The Gödel Incompleteness Theorem was a major discovery in modern logic that has consistently attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regarded it as a certain logical paradox. In Gödel’s view, Wittgenstein does not understand the incompleteness theorem, and Gödel said, “He interpreted it as a logical paradox that, but in fact, on the contrary, is a mathematical theorem of an undisputed part of mathematics (limited number theory or combinatorial mathematics).” (Wang Hao, 2009, p227). Who is right?
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9It is a mathematical theorem; see e.g. Gödel’s Incompleteness Theorems– Mauro ALLEGRANZACommented Jun 4 at 9:07
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5But see e.g. Timm Lampert, Wittgenstein and Gödel: An Attempt to Make “Wittgenstein’s Objection” reasonable (2017) as well as Timm Lampert, Wittgenstein’s “notorious paragraph” about the Gödel Theorem.– Mauro ALLEGRANZACommented Jun 4 at 9:10
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17But the theorem is NOT wrong.– Mauro ALLEGRANZACommented Jun 4 at 9:56
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11No. Wittgenstein originally thought, mistakenly, that Gödel’s proof needs the natural language interpretation of his sentence as "I am unprovable" to show that it is true but unprovable (in part, due to Gödel’s own informal remark). This mistake is reflected in the "notorious paragraph" from RFM, but Wittgenstein corrected it later while maintaining his criticism of the theorem's philosophical interpretations. History and the debate over the "notorious paragraph" are detailed in Matthíasson, Interpretations of Wittgenstein's Remarks on Gödel.– ConifoldCommented Jun 4 at 10:06
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7One thing that I think helps here is to understand the context of the Incompleteness Theorem. To my (casual) understanding, the fact that mathematics allowed for statements that were paradoxical was not news. Mathematicians were working to define a mathematical foundation which did not allow statements of that type. Gödel showed that these efforts were bound to fail. The philosophical implications of that might present some issues depending on your leanings.– JimmyJamesCommented Jun 4 at 18:46
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5 Answers
Gödel was right. O'Connor 2005 meets every known objection: it is constructive and finite, indeed it runs on commodity hardware in reasonable time; it includes Rosser's trick, it is not relative to ZFC, it addresses first-order axioms, and a large body of empirical work endorses the correctness of the proof in terms of both hardware and software. Smith 2007 rebuilds Gödel's approach using all of the tools developed by his contemporaries. Lawvere 2006 explains how Gödel's theorem is a special case of Lawvere's 1967 fixed-point theorem, and how the latter theorem is proved without any self-referential techniques; Yanofsky 2003 contextualizes Lawvere's theorem by showing that many apparently-self-referential paradoxes neatly decompose as special cases.
There is an important philosophical lesson here. Sometimes a philosopher — here, Wittgenstein — is wrong. Their reputation and prior work are irrelevant. The strength of their opinion and the beauty of their justification are irrelevant. Only the evidence presented is relevant. For mathematics, formal proof is very strong evidence, and insightful analogies are even stronger evidence; Gödel provided the (first) proof, and Lawvere provided the analogy.
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Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Philosophy Meta, or in Philosophy Chat. Comments continuing discussion may be removed.– Geoffrey Thomas ♦Commented Jun 7 at 8:56
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@JulioDiEgidio-inactive: If you have any evidence against Lawvere's theorem, we have an active discussion happening in chat and you're invited to share. The thing is that I'm honestly not inclined to talk to folks who can't work through either Smith (syntactic) or Yanofsky (semantic); the entirety of logic and maths shifted from about 1880 to 1970, and we need to keep up. It's no different from the rest of postmodernism.– CorbinCommented Jun 17 at 17:39
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@JulioDiEgidio-inactive: On a personal note: I suspect that you think of Gödel deniers as relatively harmless. This is not my first conversation with such a person; I was involved in debunking the notorious sockpuppeteer Carl Hewitt and the damage he did is still being cleaned up even though he can't ever contribute again. I recommend reading Dudley 1992 or Dudley 1996 ("The Trisectors", one of my favorites.) We must meet cranks head-on.– CorbinCommented Jun 17 at 17:43
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1@JulioDiEgidio-inactive I find it hard to read past the cursing, because there's nothing there. Maybe you could say why you object to this answer instead of spouting rude nonsense. Commented Jun 18 at 11:51
You do not understand the incompleteness theorem. It does not require "coding", and it does not depend on "actual infinity", and it does not "hide" any paradox. You cannot reject it as long as you accept basic facts about finite binary strings.
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16This answer is unnecessarily combative. The OP does not assert an understanding in their question. The OP doesn't claim to know what the correct answer is. It asks a question about two claims. Whether the OP understands the theorem or not has no bearing on whether Wittgenstein is correct in his assertion about it. Commented Jun 4 at 17:58
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18@JimmyJames The comments to the question by the OP references an article written by the author of the OP which is nonsense and demonstrates a complete lack of understanding of the incompleteness theorem. As such, this answer by user21820 is perfectly justified.– BumbleCommented Jun 4 at 20:09
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2@Bumble That's a fair point. I somehow missed that the OP and the author were the same. I still think the general idea of the question is a good one i.e. "what did Wittgenstein misunderstand about this" regardless of whether the way the OP asked is problematic. Commented Jun 4 at 20:47
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5@Joshua: I said precisely what is correct in my answer. Additionally, the theorem has nothing to do with GR. Commented Jun 5 at 4:32
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7@ZhangHong: That is utter nonsense. Formal systems do not need to be set theories. And you are deliberately ignoring the fact that the incompleteness theorem has zilch to do with actual infinity. If you do not wish to learn mathematics, that's the end. Commented Jun 5 at 4:34
In an appendix to Part I of Remarks on The Foundations Of Mathematics, Wittgenstein criticized the following argument:
I imagine someone asking my advice; he says: “I have constructed a proposition (I will use ‘P’ to designate it) in Russell’s symbolism, and by means of certain definitions and transformations it can be so interpreted that it says: ‘P is not provable in Russell’s system’. Must I not say that this proposition on the one hand is true, and on the other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable.”
(Translation by G.E.M. Anscombe; can be viewed with the original German here.)
The surrounding text seems to be a reasonable takedown of that argument. That argument is what many people believe to be Gödel's argument—enough that it is worth putting some effort into showing that it's wrong—but it's not Gödel's actual argument, which doesn't depend on a notion of truth.
Whether Wittgenstein was one of those who believed that that was Gödel's argument, I don't know. He didn't explicitly say so, nor otherwise. If he believed he was criticizing Gödel's theorem in that section then he was wrong.
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"For suppose it were false; then it is true that it is provable." is this what "many people believe to be Gödel's argument"? Commented Jun 4 at 21:45
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1@JimmyJames I think so, yes. People very often say that Gödel showed a certain proposition is true but unprovable.– benrgCommented Jun 4 at 21:48
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That seems like an unreasonably shallow interpretation and that's coming from someone who is quite sure I have a very limited understanding of this. The example I anchor on is "the set of all sets that don't contain themselves" and that Gödel showed that there must be at least one such irreconcilable statement in any logic that can incorporate arithmetic. Is that roughly on target? Commented Jun 4 at 21:53
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1@JimmyJames: Not at all. It seems that ZFC does not prove the existence of "the set of all sets that don't contain themselves", and the incompleteness theorem applies to it whether or not it does. What is your mathematical background, and do you know programming? If you tell me, then I can guide you to a complete understanding of the incompleteness theorem very quickly. Commented Jun 5 at 7:20
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Wittgenstein's criticism can be summed up as "it is not true that Goedel's proof is purely syntactic, in fact it cannot be". Wittgenstein was indeed very critical to the whole Aristotelian, Fregean, Russellian approach to logic. And, for the chronicle, some even have it that in fact Wittgenstein was quite right, or at least that his objections are quite legitimate.
But, as a word of caution, one has to read and study W. himself, not his critics, at least not to begin with, in particular the mail correspondence between W. and Russell before and up to the Tractatus is quite illuminating as to the actual debates: just maybe keep in mind that W. is not any less misrepresented and misunderstood than Marx himself is...
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Wittgenstein argues that the truth of Godel's "true and unprovable" proposition is so implausible that it cannot be used except by sophistry. Because Godel's formula itself is a paradox. Commented Jun 6 at 11:44
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1"the mail correspondence between W. and Russell before and up to the Tractatus is quite illuminating..." What is the link with the question? Godel's Theorem is later than Tractatus. Commented Jun 6 at 14:26
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@MauroALLEGRANZA "Godel's Theorem is later than Tractatus": yes, but we cannot understand Wittgenstein's objection to that theorem without knowing where it's coming from, otherwise it is not even possible to understand the terms or the analogies or the problems and the goals. Commented Jun 6 at 17:32
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@ZhangHong "Because Godel's formula itself is a paradox": I don't think so, the argument and its conclusion is not so much in question as the "standard" claim that the proof is purely syntactic, which is the result that is really at stake, since, put simply, if semantics has anything to do with it, then models in which G is false are simply false models (of arithmetic): and what that in fact entails generally in terms of Model Theory and its import. Commented Jun 6 at 17:38
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@JulioDiEgidio-inactive My paper points out that there are paradoxes in the proof of Godel's incompleteness theorem. If this paradox is necessary, then the proof of Godel's theorem is wrong. We must find out the root cause of the mistake. In my opinion, any statement that refers to itself is a cycle, an actual infinity, and inevitably brings contradictions. Commented Jun 9 at 11:36
The paradox certainly exists. Because first-order logical systems are essentially like naive set theory, they lack restrictions on the definition of propositions (as arbitrary elements in naive set theory), resulting in self-contradictory definitions such as Godel's formula. Just as we need to develop naive set theory into ZFC axiomatic set theory, we also need to improve first-order logical systems in order to avoid paradoxes.