As a result of their collaboration, the first important result in this direction was achieved. Davis, Putnam, and Robinson , showed that the problem of solvability of exponential Diophantine equations is undecidable. In , Yuri Matiyasevich added the final missing piece, and demonstrated that the problem of the solvability of Diophantine equations is undecidable.
The essential technical achievement was that all semi-decidable recursively enumerable sets can be given a Diophantine representation, i.
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As there are semi-decidable recursively enumerable sets which are not decidable recursive , the general conclusion follows immediately:. This also provides an elegant variant of the incompleteness theorems dealing with Diophantine equations:. The question of avoiding the requirement of 1-consistency here is tricky; see Dyson, Jones and Shepherson The question then arises whether there are any simple and natural mathematical statements which are likewise undecidable in chosen basic theories, e. There are now various specific statements with clear mathematical content which are known to be undecidable in some standard theories though, just how natural even these are has been disputed; see Feferman b.
Some well known, natural examples are listed below, beginning with some quite natural mathematical statements which are independent of PA , and proceeding to more and more powerful theories. It is often stated that before the celebrated Paris-Harrington theorem see below , no such natural independent mathematical statements were known. This is not, however, strictly speaking, correct. Already much earlier, around , Gerhard Gentzen see the entry on the development of proof theory had provided such a statement.
It is very natural to generalize the idea of induction from the domain of natural numbers to the domain of ordinal numbers. In set theory, such generalizations are called principles of transfinite induction. Though some constructivists may be sceptical about the legitimacy of full set theory, there are limited and more concrete cases of transfinite induction only dealing with some well-defined classes of countable ordinals that are perfectly acceptable even from the constructivist or intuitionist viewpoint. Gentzen showed that the consistency of PA can be proved if this transfinite induction principle is assumed.
Therefore, because of the second incompleteness theorem, the principle itself cannot be provable in PA Gentzen This provides a quite natural statement of finite combinatorics which is independent of PA. Perhaps an even cleaner example is Goodstein's theorem, due to Reuben Goodstein , which is purely number theoretic in nature. The theorem states that every Goodstein sequence eventually terminates at 0.
Moving now to stronger theories beyond PA , one can mention, for example, Kruskal's Theorem. This is a theorem which concerns certain orderings of finite trees Kruskal Harvey Friedman showed that this theorem is unprovable even in subsystems of second-order arithmetic much stronger than PA see Simpson There are some concrete examples of mathematical statements undecided even in stronger theories which come from the so-called descriptive set theory. This field of mathematics is related to topology and was initiated by the French semi-intuitionists Lebesgue, Baire, Borel; see the section on descriptive set theory, etc.
It studies sets which possess relatively simple definitions in contradistinction to the ideas of arbitrary sets and various higher power-sets, which the semi-intuitionists rejected as meaningless called projective or analytic sets. Classically these were defined as the sets that can be built up from a countable intersection of open sets by taking continuous images and complements finitely many times; they coincide with the sets which are definable in the language of PA 2. A Borel function is defined analogously see, e.
Harvey Friedman has established the following theorem: roughly, if S is a Borel set, then there exists a Borel function f such that the graph of f is either included in or disjoint from S.
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Friedman showed that this simple-sounding theorem is not provable even in full second-order arithmetic PA 2 , but proving it necessarily requires the full power of ZFC see Simpson Further, it was a traditional question of descriptive set theory a question which can be formulated in the language of second order arithmetic whether all projective sets see above are Lebesque measurable. This remained an open problem for many decades, and for a good reason: it turned out that the statement is independent even of the full ZFC set theory see Solovay However, this case is very different.
In all the above independence results the relevant statements are still theorems of mathematics, taken as shown to be true the last case, which requires large cardinal axioms that go beyond ZFC , is more controversial; still, at least many set-theoreticians find such axioms plausible. And with the first incompleteness theorem itself, the truth of the unprovable statement easily follows, given that the assumption of the consistency of the system is indeed correct.
However, in the case of Cohen's result, there is absolutely no indication whether CH should be considered true, false, or perhaps lacking a truth-value. The possibility of incompleteness in the context of set theory was discussed by Bernays and Tarski already in , and von Neumann, in contrast to the dominant spirit in Hilbert's program, had considered it possible that logic and mathematics were not decidable.
Hilbert , on the other hand, had assumed that Peano Arithmetic and other standard theories were complete. In his attempted proof, he needed the notion of truth.
Murawski This also easily yields a weak version of the incompleteness result: the set of sentences provable in arithmetic can be defined in the language of arithmetic, but the set of true arithmetical sentences cannot; therefore the two cannot coincide. Moreover, under the assumption that all provable sentences are true, it follows that there must be true sentences which are not provable. This approach, though, does not exhibit any particular such sentence. In particular, even the notion of truth was considered as suspicious or even nonsensical at the time, at least by some logical positivists e.
This led to the incompleteness theorems in the form that they are now known. Some important figures in the field of logic and the foundations of mathematics quite quickly assimilated the results and understood their relevance, but there was also quite a lot of misunderstanding and resistance for detailed accounts of the reception, see Dawson ; Mancosu John von Neumann, who was in the audience and was at the time working in the context of Hilbert's program, immediately understood the great importance of the result.
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The word quickly started to spread of these results which apparently had great importance for the foundations of mathematics—though views on what really was the moral varied. Paul Bernays, perhaps the most important collaborator of Hilbert, showed great interest in the results, though he first had difficulties in understanding them properly.
He also suggested that his methods would be applicable to standard systems of set theory however, it was only after the satisfactory characterization of decidability and the Church-Turing thesis a few years later that it was possible to give a fully general formulation of the incompleteness theorems see above ; this was first done in Kleene Zermelo seems to have had serious difficulties in understanding the relevant concepts and results. He obtained abstract versions of incompleteness results apparently already in In particular, he observed that his methods would provide a statement undecidable in Principia Mathematica.
It was reported much later in Post In , Hilbert and Bernays' second volume of Die Grundlagen der Mathematik appeared, including a detailed proof of the second incompleteness theorem. However, in more philosophical circles, some resistance remained. The dominant initial reaction was that Wittgenstein simply failed to understand the result.
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To begin with, they pose, at least prima facie , serious problems for Hilbert's program this issue is discussed in some detail in the section on the impact of incompleteness in the entry on Hilbert's Program. Quine and Ullian , for example, consider the traditional philosophical picture that all truths could be proved by self-evident steps from self-evident truths and observation. Shortly afterwards, J. More recently, very similar claims have been put forward by Roger Penrose , John Searle has joined the discussion and partly defended Penrose against his critics.
Crispin Wright , has endorsed related ideas from an intuitionistic point of view for criticism, see Detlefsen The anti-mechanist's argument thus also requires that the human mind can always see whether or not a given formalized theory is consistent. However, this is highly implausible cf. Davis Lucas, Penrose and others have attempted to reply to such criticism see, e.
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III, in It is not a necessary consequence of incompleteness theorems. His fundamental reasons for disliking the latter alternative are much more philosophical. He argued that it is consistent with all the facts that I am indeed a Turing machine, but that I cannot ascertain which one. For some critical discussion, see Chihara and Hanson There are several introductory textbooks in mathematical logic which give a good exposition of the incompleteness theorems and related topics; for example:.
The more philosophical aspects around the incompleteness theorems are surveyed in the following two sources:. The latter also provides a very accessible informal and yet reliable explanation of the incompleteness theorems. The author would like to thank Richard Zach for his careful and valuable comments on the earlier drafts of this entry. The author and SEP editors would like to thank Richard O'Callaghan for bringing to our attention that, in an earlier version of the supplement on the Diagonalization Lemma, the definition of the substitution function, although standard, didn't suffice for the purposes of the proof.
Raatikainen uta. Introduction 1. The First Incompleteness Theorem 2. The Second Incompleteness Theorem 3. Results Related to the Incompleteness Theorems 4. Philosophical Implications—Real and Alleged 6. Barkley Rosser in , the first theorem can be stated, roughly, as follows: First incompleteness theorem Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.
A rough statement is: Second incompleteness theorem For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself. The First Incompleteness Theorem In this section, the main lines of the proof of the first incompleteness theorem are sketched. In sum, we have: The Representability Theorem In any consistent formal system which contains Q : A set or relation is strongly representable if and only if it is recursive; A set or relation is weakly representable if and only if it is recursively enumerable.
Then Cons F is not provable in F. Tarski's Undefinability Theorem Let F be a consistent formalized system that contains a sufficient amount of arithmetic. Therefore, it can be concluded: Church's Theorem First-order predicate logic is undecidable.
As there are semi-decidable recursively enumerable sets which are not decidable recursive , the general conclusion follows immediately: MRDP Theorem There is no general method for deciding whether or not a given Diophantine equation has a solution. This also provides an elegant variant of the incompleteness theorems dealing with Diophantine equations: Corollary For any 1-consistent axiomatizable formal system F there are Diophantine equations which have no solutions but cannot be proved in F to have no solutions.
He stated that given any machine which is consistent and capable of doing simple arithmetic, there is a formula it is incapable of producing as being true … but which we can see to be true. Barwise ed. There are several introductory textbooks in mathematical logic which give a good exposition of the incompleteness theorems and related topics; for example: Boolos, G. Enderton, H. Van Dalen, D. Two books that are dedicated to the incompleteness theorems are: Smullyan, R. Smith, P. Another useful book on the incompleteness theorems and related topics is: Murawski, R.
Dordrecht: Kluwer. The more philosophical aspects around the incompleteness theorems are surveyed in the following two sources: Raatikainen, P. Detlefsen ed. Awodey, S. Parrini et al. Awodey and C. Klein eds. Barzin, M. Benacerraf, P. Bezboruah, A. Boolos, G. Jeffrey, , Computability and logic , 3 rd revised edition, Cambridge: Cambridge University Press.