Is Kurt Gödel's Incompleteness Theorem a "cheap trick"?

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I found a throw-away critique of Kurt Gödel's Incompleteness Theorem in an essay about Deconstruction:

The basic enterprise of contemporary literary criticism is actually quite simple. It is based on the observation that with a sufficient amount of clever handwaving and artful verbiage, you can interpret any piece of writing as a statement about anything at all. The broader movement that goes under the label "postmodernism" generalizes this principle from writing to all forms of human activity, though you have to be careful about applying this label, since a standard postmodernist tactic for ducking criticism is to try to stir up metaphysical confusion by questioning the very idea of labels and categories. "Deconstruction" is based on a specialization of the principle, in which a work is interpreted as a statement about itself, using a literary version of the same cheap trick that Kurt Gödel used to try to frighten mathematicians back in the thirties.

Now this statement strikes me as perfectly apt, but not sufficiently grounded. I suspect that it's correct, but there isn't enough there to know.

So is the Incompleteness Theorem a "cheap trick" or is it a serious argument that propels philosophy forward? (I presume that the theorem is perfectly valid and valuable in mathematics where it originated.)

Jon Ericson

Posted 2011-06-08T19:18:30.400

Reputation: 6 843

It is well understood that the truth of a system cannot be ascertained within the system itself -- Could you make a metal detector out of metals alone? Godel shows that mathematics is no exception. – sova – 2011-06-17T02:38:00.410

@sova: A magnetized piece of iron is a metal detector after a fashion. Barring the example, however, that's a reasonable summary of the idea, I believe. – Jon Ericson – 2011-06-17T17:24:15.560

@Jon: +1. A very good question! with very good answers! – None – 2011-06-22T16:07:55.327

I created a relevant chat to discuss the exactly how how Gödel's Incompleteness Theorem is “cheap trick”

https://chat.stackexchange.com/rooms/109314/defining-godel-incompleteness-away

– polcott – 2020-06-15T22:00:22.737

The key "cheap trick" aspect of Gödel's Theorem is its foundational basis: A theory T is incomplete if and only if there is some sentence φ such that T ⊬ φ and T ⊬ ¬φ.

Every formal system capable of representing self-contradiction is defined as "incomplete" on the basis that it can express self-contradiction therefore making a sentence and its negation unprovable.

Instead of saying that the self-contradiction of the liar paradox sentence: "this sentence is not true" makes the liar paradox ill-formed we decide that English is "incomplete" because English can express the liar paradox. – polcott – 2020-06-18T03:10:01.990

3In the interest of improving the question, anyone care to comment on why there are downvotes to the question? I'm guessing the phrase "cheap trick" is the problem. – Jon Ericson – 2011-06-08T23:21:27.800

5No idea why there are downvotes; it's best not to pay too much attention to them. If someone has a useful opinion regarding an actionable way that your question can be improved, they'll leave a comment. Otherwise, just keep on doing what you're doing. A +1 from me. – Cody Gray – 2011-06-09T05:04:48.837

4I'm pretty sure only those who doesn't like it would call it a "cheap trick". That's a typical rationalization put forward when reality bites. – Lennart Regebro – 2011-06-09T07:48:03.413

1@thei: What I meant was a) in the context of the article, the critique of Gödel was helpful in understanding the critique of "Deconstruction" and b) the idea that Gödel Incompleteness is a trick fits with every account I've read. And of course those statements are commentary on why I ask the question. The next paragraph contains the question, which has been answered once so far. (And your question is nothing like mine.) – Jon Ericson – 2011-06-09T17:41:19.697

1I can't understand why, if it was a "cheap trick", is still used by many people to stylize their literary works, including the criticizer himself.. – johan.i.zahri – 2011-06-10T17:22:36.407

@johan.i.zahri: You probably need to read the article I linked to: it's a frontal assault on the concept of "Deconstruction". I see now that the passage I quote is not typical of the brutal attack, but just the bit where the author defines what deconstruction does. – Jon Ericson – 2011-06-10T17:26:41.323

@Jon Ericson: well isn't he himself using some "unprovable" arguments/assumption such as "Engineering and the sciences have, to a greater degree, been spared this isolation " if he himself use this "cheap trick", isn't it hypocritical of him? – johan.i.zahri – 2011-06-10T18:15:47.753

1@johan.i.zahri: I don't see how using "unprovable" arguments, if they are in fact unprovable, would be an example of the "cheap trick". The trick is to interpret something as making a statement about itself and use that interpretation to undermine the work itself. The point of the quote you pulled is that engineering and the sciences resist deconstruction because they can normally be tested against the physical world. – Jon Ericson – 2011-06-10T18:45:19.367

@Jon Ericson:wasn't it that the reason he is able to undermine the work because of the unprovabality caused by the interpertation? – johan.i.zahri – 2011-06-10T19:17:50.470

@johan.i.zahri: I see your point now. That, I suppose, would be part of his attack. But the more critical thrust of his argument seems to me that certain academic circles have become "epistemologically challenged" as Chip calls it. He defines the phrase as "a constitutional inability to adopt a reasonable way to tell the good stuff from the bad stuff." The problem is less with results as it is with methods. – Jon Ericson – 2011-06-10T19:29:19.907

4Very good job on attacking the fallacies behind the "deconstruction" movement. But unfortunately the person in question did not understand the theorem, probably. I've seen doctors in the humanities who can't even understand a simple Cartesian x-y linear graph. Instead, they are trained to despise everything "Cartesian". – Rodrigo – 2015-11-06T18:33:08.117

@To anyone: Is there any evidence in any of Hilbert's writings on logic and foundations that the 'arithmetization of mathematics' has anything to do with 'coding' in the Goedel sense? If there is, please cite the reference (I think that won't get you into any trouble with the Hilbert-Bernays project). This would go a long way in answering Jon's question. – Thomas Benjamin – 2015-11-23T23:57:54.647

@JonEricson don't understand the downvotes either – fifaltra – 2015-12-21T15:03:14.930

Answers

81

Gödel himself worried that his incompleteness theorems were a kind of cheap trick, just a hidden trivial version of the liar paradox, but using "this statement is not provable" instead of "this statement is false." So I think the question is a good very one.

And although I have huge admiration for the theorems, let me describe another sense in which the first incompleteness theorem can be viewed from the modern perspective as a cheap trick: it is just the halting problem in disguise.

Let me explain. It is comparatively easy to prove (see below) that the halting problem is undecidable, that is, there is no computable procedure that can reliably determine whether a given program/input pair will lead to a halting computation. Suppose now that T is a true theory with a computably axiomatizable list of axioms. If T were complete, then we could solve the halting problem in the following way: given program p and input x, we systematically search through all possible proofs from T of either the statement asserting that p does halt on x, or of the statement asserting that p does not halt on x. If T is true and complete, then we will eventually find such a proof on one side or the other. Thus, we will be able to say in finite time yes-or-no whether p halts on input x. This contradicts the undecidability of the halting problem. So T must not be complete after all. In other words, there will be true statements not provable in T. One can use the proof to show that there are such statements of the form, "such-and-such program does not halt on such-and-such input." The statement is true, in the sense that that program does not halt on that input, but we are unable to prove this statement in T.

This proof of the incompleteness theorem allows one to dispense with the usual arguments via the Gödel-fixed point lemma, which can sometimes be confusing, and reveals the incompleteness theorem instead simply as a version of the halting problem. Indeed, many readers may believe that the self-referential aspects of the fixed-point lemma lay at the heart of the incompleteness phenomenon, but this proof seems completely to avoid self-reference (well, it confines the self-reference aspect to the proof of the undecidability of the halting problem itself).

So what was Gödel's real achievement? Perhaps the most important idea that he had in his theorems was the arithmetization of syntax, the idea that assertions of number theory can be viewed as assertions about assertions. This idea is profound, and I used it above in the halting problem argument, in presuming that the assertion that a program halts or does not halt is expressible as a statement that might be proved or refuted in T. The arithmetization idea has now been woven completely into the modern perspective, as we all know that the philosophical articles that we write on our computer, as well as photos, music, videos and so on that we have there, are represented ultimately with zeros and ones inside the computer, and so it is an easy step for us to think of an article as really a very long sequence of bits, essentially an enormous number. And this is the essence of arithmetization.


Proof that the halting problem is undecidable. If there were a computable procedure to reliably determine whether a given program/input halts, then design a new program q that on input p first asks whether p halts on input p, and then performs the opposite behavior itself. It now follows that q halts on input q if and only if it doesn't, a contradiction.

JDH

Posted 2011-06-08T19:18:30.400

Reputation: 3 508

You wrote, "Gödel himself worried that his incompleteness theorems were a kind of cheap trick, just a hidden trivial version of the liar paradox, but using "this statement is not provable" instead of "this statement is false."..." - I think that it will be better to add references/arguments which supports the view that Gödel's incompleteness theorems were not a kind of cheap trick, just a hidden trivial version of the liar paradox. – None – 2017-07-29T07:58:56.650

3"Perhaps the most important idea that he had in his theorems was the arithmetization of syntax, the idea that assertions of number theory can be viewed as assertions about assertions." <--- This – Dennis – 2013-01-08T07:17:54.110

This is a great answer. The only problem I have with it is that Gödel's theorem (published in 1931) is described as Turing's theorem (published in 1937) "in disguise." But Gödel did not have a crystal ball, so I find this claim rather incredible.

– Kevin – 2018-07-31T18:05:51.230

2I had written, "viewed from the modern perspective...". Of course, Gödel himself did not know of the undecidibility of the halting problem at that time, and indeed, he had later expressed amazement that Turing was able to provide a satisfactory formal notion of computability, which is something that Gödel had attempted to do with his primitive recursive functions, but failed. – JDH – 2018-07-31T18:50:12.653

4Brilliant answer! Now I must find my copy of GEB and see if it refers to the halting problem. I feel that it must. In addition, your explanation of "Gödel's real achievement" satisfies the implied question about why do we still care about the man's achievements. I've accepted you answer, since it fully answers my question. – Jon Ericson – 2011-06-21T18:04:00.990

1Wow, thanks! But the other answers are good too. Meanwhile, we should track down the reference for Goedel's worries that his results were just another cheap paradox. I've heard this in talks, but I don't recall the citation. – JDH – 2011-06-21T18:07:25.703

2Yes, that would be nice to have. (I like the other answers too. I might be a bit biased toward one of them...) – Jon Ericson – 2011-06-21T18:10:09.503

I think this argument is too general, given that Godel's incompleteness theorem only applies to systems which contain arithmetic, and as the wiki notes, "simpler" systems like the Tarski axioms for Euclidean geometry are both complete and consistent. To show that Godel's incompleteness theorem for any computable extension of arithmetic's Peano axioms follows from the halting problem, I believe you first have to prove the Peano axioms are Turing-complete, which isn't trivial.

– Hypnosifl – 2020-05-11T00:17:08.293

@Hypnosifl: On the one hand, I'm skeptical that you can axiomatize a Turing machine in anything substantially less powerful than ZFC (but maybe you could do the lambda calculus instead?). On the other, I think JDH's argument still stands, because ultimately, these various theorems are all just more elaborate forms of Russell's paradox and/or Cantor's diagonal argument. – Kevin – 2020-06-13T23:41:18.547

@Kevin - It says on this page that the Peano system is computationally universal, i.e. Turing-complete. But my objection to JDH's argument was that it was too general, it just says 'Suppose now that T is a true theory with a computably axiomatizable list of axioms' but it doesn't specify that T must be a Turing-complete theory, which means there is nothing in the argument that says T can't be the system consisting of the Tarski axioms for Euclidean geometry (which is known to be complete and consistent).

– Hypnosifl – 2020-06-14T00:36:37.133

@Hypnosifl: Strictly, you are conflating Turing-completeness (a property of computational models) with the ability to model and axiomatize a Turing machine (a property of axiomatic systems). The former can compute the output of a Turing machine that halts. The latter can prove more general statements such as "Machine M halts on input I." Regardless, I still think that JDH's argument is reasonable since it's not apparently intended to be a strict mathematical proof, but rather a "compare and contrast" between Godel and Turing. – Kevin – 2020-06-14T00:41:51.507

@Kevin - Ah, my terminology may have been off--but would the idea that the Peano system is "universal" mean that for any question about whether a given algorithm halts on some input, there is a finite procedure you can use to generate a specific WFF in arithmetic, such that the question of whether that formula is provable in the Peano system is equivalent to the question of whether the original algorithm/input combo halts or not? I take your point that JDH may not have been going for a real proof, but I think if you add this kind of assumption about the universality of Peano it could be one. – Hypnosifl – 2020-06-14T03:22:12.277

The Halting Problem is also a cheap trick in that it was intentionally designed to be analogous to this question: "Will your answer to this question be no?" Instead of actually placing any limit on computation we can easily dismiss the English version as a self-contradictory error. Likewise with its Halting Problem counter-part. – polcott – 2020-06-14T23:48:16.177

@polcott: Unfortunately, the Halting Problem cannot be "cabined in" to the extent that you suggest. See for example Rice's theorem, a straightforward corollary of it (TL;DR: It's undecidable whether a given program behaves in a given manner, unless the behavior is trivial).

– Kevin – 2020-06-15T05:42:15.750

@Kevin I have spent about 12,000 hours on this since 2004. Once halting is decidable Rice fails. The cheap-trick of the halting problem counter-examples is that they simply make sure to take the opposite action of what the halt decider decides. This is precisely the same trick with the above simple English. – polcott – 2020-06-15T05:46:58.307

@polcott: In practice, most programming languages are not limited in a fashion that makes them decidable (e.g. every nontrivial type system allows casting), so what you suggest sounds more theoretical than actual to my mind. – Kevin – 2020-06-15T05:53:29.240

@Kevin I am nearly finished with writing an x86 based program that matches the Peter Linz Ĥ template at the bottom of page 319 such that Ĥ correctly decides halting on itself: http://www.liarparadox.org/Peter_Linz_HP(Pages_315-320).pdf

– polcott – 2020-06-15T05:57:40.830

@polcott: The industry is rather sluggishly adopting Rust at the moment, which is decidable in many useful ways, outside of its unsafe blocks. Given the rather long time it took for Rust to be accepted (and the fact that no other programming language has since implemented a borrow checker, to my knowledge), what makes you think your language will be more successful? To my mind, if industry shows no interest in using it, then it is of no practical consequence. It may have theoretical significance, but then, so does Turing. – Kevin – 2020-06-15T05:59:46.810

@Kevin I have shown how to decide the general halting problem counter examples with a fully operational sufficiently Turing complete language. The actual source code of the halt decider is written in "C". – polcott – 2020-06-15T06:14:13.400

Wonderful connection to the Halting problem! – Michael – 2013-10-01T00:04:55.710

I'm sure I'm not the first to have attempted to exchange the words "good" and "very" in your second sentence, only to have been thwarted by the "edits must exceed 10 chars" rule. Only you can make this edit. – samerivertwice – 2020-09-04T10:23:20.673

I'd say that the difference in the argument of this answer is that it presents an existential proof (there is a sentence G that is true but can not be proven) versus a constructive proof of that very sentence! We all know that Goedel was a constructivist / intuitionist. See https://en.wikipedia.org/wiki/Intuitionistic_logic.Perhaps. His construction hardly qualifies as a cheap trick but shows extreme deductive powers.

– Cuc – 2020-09-09T06:38:27.230

@JDH I’m reading GEB rn. I don’t think it directly mentions the halting problem. But I can’t remember (taking me months to digest it). I know Stephen Wolfram definitely draws a line between the two quite frequently – Connor McCormick – 2021-02-13T05:43:13.617

2

The undecidability of the Halting problem proves only the weaker form of the Goedel theorem, see http://en.wikipedia.org/wiki/Undecidable_problem#Relationship_with_G.C3.B6del.27s_incompleteness_theorem

– Slaviks – 2014-04-17T17:46:59.060

1xtian, I don't really agree with all your that-zapping edits, nor with your insertion of a split infinitive. But oh, well. – JDH – 2011-10-09T01:52:43.367

You say : "so it is an easy step for us to think of an article as really a very long sequence of bits, essentially an enormous number", but what matter here is not the sequence of symbol, it's the representation it will generate into a mind which intrepret it. – psychoslave – 2015-02-09T01:59:09.590

Your answer interests me. Can we say if the halting problem falls that Godel must fall with it? – Joshua – 2015-05-18T00:21:14.253

The problem comes from the halting problem being not wholly undecidable because it contains a hidden assumption. If you can find the assumption and resolve it, you will be able to see how to resolve also your paradoxical input. – Joshua – 2015-09-25T20:13:18.063

The "fixed-point lemma" is just a an easy way to find a statement that cannot be proven or disproven for some good reason. There are gazillions of possible statements, and it wouldn't be unexpected if some cannot be proven or disproven, for no particular reason whatsoever. – gnasher729 – 2016-10-19T14:16:38.850

If the arithmetization of syntax makes for such an important idea, then it would have to have some sort of rational justification in the first place. But, I will assert that Goedel numbers are constants. Each constant symbol stands for but one object over time. Some of the symbols which get arithmetized though are variables. Each variable stands for at least two objects over time. Thus, Goedel numbering is inadequate to analyze the structure of variables. Since variables are more essential to logic and mathematics than constants, the arithmetization of syntax is NOT so important. – Doug Spoonwood – 2016-10-31T23:40:32.363

66

Gödel's Incompleteness theorems are not cheap tricks in any sense of the phrase. If you want to call an ingenious method that no-one else anticipated a 'trick' then so be it - but it is in no way cheap. Let's review what Gödel proved in his two so-called incompleteness theorems. I will state the theorems informally but note that every single term in the statement has a formal and perfectly determinate counterpart:

Gödel's First Incompleteness Theorem (G1T) Any sufficiently strong formalized system of basic arithmetic contains a statement G that can neither be proved or disproved by that system.

Gödel's Second Incompleteness Theorem (G2T) If a formalized system of basic arithmetic is consistent then it cannot prove its own consistency.

Now, as I see it, you are asking two questions:

  1. Are these theorems 'serious arguments'?
  2. Do they propel philosophy forward?

The answer to both questions is yes. I answer them in turn:

  1. The argument itself is metamathematical which means that it employs a meta-language to prove things about the object language of ordinary mathematics.

    The way Gödel does this is he takes his metalanguage to be one that includes intuitive notions of arithmetic (the natural numbers) together with an understanding of what primitive recursive functions on the natural numbers are. Using this meta-language he proves that any formalization of basic arithmetic can capture its own provability relation. He first defines what is called a Gödel numbering scheme in which every formula of the language in our formalization is assigned a unique number (in our metalanguage.)

    He then proves that there is a formal one-place open formula NotProv(x) that can be interpreted to mean "x is unprovable" where x is a numeral in the formalization (remember that what is under consideration is a formal system of basic arithmetic so it will contain the equivalent of intuitive numbers, i.e. numerals) that correspond to a given sentence in the language via the Gödel numbering.

    Now, given NotProv(x) we can do what Gödel called a diagonalization, namely apply NotProv(x) to itself, i.e. take x to be the numeral corresponding to the formula NotProv(x). Call the resulting sentence G = NotProv(NotProv). And since NotProv(x) says that 'x is unprovable' you can see that G says 'I am unprovable'. And something that says it is unprovable cannot be proved nor disproved.

    This is a very quick and informal way to present the argument - one would normally have to distinguish between the semantic and the syntactic versions of the theorem. But the point is that as you can see there is serious and rigorous work going on here.

    The proof of (G2T) is similar. Using NotProv(x) you can define a 'consistency sentence' for your given formalization by writing NotProv(0=1), i.e a sentence that says 'No contradiction is provable' which is equivalent to 'This system is consistent.' And by a similar but more technically subtle argument you can argue that this sentence is unprovable, given that the system is in fact consistent.

  2. The second theorem is arguably more epoch-making than the first because it spelled the end of Hilbert's Program. This is a major philosophical shift in the philosophy of mathematics, essentially spelling the end of the philosophical school of formalism.

    Furthermore, people have argued that (G1T) proves that we can never fully capture arithmetical truth because a further consequence of (G1T) is that the sentence G is actually true and hence we can conclude that any formalization of arithmetic will contain statements which we can see are true but which are not in fact provable in that system.

    This has led people like Michael Dummett (an intuitionist) to label arithmetical truth as 'indefinitely extensible' (cf. Dummett 'The Philosophical Significance of Godel's Theorem'.) People like Lucas and Penrose have used both (G1T) and (G2T) to argue in favour of what is called an anti-mechanist thesis, i.e. that minds cannot be machines (cf. Penrose 'The Emperor's New Mind' and Lucas 'Minds, Machines and Godel'.)

In general, I have to say that the philosophical impact of both (G1T) and (G2T) cannot be overstated. They were events of monumental significance for analytic philosophy, for the philosophy and practice of mathematics as well as for theories of computation and machines. Most people (especially idiotic and ignorant continental postmodernists who have made it a sport to abuse mathematics in their pursuit of alternative vocabularies) fob them off as tricks because they have not bothered to look at the actual technical details involved and think that the idea of the proof gives them a perfect grasp of its implications. Popularizations don't really do the theorems justice.

If you are interested I recommend you go through the whole argument - the moment of revelation when it clicks together is as near an aesthetic experience as you're ever likely to have doing formal logic.

Chuck

Posted 2011-06-08T19:18:30.400

Reputation: 3 238

1Your answer was excellent, but there was one part I found difficult to understand. For the diagonalisation, I think it would be clearer to say that we are using this technique to show that there must exist a k such that NotProv(k)=k. Then, since G=NotProv(NotProv(k))=NotProv(k), G=NotProv(G) – Casebash – 2011-06-16T10:44:31.807

2It's worth emphasising the point that these theorems rely on your being able to recognise (somehow) that NotProv(x) is indeed true. That "somehow" is fascinating. – Seamus – 2011-06-16T10:52:10.630

1JUst to dispel any 'magic' that people might take away from the above description. Chuck says "...we can conclude that any formalization of arithmetic will contain statements which we can see are true but which are not in fact provable" should be followed always by "in that system". It definitely is provable in the meta-system. That is how we know it is true. – Mitch – 2011-06-17T02:12:37.057

3@Mitch I added 'in that system' even though I think it is clear from the sentence that we are talking about a particular formalization and its provability. But it is certainly not true that the Godel sentence for a system S is 'provable' in the meta-system in any way that can be compared to what we usually call a mathematical proof. Because note that the argument depends on the conditional 'If S is consistent then G is true' - and so a proof of the consistency of S would be needed. But for very many S (and arguably for all) we have no proof of consistency, merely good reasons to believe in it – Chuck – 2011-06-17T08:32:44.903

@Chuck: Foucault & others rejected the term postmodernist, and the term has been used largely pejoratively, rather positively or assertively, and by people who have not read and do not understand the significance of the continental tradition. So, boing-flip, back at you. – CriglCragl – 2020-04-22T20:55:52.103

2I find it amusing that this answer takes a swipe at postmodernist just as Chip Morningstar did taking an opposite position on Gödel. I will need to reread this answer, which seems very compelling. Thank you. – Jon Ericson – 2011-06-08T20:26:28.473

21It's not that amusing if you take into account that postmodernism is so ironically meta-clever that you can be postmodernist by rejecting postmodernism – Chuck – 2011-06-08T20:56:40.823

6

As no one else has yet taken the other side, I'll try my hand at devil's advocate. Keep in mind that I am not a mathematician so the answer will likely contain mistakes and I'm not committed to this view, but am interested in seeing the debate become a debate. Further, my understanding of Gödel's work comes largely from my reading of Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter. As such, this may be a criticism of Hofstadter's book rather than of Gödel Incompleteness. Caveat lector!

First, I accept its value in mathematics. (How can I not?) I will point out that Gödel's work did seem to end the Principia Mathematica project, which might be what the article means by "frighten[ing] mathematicians back in the thirties." In addition, the Wikipedia article on Foundations of mathematics suggests that the incompleteness theorems have diverted mathematics from Hilbert's program of formalism:

In a sense, the crisis has not been resolved, but faded away: most mathematicians either do not work from axiomatic systems, or if they do, do not doubt the consistency of ZFC, generally their preferred axiomatic system. In most of mathematics as it is practiced, the various logical paradoxes never played a role anyway, and in those branches in which they do (such as logic and category theory), they may be avoided.

Second, I accept that the Theorems are, in fact, true. For this I'm greatly indebted to GEB, which may be a popularization, but also produced in me something akin to an "aesthetic experience". The remarkable idea that a formal system can be made to evaluate itself and that such a self-referential operation implies that the system will thereby be rendered incomplete took hold of me as I read and understood it. Further, the concept seems inescapable, because it is.

So what we are left with is the application of Gödel Incompleteness outside of mathematics.
And the more that I think about it, the more that I think, "So what?" Obviously it's of great help if you are in a dialog with someone who wants to create a complete, consistent, self-validating system of thought. But as we are all postmodern in the chronological sense, that doesn't seem to be an issue all that often. And of course Gödel's work will be invaluable to those who are looking for the limits of Artificial Intellegence or who wonder if there is any mechanical model that can simulate a mind.

When I try to make sense of the ideas in the context of the intellectual landscape, I feel like I'm waking from a beautiful dream. It was profound and compelling when I was under its spell, but now I shake off drowsiness and wonder how the core idea is any different from the Epimenides paradox:

They fashioned a tomb for thee, O holy and high one
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.

– Epimenides, Cretica

Surely, an interesting puzzle, but not really something upon which to build a philosophical argument upon. Which makes me think that Gödel is often cited by non-mathematicians because he's a famous mathematician with an umlaut in his name. And that, I think we can all agree, would be a cheap trick.

Jon Ericson

Posted 2011-06-08T19:18:30.400

Reputation: 6 843

Thanks for your answer Jon. Could you explain the correlation b/w the 'All Cretans are liars' paradox and Gödel? (BTW:, GEB has been on my 'must read' list for a long time now. Perhaps the time has come to finally read it.) As I understand it, the Prophet of Crete just committed a self-referencial fallacy. In that sense it's necessarily false. You could use that as a self-contradictory statement in a logical argument to provide you with a proposition that is always false. And then could use conjunction with a true statement (perhaps a tautology) and have a conjunction that is false. – boehj – 2011-06-15T00:31:17.807

Also, I should add, that I agree wholeheartedly that poor old Gödel has been used improperly by so many that now whenever one sees the name, one should always be on higher alert. – boehj – 2011-06-15T00:34:23.530

@boehj: The way I understand the ideas in GEB, Epimenides (or perhaps St. Paul if Epimenides is referring to other Cretans) isn't so much committing a fallacy, but making a statement whose truth value cannot be determined (a paradox). To borrow from Chuck's answer: it's like finding a statement in a formal system that claims "I am unprovable". If you are determined to have a complete and consistent system, such a statement is a problem. Otherwise, it's about as troubling as an ancient Cretan claiming to be a liar. (I guess my answer fits into a comment with space to spare. ;-) – Jon Ericson – 2011-06-15T17:01:36.930

What about the impact on the unification project in physics, as expressed by Hawking? http://www.hawking.org.uk/godel-and-the-end-of-physics.html It seems strange to quote SEB, and not note GI implies minds are strange-loops rather than Turing Machines. I suggest https://en.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma as a more colloquial phrasing of Godel's insights.

– CriglCragl – 2020-04-22T21:03:32.053

The Epimenides Paradox really isn't one. Consider a more formalized phrase: "All Cretans are always liars" pronounced by a Cretan. Now, if his phrase were true, it contradicts itself. Therefore, it can't be. But can it be false? Let's see. The negation of the phrase becomes: "There is a Cretan that sometimes tells the truth." There is no contradiction in this! So, sharing an untruth, Epimenides knows that he doesn't always speak the truth, even if he tried, but that at least he is aware of it. In that sense, he is self-critiquing, but even more critiquing those who see a paradox in everything. – Cuc – 2020-09-09T06:50:23.780

3

The basic enterprise of contemporary literary criticism is actually quite simple. It is based on the observation that with a sufficient amount of clever handwaving and artful verbiage, you can interpret any piece of writing as a statement about anything at all.

This is a degeneration of Derridas Deconstruction which could be viewed as an attack on the then dominant (& stagnant) school of Structuralism or a way past it. To use a mathematical analogy: mathematics (in one sense) is about axiomatic systems, but this does not mean that any axiomatic system is of equal value. Likewise not every interpretation of a piece of writing is of equal value. Judgements of taste must still be made.

The broader movement that goes under the label "postmodernism" generalizes this principle from writing to all forms of human activity, though you have to be careful about applying this label, since a standard postmodernist tactic for ducking criticism is to try to stir up metaphysical confusion by questioning the very idea of labels and categories.

Postmodernism is a questioning and reaction of Modernism; in the same way that Romanticism was a reaction to early Modernism. From some point in the future looking back it may be seen as part of Modernism. Its really too early to say (though of course one does).

"Deconstruction" is based on a specialization of the principle, in which a work is interpreted as a statement about itself, using a literary version of the same cheap trick that Kurt Gödel used to try to frighten mathematicians back in the thirties.

Deconstruction is roughly about inverting dominant modes of interpretation, in various modes, and its not a new technique: after all Marx inverted Hegel to present a critique of Capitalism. One could say that Deconstruction is both a literary & political tool.

Godels theorem, from a mathematical logic perspective is not a cheap trick, but certainly it has been used as a cheap trick by philosophical & mathematical hustlers. Paradox & antinomies have been used by serious philosophical thinkers, such as Hegel and Kant (in passing only) in the West; and by Nagarjuna and Daoism in the East.

Godels achievement, in context, is one part of the reinvigoration of formal logic since Frege, he introduced new techniques and questions into mathematical logic. However most popular expositions miss the importance of Paradox and tying it into the larger framework of Paradoxical thought in Philosophy - they settle for an exposition of Godels proof, whereas his main ideas are explicable in fairly simple terms - as they should be - and they do not give the larger & broader picture of Mathematical Logic: categorical Logic, intuitionist logic, inconsistent mathematics, paraconsistency and so on.

There is an incredible amount of verbiage about Godels Theorem, important though it is, which should be contemplated alongside the incredible amount of verbiage around Deconstruction, important though that is.

One of the elements of Badious Programme is to prune back this verbiage & metaphysical idiocy by making mathematics the site of ontology. But one should note that his book Being & Event references the Event of Derrida in the paper he presented at Columbia University which was to consolidate Structuralism but actually became a springboard for Deconstruction.

Although, Godels Theorem is presented usually as a death-knell of Mathematical Logicism, there has been found ways past it; certain parts of his programme has been completed. For example Gentzens proof of the consistency of PA, paraconsistent logic helps overcome contradictions in the rational architecture of mathematics by localising them.

There appears to be a general tendency towards Logical Pluralism which might be considered the outcome of the Logical Monism of Hilberts programme after a century of thought.

So far from Post-Modernism being inconsequential, one can see that the grand narrative of logical monism which may be seen as part of the modernist project has become Post-Modern by moving towards Logical Pluralism. Not the One but the Multiple.

Mozibur Ullah

Posted 2011-06-08T19:18:30.400

Reputation: 1

Great answer, thanks for addressing these passages in the op. – CriglCragl – 2020-04-22T21:10:18.423

2

I think you are right : the Incompleteness Theorem is a perfectly valid mathematical result and is of GREAT value in mathematics where it originated.

Regarding his "philosophical significance" ... the discussion is impressive and the conclusion is still missing.

This - I think - is a common pattern : in XVII century the pooof of Law of Gravitation by Newton (a perfectly valid mathematical result proved from Newton's axioms (the Law of Motion) and with a good fitting with empirical evidence) give birth to a big discussion between philosophers (newtonians vs leibnitians) about the nature of force (are them really existing ?), absolute space, presence of God in the physical world ...

The same with regard to Quantum Mechanics laws and determinism, etc.

So the same hold for Godel's Theorems : INSIDE Mathematics, they give us a lot of information. OUTSIDE Mathematics, they suggest ideas regarding (for example) human mind and knowledge, but is very difficult to think that (as in previous historical examples) they can "solve" big philosophical problems.

Mauro ALLEGRANZA

Posted 2011-06-08T19:18:30.400

Reputation: 33 575

1

No it's not a cheap trick if you want to understand whether something is true or both true and provable. For example can you prove that you don't have a proof? If you can then you have a proof and the proof is false. So it might be true that there is no proof though if you try to prove it it's proving the opposite of what you want to prove.

But yes it's a cheap trick since consistency not is a sufficient feature. Consistence very well could be a necessary feature but you can make a counterexample which disproves that being consistent is enough information.

Consistency only seem to mean that you can't prove a false statement and that what you prove must also be true.

Proving yourself to be a liar by not telling the truth is an old paradox that defies the law of the excluded middle and one solution to self-references is to avoid self-references completely so that everything true can be proven and vice versa everything provable is true.

For example a tautology is true (A=A is true) but a tautology doen't prove anything. So A=A is true and doesn't prove anything. Therefore typically false statements ("Peter is not being himself" is like "A is not A") can be more provable than exact truths (A=A) due to similarity instead of equality.

Niklas R.

Posted 2011-06-08T19:18:30.400

Reputation: 570

1

I will try to analyze this argument using proposition dependency. But why must dependency of proposition? Because proposition must be associated with existences or it's meaningless, and how an existence related to other existence is through a dependency.

Proposition dependency:

  • A proposition is constructed to understand realities (existences). Existences can be perceived by us because of their functionality, therefore nodes of a proposition exist as functions.

  • Anything that exists has functionality. There are two possibilities; dependence upon something else (A->B) or "not" dependence upon something else (A|B).

  • Therefore, a 'proposition' consists of nodes of functions that form a series of dependency

    Terms:

    • Cause = (c)
    • Caused = (cd)

Liar Paradox

An example of the use of dependency of proposition can be implemented to analyze this issue, a liar paradox.

Liar paradox, "He is telling the truth that He is lying, therefore He is not lying."

Syllogism

  1. H then T = (c1) -> (cd1) (If there is him, then, there is telling something)
  2. T then Ac = (cd1) -> (cd2) (If there is telling something, then, there is action from himself)
  3. H then Ac = (c1) -> (cd2) (If there is him, then, there is action from himself - telling the truth)
  4. Ac then Ev = (cd1) -> (cd3) (If there is action from himself, then, there is another event which is never happened as he told - he is lying)
  5. H then Ev = (cd1) -> (cd3) (Therefore, If there is him, then, there is another event which is never happened as he told - he is lying)

    • "He is not lying" is not contradict with "He is lying (H then Ev), because "He is not lying" is pointing to (H then Ac).

    • H then Ac = (cd1) -> (cd2) is line with H then Ev = (cd1) -> (cd3)

Therefore there is no contradiction & paradox here.

Incompleteness Theorem

Incomplete because there is a kind of proposition that left behind to be proved.

I don't understand fully about how Godel made argumentation with his Godel's number and more, but i tried to understand the essence of what did Godel mean by incompleteness theorem. Through my simple understanding about Godel's incompleteness theorem, i tried to deepening further to see a clear distinction and put it in appropriate places.

Kurt Godel Logical Framework

Suppose there is a programming system that has ability to prove any proposition, therefore:

  1. A proposition is always provable (by a programming system)
  2. "G" is a proposition
  3. Therefore, "G" is always provable"

    • "G" is unprovable proposition
    • Therefore "unprovable proposition is provable"

Syllogism

  1. (All)P are provable
  2. G is P
  3. G is provable

    • G = unprovable proposition

Consequences

  • If G is provable then = unprovable proposition is provable = INCONSISTENT.
  • If G is unprovable then = unprovable proposition is unprovable = INCOMPLETE (because there is a proposition left behind that is unprovable)

Dependency of Proposition for Incompleteness Theorem

Now, we try to place this incompleteness theorem issue to a dependency of proposition to learn something whatever it is.

Kurt Godel Logical Framework

Syllogism

G = Unprovable Proposition

  1. (All)P then Pr = All(cd1) <- (c1)

    (If there are all propositions, then, those are provable)

  2. (several)P then G = several(cd1) -> (cd2)

    (If there are some propositions, then, several of propositions are typical G)

  3. G then Pr = (cd2) <- (c1)

    (If some of propositions are typical G, then, those are provable)

    • G = unprovable proposition

Consequences

INCONSISTENT

  • If G is provable then = unprovable proposition is provable = INCONSISTENT.

From here i will use dependency on proposition to make us see a clear distinction for possible arrangement (easier then using syllogism).

  • (several)P -> ~Pr = several(cd1) | (c1) = "unprovable proposition"

    (a proposition has no relation with provable)

  • {(several)P -> ~Pr} then Pr = several(cd1) | (c1) <-> (c1) or (c1) -> several(cd1) | (c1)

    (If there are some propositions that has no relation with provable, then, those are provable) = (If there are some propositions that has no relation with provable, then, those proposition has relation with provable)

  • From syllogism asserts that there is contradiction

  • From dependency of proposition, {several(cd1) | (c1) <-> (c1)} or {(c1) -> several(cd1) | (c1)} asserts

    • several(cd1) | (c1) <-> (c1) = several(cd1)
    • (c1) -> several(cd1) | (c1) = (c1) -> several(cd1) = several(cd1) <- (c1)

There is contradiction (according to syllogism) and there is no inconsistency here (according to DOP).

INCOMPLETE

  • If G is unprovable then = unprovable proposition is unprovable = INCOMPLETE.

From here i will use dependency on proposition to make us see a clear distinction for possible arrangement (easier then using syllogism).

  • (several)P -> ~Pr = several(cd1) | (c1) = "unprovable proposition"

    (a proposition has no relation with provable)

  • {(several)P -> ~Pr} then ~Pr = several(cd1) | (c1) | (c1) or (c1) | several(cd1) | (c1)

    (If there are some propositions that has no relation with provable, then, those are not provable) = (If there are some propositions that has no relation with provable, then, those proposition has no relation with provable)

  • From syllogism asserts that there is no contradiction

  • From dependency of proposition, {several(cd1) | (c1) | (c1)} or {(c1) | several(cd1) | (c1)} asserts

    • several(cd1) | (c1) | (c1) = several(cd1)
    • (c1) | several(cd1) | (c1) = (c1) | several(cd1) = several(cd1) | (c1)

There is no contradiction (according to syllogism) and there is no inconsistency here (according to DOP).

Electrical Circuit of Reasoning

To make this assertion clear enough to be understood, i am going to use popular example,

  • A proposition is (the light) and provable is (switching on)
  • (Unprovable proposition) is equal to (the light that can't be switched on)

  • Unprovable proposition that is provable = A light that can't be switched on was trying to be switched on

    • A light that can't be switched on was trying to be switched on, therefore no light was on.

    • The key understanding in this case, is that a system still had ability to test a connection (ability to prove, ability to send electricity), but since a target (unprovable proposition) can't be attempted to switched on, then the light (unprovable proposition, the light that can't be switched on) is still off. But it didn't assert that a system was failed to run its fully functional.

    • The failure to aware this, it's because on semantically level, one proposition to another may become ambiguous, with no clear distinction about its own barrier. But by associating it to existences (beyond semantically level). We finally found that there is no consistency and there is no incompleteness as asserted by Godel Incompleteness Theorem.

Indeed we may be understand (through another direction) the truth that if we want to make a well defined statement, then it must be completed but inconsistent and a statement is consistent but it's not complete. But Kurt Godel's theorem has no related with incompleteness and inconsistency.


Please, refer to this link for better understanding: Dependency of Proposition.

Seremonia

Posted 2011-06-08T19:18:30.400

Reputation: 730

1

It is an important challenge.

According to the Theory of Types, every proposition belongs to a certain order. In original Gödel sentence G, the order of G is ambiguous.

The original sentence

 ~(T -> G)

should be a simultaneous assertion of multiple sentences of this kind:

 ~(T -> Gn), where n is a natural number, 
             and Gn stands for "G of the nth order."

Let G stand for the series of "is not provable by T": G1, G2, G3, ..., Gn, ...

Let S stand for the correlation function "is not provable by T"

Then G2 = S(G1); G3 = S(G2); G4 = S(G3); ... Thus

 G = G1, G2, G3, G4, ... 
S;G= G2, G3, G4, G5, ...

where ; is a PM notation for applying correlation function on a series; G cannot be treated as a class because no two G's are of the same order (there is a lot of room for expansion in the theory of types, as of today and by the theory of types, I can't even utter phrases like "two G's".).

Since there is no G1, S;G and G are the same series. (I realize that G1 is a problem; if logical types can also be extended downwards ad infinitum, then this solution is complete.)

And G should be read as:

This series of statements cannot be proved by theory T.

Notice that the input "this series of statements" and the output of the above sentence are the same. It appears to be self-referential, but it is not individually; it is the same series shifted to the right; in a similar fashion, adding 1 to the series of integers results in the same series.

If the order of G is spelt out, G is a simultaneous assertion of the following sentences:

The first order statement G1 in this series cannot be proved. 
- this is the 2nd order statement G2, which is False
  because there is no G1. Gn is a statement about a statement, 
  and a statement about a statement is at least 2nd order.


The 2nd order statement G2 in this series cannot be proved.
- this is the third order statement G3, which is True 
  because we just showed G2 is false.


The 3rd order statement G3 cannot be proved.  - G4
---False, we just showed G3 is true.

    So on so forth.

See Liar's paradox in Principa Mathematica

On the other hand, if G is taken literally, it is nonsense according to the theory of types:

No proposition can say anything about itself, because the propositional sign cannot be contained in itself (that is the "whole theory of types").

Wittegenstein, Tractatus 3.332

George Chen

Posted 2011-06-08T19:18:30.400

Reputation: 2 102

2When people say that G "refers to itself" what they mean is that there is a number, the Godel number N, which occurs in it, and this number happens to be the number of the formula expressing G. In fact, even this is inaccurate, by Godel's construction N is not the number of G, but of G' which is equivalent to G (provably in PM or PA). There is no conflict with the theory of types, G and G' are well-formed formulas in PM. And Godel's proof does not depend on taking G or G' literally or otherwise, although he gave this impression in the preface to the first paper when explaining motivation. – Conifold – 2016-10-19T20:07:03.857

@Conifold - There is no such thing as "well-formed formulas" in PM; that is formalists' phraseology and makes no sense in PM. Your comment betrays that you do not have a clue about the problem. – George Chen – 2016-10-19T22:42:33.913

1

PM describes language for propositions in the Introduction, but this is not a matter of phraseology. G is neither formed nor interpreted the way you describe, it does not have itself as a constituent. Self-reference interpretations and Liar analogies come from popular expositions of Godel, not from his construction of G. Did you see Rodych's paper? He is sympathetic to Wittgenstein philosophically, and argues that his objections are not affected by this original technical misunderstanding (which he later acknowledged).

– Conifold – 2016-10-20T01:35:20.733

-1

The theorems had serious implications. They pretty much killed Logical Positivism, thus proving -- again -- that it is impossible to have a 100% rational system of beliefs (rational means explainable through logic and reason alone).

The latter was known at least since Descartes' cogito ego sum, which, strictly speaking, limited our knowledge to the existence of us ourselves (of our thinking Self) -- and, hence, completely deprived of freedom.

Fortunately, we have another option, and in practical terms, it is just as good as 100% rationality. We can fix it by making a sole and almost natural assumption about us a) being awake, and b) capable to figure it out. (so natural, few are aware there even was an assumption). Specifically, we assume is that In other words, we assume the existence of objective and explainable reality, which we all share and are a part of.

That is also Søren Kierkegaard's "leap of faith" -- it is, actually, because the objective and explainable through lógos reality was, in very ancient times, referred to as God. With that as our First Premise, we can explain our experiences through reason alone.

 

  In the beginning was the Lógos, and the Lógos was with God, and the Lógos was God. It was with God in the beginning. Through it everything was made; without it, was made nothing. In it was life, and that life was the light of men. And the light shined in the darkness, yet the darkness did not comprehend it. -- John 1:1-5

Yuri Alexandrovich

Posted 2011-06-08T19:18:30.400

Reputation: 423

-3

How Gödel's Incompleteness and Tarski Undefinability are a"Cheap trick"

14 Every epistemological antinomy can likewise be used for a similar undecidability proof. Godel, Kurt 1931. On Formally Undecidable Propositions of Principia Mathematica And Related Systems I. page:40

Antinomy (Greek αντι-, against, plus νομος, law) literally means the mutual incompatibility, real or apparent, of two laws. It is a term often used in logic and epistemology, when describing a paradox or unresolvable contradiction. https://www.newworldencyclopedia.org/entry/Antinomy

The conventional definition of incompleteness: A theory T is incomplete if and only if there is some sentence φ such that (T ⊬ φ) and (T ⊬ ¬φ).

In other words he was saying that any self-contradictory sentence will prove incompleteness and according to the definition of incompleteness he was correct.

Does it really make sense to decide that a formal system is incomplete on the basis of its inability to prove self-contradictory sentences?

polcott

Posted 2011-06-08T19:18:30.400

Reputation: 286