truth". Perhaps this is because I'm a mathematician. If by "trivial" you mean "not very interesting to most people who aren't mathematicians" then you might be right.
As a technical result, Goedel's theorem is comparatively trivial. What makes it trivial is that what the above says about number theory is obvious if you believe that number theory is at least as complex as geometry (re: the ParallelLinesPostulate). At least "number theory can't be resolved by a finite set of axioms" is new and non-trivial. -- rk
I did know the basics of Goedel's theorem but not much beyond that. I know that the theorem rests on constructing a sentence that is "undecidable" but as much as people seem to be in awe of this feat, that's simply not a big deal. Every axiom in every axiomatic set is "undecidable" from the other axioms. If it were otherwise, the axiom would be redundant or the system inconsistent. So what the theorem does is prove that you can always generate new axioms. Ho hum. And that's why I maintain Goedel's theorem is ... unremarkable at best. Now, if the theorem specifies the exact conditions under which you can generate new axioms indefinitely then that's an important result, but only a technical achievement instead of a fundamental one.
If you think it's uninteresting that "you can always generate new axioms", then I suggest that that's a fact about you much more than about Goedel's theorem. If it seems unremarkable now, that's at least partly because Goedel's theorem has been known for something like 70 years. And I'll tell you something else, that Shakespeare bloke was really full of cliches. :-)
Quantum mechanics is both a historical achievement and a fundamental one. I have no problem saying that Goedel's theorem is an important historical achievement but I have doubts about its being a fundamental one. -- rk
And actually, this is why Goedel's theorem shows "truth" to be nonsense; because every time you construct a Goedel sentence, you're given another choice (either accept the sentence or one of its negations as an axiom) and your set of "true" statements changes yet again. And since the choice you made was arbitrary in the first place, it means your new set of "true" statements is arbitrary. But then people go on to claim that "obviously that statement is true even if it can't be proved" and they don't realize that the reason they consider the statement "true" is because they're resolving it using their own reasoning, their own implicit internal axiomatic system instead of the given system. -- rk
Interjection (ie doesn't follow what chronology exists) - the above is quite a bit of an overstatement! Goedel's theorem does not show Truth to be nonsense (though it may still be), it shows axiomatization of Truth to be nonsense. But if you wanted, you could take truth to be composed of many different axiomatic systems - that is, just say that there are two different types of "real" things, Fk(k)-sets and ~Fk(k)-sets, just like there are two different types of "real" geometry (and then wonder which one the universe acts like)... that is to say that instead of having an absolute truth composed of statements, you could have one composed of implications. It's a little strange to make it absolutely true that you can't have any absolute truths... -- JoshuaGrosse
The idea that "every time you construct a Goedel sentence, you're given another choice" has no obvious justification. In what sense can we "choose" whether or not a formal system is consistent? -- TorkelFranzen
Responding dialog has been saved and removed. In essence, TF claimed that truth must always arise from within the context of an interpretation, and the standard interpretation of T precluded ~G. He repeatedly failed to comment on why this was any different then demanding geometry to be Euclidean, and I don't think he will, because by the end he was only being self-admittedly insulting. He did, though, leave these links:
INVITATION: For a number of reasons, a wiki page is not a very good place to conduct a discussion of this kind. Please feel free to open the subject in sci.logic or sci.math or one of the philosophy groups, where we can deal with it in a more orderly and coherently documented fashion. -- TorkelFranzen
The people I've talked to about Goedel's theorem understand very well that if they consider the Goedel sentence to be "true", it's because of their own reasoning rather than the given system. That's the whole point.
[The immediately following text was written earlier than the immediately preceding text. The chronology here is becoming incomprehensible. It ought to be fixed, but probably not until the presently-active discussion is done with.]
"Fixed" as in cemented in place, or "fixed" as in refactored? Constantly fixing chronology either way on a wiki is like trying to chase water with a butterfly net.
Discussion stimulator: what if one treats "reality" as a wholly self-contained form of mathematical notation describing itself? -- Julian_Morrison
What on Earth does it mean for mathematical notation to be "wholly self-contained"? -- JoshuaGrosse
I don't know, but if it has any relation to Goedel's incompleteness theorem, it must mean something like this: "There is a subset of it that can symbolize the whole system". Like the logical system of the integers if "wholly self-contained", and you can assign Goedel numbers to sentences (thus it symbolizes the system within itself). Any mathematical system that is self-contained in this sense has a statement that is true (I will explain what that means later) but can't be proved - that is essentially Goedel's theorem. The statement in the system of integers goes like "The Goedel number of this statement cannot be created by transformations of the Goedel numbers of the axioms [meaning it cannot be derived from the axioms]". If it is provable, than it is "correct" is some sense of the word, so it cannot be proved, and we have a contradiction. If it isn't provable, it is "correct", because the assertion in the statement is correct. Therefore the statement is correct, but cannot be proved. A more formal definition of "correct" of "true" goes like this: a statement is true if, when we add it to the axioms we don't have a contradiction, and if we add the statement's negation to the axioms, we have a contradiction. I hope it helps someone understand the real meaning of the Theorem, though I wrote it a little unclearly. -- AmirLivne
Any discussion herein is within the realm of someone's conceptive powers, since we would have to be everything to think about everything (as our senses are limited and there is at least an infinite amount of information unavailable to us. [multiple infinities is a condition of existence, as between 1 and 2 you can parse the amount into an infinite number of splices, and there are an infinite number of numbers and one can add something to infinity to make it even bigger] Then whatever we deem to think is limited. I think it's wise to instead of insisting on positing something that is beyond your or my limited intelligence, powers of observation, or even what has been observed such as truth it's better to say "this seems to be true based on what < I or [ collective] we > know at the moment. Unless we have complete knowledge we cannot make complete truth. Due to the nature of things there will never be end to what can be learned, therefore truth, as a total conserved doctrine, as we know it is incomplete and arguments about it are pointless. -- Stewart Alexander
Why pointless? How does that follow?
GoedelsIncompletenessTheorem has no significant consequences at all. There's one tangible consequence: mathematicians and computer scientists now know that any formal system has limitations in completeness or in expressiveness and they have to live with it.
There should be another consequence: philosophers should realize the same. Instead materialists, idealists, physicalists and whatever they call themselves still ridicule each other, calling the theories of the others incomplete or inconsistent or nonsense or any combination thereof. The incompleteness theorem shouts at them that the world is incomplete and turing machines sometimes don't halt and we cannot know it in general and they simply don't listen and continue on their futile quest for a theory, the non-existence of which has already been proved.
Consequences? Yeah, in a rational world, maybe. Not here, though.
#goedel.py
def stmt(s):
print s
return stmt(not s)
#"this statement is false"
stmt(False)
No no no; that's the CretanParadox or RussellParadox, which is not the same as Goedel's theorems.
Looks to me like a StackOverflow... unless you're on StacklessPython, in which case it runs forever...
Or if one assumes tail recursion optimization, and in any case the intention is plain.
Which reminds me, CortlandKlein? has found the best blonde joke ever; see his blog: http://pixelcort.com/165 -- get back to me after you finish reading the joke.
According to http://securebar.secure-tunnel.com/cgi-bin/nph-freebar.cgi/110110A/http/www.cs.auckland.ac.nz/CDMTCS/chaitin/unm2.html on the BerrysParadox : 'Gödel said, "It doesn't matter which paradox you use." He had used a paradox called the liar paradox'. According to http://en.wikipedia.org/wiki/Liar_paradox, that is precisely the one above. If true, it was the basis of his incompleteness theorem. And the above shows the consequences of not having one. :-)
What's your point? I said that the paradox is not the same thing as Goedel's theorems, and I said that because the above implied otherwise by being named "goedel.py". You seem to be violently agreeing.
It's also false that you (or most of the people on this page) could use any paradox; Goedel could, because he knew what he was up to. Most people would just replace his proof with the paradox and think they'd done what Goedel had done.
There was never any suggestion that it was the same. This page is about consequences and explorations of the concept. You have no idea regarding my capabilities, so it is premature for you to make claims about I could or could not use. Not that I am claiming any but you need to make a distinction between the Theorems and their proofs. Still, because the Theorems themselves are concerned with truth and provability, they are inextricably entwined.
As I said, the fact that the above code was labelled "goedel.py", rather than e.g. "LiarParadox?.py", is in fact the suggestion that the paradox and Goedel's theorems are the same, and that was the misconception I was correcting, and I don't see anything relevant to that in anything you've said; your comments all appear to be a non-sequitur to my point.
The only reason that it some sense it doesn't matter which paradox is used, is because it all boils down to the same thing, as used by Goedel. It makes a great deal of difference what sort of statement is constructed. There's a frequent confusion that Goedel merely constructed a statement that was independent of the axioms underlying the embedding system, which would mean that that either that statement or its inverse could equally well be added to the system and still result in a consistent system. But that's not what he did.
''My argument is appropriate to your point because the liar paradox (in the form of "this statement is not provable" using GoedelNumber? encodings of statements and their proofs from arithmetic) was originally the approach Goedel took to prove the incompleteness theorem. The fact that other paradoxes can be used is an aside and not my main point; the paradox I tried to illustrate in the original code was what Goedel actually used. It might not be evident in the final form of the theorem, but anyone familiar with the proof would know that it is related. Your initial point gave the impression there is no connection whatsoever. To go from "no, no, no" to blonde jokes was for me a good example of 'from a contradiction you can derive anything', it would make anyone smile. The original code was a result of an analogy I saw recently using a card with contradictory statements on each side to illustrate the ambiguous nature of the paradox. I tried to translate this to a simple program and thought I would share it, not putting it in the main GoedelsTheorem page precisely because it is just an experiment. Since the theorems and their proofs are also related to the HaltingProblem, I thought the fact that it is infinitely recursive that might also foster some discussion, which it did regarding stackless python. Compare http://www.cs.auckland.ac.nz/CDMTCS/chaitin/georgia.html Maybe I could have named it better, that should have been your initial point. But to dismiss it as completely unrelated really has no basis (and the fact that that is why the program does not terminate is ironic). My origininal point, which was not meant to be anything profound, was to share a few lines of code that might make both the theorems and the idea around their proofs, which is quite complex, a little more accessible. For 'most of the people on this page' ''
For cryin' out loud..."Maybe I could have named it better, that should have been your initial point". That WAS my point, or an equivalent restatement, anyway.
"But to dismiss it as completely unrelated really has no basis" -- don't be ridiculous, I didn't anywhere say that paradox and Goedel's Theorems are unrelated!
As for "The original code was a result of an analogy I saw recently using a card with contradictory statements on each side to illustrate the ambiguous nature of the paradox. I tried to translate this to a simple program and thought I would share it" -- you don't seem to realize that those two-sided cards are a truly ancient and stale joke, well-known simply as a joke to zillions of non-technical people, and that the code you offered is also incredibly ancient and well-known. How could something so simple be otherwise?
"I thought the fact that it is infinitely recursive that might also foster some discussion" -- infinite recursion, on this or any topic, is once again far too much of an ancient stale subject to expect it to provoke discussion. It's just not interesting.
"My origininal point, which was not meant to be anything profound, was to share a few lines of code that might make both the theorems and the idea around their proofs, which is quite complex, a little more accessible. For 'most of the people on this page'" -- you failed. And for you to think any such thing simply makes it all the more questionable that you understand the theorems as well as you think you do. The Liar's paradox is trivial to get the basic point. Goedel's theorems are difficult to understand. Have you worked through (an English translation of) them?
"To go from "no, no, no" to blonde jokes was for me a good example of 'from a contradiction you can derive anything'" -- you are very tiresome. You obviously didn't follow the link to see the joke; it was not what you thought it was, so you missed the point, and the humor.
Look, this is all pointless. Rename your code and move on.
Yes it is pointless but since you are liberal with your opinions I will just add. 1 - you are equally tiresome. 2 - I maintain that the code is correctly named as you could add GoedelNumber? and other related classes to futher explore the proof, theorems and concepts. The first statement was just a start. Will I promise to add more? No. Maybe I did fail even putting up what I did but it did show you did not know quite as much as you thought either. Yes, yes, yes.
People focusing on the apparently paradoxical aspects of the incompleteness theorem often seem to miss the primary implication of the theorem, which relates to the problem Goedel was working on. It all actually comes down to the goal of automated theorem proving, one of the key goals of the HilbertProgram?. Goedel was trying to find an algorithm or set of algorithms by which all theorems of a given formal system could be generated, while discarding all false assertions (or alternately, a test by which any proposed theorem could be proven or disproven - which was incorrectly believed to be the same thing, and is also what connects it to the HaltingProblem).
What Goedel found instead was a proof that no such algorithm is possible for any formal system which (if I understand the strength criterion correctly) is capable of expressing Peano arithmetic.
The paradox aspect isn't that strange; proof or disproof by contradiction is a basic technique in mathematical logic. What he showed, however, was that certain assertions cannot be resolved as true or false, and more importantly, that any method for mechanically generating theorems would either find at least one such assertion, or else would be unable to generate at least one true theorem of the system. -- JayOsako
Interrelationship with GoedelsIncompletenessTheorem , GeneralSystemsTheory? and Human's InterSubjectivity.: go to InterSubjectivity
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