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Shaken Foundations or Groundbreaking Realignment? A Centennial Assessment of Kurt Gödels Impact on Logic, Mathematics and and Computer Science Computer Science John W. Dawson

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Die Arbeit über formal unentscheidbare Sätze würde wie ein Erdbeben empfunden. Sätze würde wie ein Erdbeben empfunden. Sir Karl Popper, Der wichtigste Beitrag seit Aristoteles Sir Karl Popper, Der wichtigste Beitrag seit Aristoteles Gödels theorems produced a debacle. Gödels theorems produced a debacle. … Uncertainty and doubt concerning the … Uncertainty and doubt concerning the future of mathematics have replaced the certainties and complacencies of the past … [so that] scientists must … be concerned about what mathematics can be confidently employed …. Morris Kline, Mathematics: The Loss of Certainty Morris Kline, Mathematics: The Loss of Certainty

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If we take the foregoing assess- ments at face value, we might expect the mathematical land- scape in the aftermath of Gödels incompleteness theorems to have resembled that of San Francisco ten days before his birth: If we take the foregoing assess- ments at face value, we might expect the mathematical land- scape in the aftermath of Gödels incompleteness theorems to have resembled that of San Francisco ten days before his birth:

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But others, including Gödel himself, have taken a much more optimistic view of those theorems:

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… what Gödels theorem actually tells us … can be viewed in a much more positive light, namely that the insights that are available to human mathematicians … lie beyond anything that can be formalized as a set of rules. … what Gödels theorem actually tells us … can be viewed in a much more positive light, namely that the insights that are available to human mathematicians … lie beyond anything that can be formalized as a set of rules. Sir Roger Penrose, Shadows of the Mind Sir Roger Penrose, Shadows of the Mind I have come to cherish incompleteness for the I have come to cherish incompleteness for the support it lends to mechanism …, and to Turings support it lends to mechanism …, and to Turings thesis in particular. thesis in particular. Judson Webb, Mechanism, Mentalism, and Meta- Judson Webb, Mechanism, Mentalism, and Meta- mathematics mathematics

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So what impact has Gödels work had 1. On mathematical logic? 2. On mathematics in general? 3. On computer science?

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Gödels impact on logic Simon Kochen has recalled that during his oral examination for the doctorate, Steve Kleene asked him: Name five theorems of Gödel. The point of the question was to stress how seminal Gödels work has been for the development of logic. For example:

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1. In his doctoral dissertation, Gödel stated and proved the Completeness Theorem for first-order logic (Every consistent first-order theory has a realization); and in its published revision he also proved the (countable) Compactness Theorem two cornerstones of Model Theory. two cornerstones of Model Theory.

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2. Gödels Incompleteness Theorems restricted the scope of Hilberts Proof Theory, and introduced notions (primitive recursion) and techniques of proof (arithmetization of syntax) that became central to Recursion Theory.

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3. By showing that the Axiom of Choice and Cantors Continuum Hypothesis could be assumed without contradiction, Gödels consistency results in Set Theory muted controversy over the former and (toge- ther with Cohens work) led to a search for axioms to decide the latter. Once again, Gödels proofs introduced a new concept (that of inner model).

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4. Gödels Dialectica interpretation, based on the notion of computable function of finite type, provided a new consistency proof for arithmetic and opened new directions in Intuitionistic Proof Theory. (Earlier, his negative translation of classical arithmetic into its intuitionistic counterpart arithmetic into its intuitionistic counterpart showed that if classical arithmetic is incon- showed that if classical arithmetic is incon- sistent, so is intuitionistic arithmetic.) sistent, so is intuitionistic arithmetic.)

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More generally, before Gödels work: The conception of quantifiers was in flux. (Some, e.g. Zermelo, regarded them as infinitary conjunctions.) The conception of quantifiers was in flux. (Some, e.g. Zermelo, regarded them as infinitary conjunctions.) There was distrust of semantic notions, and of proofs involving transfinite con- structions. There was distrust of semantic notions, and of proofs involving transfinite con- structions. The importance of the distinction between first-order and higher-order logic was not appreciated. The importance of the distinction between first-order and higher-order logic was not appreciated.

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Within logic, Gödels results: Helped to establish first-order logic as a tractable domain for metalogical study. Helped to establish first-order logic as a tractable domain for metalogical study. Demonstrated the connection between truth and provability in first-order theories, as well as the need for distinguishing between them in the context of arithmetic. Demonstrated the connection between truth and provability in first-order theories, as well as the need for distinguishing between them in the context of arithmetic. Showed the fruitfulness of transfinite methods in set theory, and focused the attention of set theorists on the rank hierarchy. Showed the fruitfulness of transfinite methods in set theory, and focused the attention of set theorists on the rank hierarchy.

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Philosophically, Gödels results: Showed the untenability of the logistic thesis that all of mathematics is subsumed within one all-embracing system of logic. Showed the untenability of the logistic thesis that all of mathematics is subsumed within one all-embracing system of logic. Caused a redirection of formalist efforts in proof theory away from finitary methods. Caused a redirection of formalist efforts in proof theory away from finitary methods. Clarified what sorts of consistency proofs may be meaningfully sought; and Clarified what sorts of consistency proofs may be meaningfully sought; and Led him to champion mathematical Platonism as a foundational philosophy. Led him to champion mathematical Platonism as a foundational philosophy.

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Reactions to the incompleteness theorems Initially, there was widespread conster- nation concerning their implications, and Initially, there was widespread conster- nation concerning their implications, and Many mathematicians exhibited lack of understanding of details of the proofs. Many mathematicians exhibited lack of understanding of details of the proofs.But: Acceptance followed quickly, and Acceptance followed quickly, and In their wake, controversy over competing foundational philosophies largely died down, apart from Intuitionism. In their wake, controversy over competing foundational philosophies largely died down, apart from Intuitionism.

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Outside of logic Gödels theorems attracted little notice. Then as now, most mathematicians regarded foundational issues as the province of logicians questions largely irrelevant to their own research. Gödels theorems attracted little notice. Then as now, most mathematicians regarded foundational issues as the province of logicians questions largely irrelevant to their own research. Gödels results have had hardly any effect on mathematical practice. In particular, Gödels results have had hardly any effect on mathematical practice. In particular,

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The quest for natural examples of unde- cidable arithmetical statements (those arising outside a logical context) goes on. Though progress has been made toward that goal (by Harvey Friedman and others), the results remain debatable. Many mathematicians, however, do seem to have adopted Platonism as their working philosophy.

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Works by Gödel of special relevance to computer science 1. Completeness of first-order logic (1930) 2. Incompleteness of formal number theory (1931) 3. 1934 IAS lectures on formally undecid- able statements (first published 1965) 4. On the lengths of proofs (1936)

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5. Some basic theorems on the founda- tions of mathematics, and their philo- sophical implications (Gibbs Lecture, *1951) 6. Letter to von Neumann (*1956) *First published posthumously in Gödels Collected Works in Gödels Collected Works

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Significance for computer science 1. The completeness theorem established that the rules of inference developed prior to Gödels work were adequate for deriving all logical consequences of a set of axioms. A computer incorporating only those rules could thus carry out all such derivations.

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2. The incompleteness theorem, recast by Alan Turing as the Halting Problem, established limits on what computers can do. But for computer science its proof was of greater significance, because: a. It provided a precise definition of the class of (what are now called) primitive recursive functions. b. In it, object language statements were distinguished from metalinguistic ones.

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c. It introduced the idea of representing one data type (sequences of strings) by another (numbers). In addition, as Martin Davis has remarked, the very structure of the incompleteness proof strongly resembles that of a computer program.

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3. In his 1934 lectures Gödel defined the notion of general recursive function that Alonzo Church adopted in the statement of his Thesis (1935). 4. The length-of-proof paper gave what is apparently the earliest statement of aspeed-up theorem. 5. In his Gibbs lecture, Gödel addressed the question whether the power of the human mind exceeds that of any finite machine.

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6. In his 1956 letter to von Neumann (then terminally ill with spinal cancer), Gödel posed a question equivalent to what is now called the P = NP problem the central question in theoretical computer science today. No earlier statement of that problem has been found. But the letter had no influence on the develop- ment of computer science, since it was not known until the 1990s.

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So how do the assessments of Gödels impact quoted at the outset bear up under critical scrutiny?

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Re the quotation from Popper: If the incompleteness theorems were per- ceived by some as an earthquake, its shock waves were felt most acutely by those at its epicenter: the formalists of Hilberts school (towards whom Gödel tailored his proof, in which he eschewed semantic methods). Elsewhere the reverbe- rations attenuated rapidly.

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Nota bene: Gödel himself started out as a follower of Hilberts program, not as an iconoclast. He discovered his incompleteness theorems in the course of trying to give a consistency proof for analysis relative to arithmetic proof for analysis relative to arithmetic in furtherance of Hilberts program. But he soon realized that arithmetic truth, unlike provability, is not definable in arithmetic.

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Gödel to Yossef Balas (p. 1 of unsent draft)

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Gödel to Balas (p. 2) ( truth is not expressible in the same language)

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Thus Gödel himself was among those most affected by his own discoveries. How did he respond? By adopting a Platonistic view of mathematical truth while remaining faithful to Hilberts belief that In mathe- matics there is no Ignorabimus.

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Specifically, Gödel shared the view expressed by Penrose But unlike Penrose, he did not claim that his incompleteness theorems proved the super- iority of human over machine intelligence. He only asserted that they implied a disjunc- tive conclusion:

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(From Gödels Gibbs Lecture) Either … the human mind infinitely surpas-Either … the human mind infinitely surpas- ses the powers of any finite machine, or else there exist absolutely unknowable Diophan- tine problems. Gödel thought the former more likely.

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In contrast to the quotation from Webb: Late in his life Gödel published a brief remark entitled A philosophical error in Turings work. In it he disputed Turings contention that a machine was capable of carrying out any humanly computable procedure. Turing, he said, had disregarded that mind, in its use, is not static, but con- stantly developing.

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As for Morris Klines startling assertion, we can perhaps agree that we no longer harbor some of the certainties and complacencies of the past. In particular, I suspect there are few today who share Hilberts idealism, or Klines apparent belief that there should be a single overarching theory embracing all of mathematics. But mathematics is hardly plagued by uncertainty and doubt about its future.

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Contrary to Kline Gödels results altered the mathematical landscape, but they did not produce a debacle. Gödels results altered the mathematical landscape, but they did not produce a debacle. There is less controversy today over mathematical foundations than there was before Gödels work. There is less controversy today over mathematical foundations than there was before Gödels work. There is no reason to regard mathematics as any less secure than before, and recent progress in settling long-standing ques- tions gives no cause for lamentation over its future prospects. There is no reason to regard mathematics as any less secure than before, and recent progress in settling long-standing ques- tions gives no cause for lamentation over its future prospects.

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References: Books Kurt Gödel, Collected Works (5 vols.), ed. Solomon Feferman et al. Oxford University Press, 19862003. Torkel Franzén, Inexhaustibility: A Non- exhaustive Treatment. Association for Symbolic Logic (A K Peters), 2004 Torkel Franzén, Gödels Theorem: An Incomplete Guide to its Use and Abuse. A K Peters, 2005. A K Peters, 2005.

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References: Articles J. W. Dawson, The reception of Gödels incompleteness theorems, pp. 74–95 in S.G. Shanker (ed.), Gödels Theorem in Focus, Croom Helm, 1988. J. W. Dawson, The reception of Gödels incompleteness theorems, pp. 74–95 in S.G. Shanker (ed.), Gödels Theorem in Focus, Croom Helm, 1988. P. Mancosu, Between Vienna and Berlin: The immediate reception of Gödels incompleteness theorems, History and Philosophy of Logic 20 (1999), 33–45. P. Mancosu, Between Vienna and Berlin: The immediate reception of Gödels incompleteness theorems, History and Philosophy of Logic 20 (1999), 33–45. I. Grattan-Guinness, The reception of Gödel's 1931 incompletability theorems by mathemati- cians, and some logicians, up to the early 1960s. (Vienna conference proceedings, to appear) I. Grattan-Guinness, The reception of Gödel's 1931 incompletability theorems by mathemati- cians, and some logicians, up to the early 1960s. (Vienna conference proceedings, to appear)

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