Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system containing basic arithmetic. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e., an algorithm) is capable of proving all truths about the arithmetic of the natural numbers. For any such formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.
Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.
Formal systems: completeness, consistency, and effective axiomatization
The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and effectively axiomatized, these concepts being detailed below. Particularly in the context of first-order logic, formal systems are also called formal theories. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation (or rules of inference) that allow for the derivation of new theorems from the axioms. One example of such a system is first-order Peano arithmetic, a system in which all variables are intended to denote natural numbers. In other systems, such as set theory, only some sentences of the formal system express statements about the natural numbers. The incompleteness theorems are about formal provability within these systems, rather than about "provability" in an informal sense.
There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.
A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2004, p. 112).
This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC).
The theory known as true arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent, and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.
A set of axioms is (syntactically, or negation-) complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms (Smith 2007, p. 24). This is the notion relevant for Gödel's first Incompleteness theorem. It is not to be confused with semantic completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his completeness theorem, Gödel proved that first order logic is semantically complete. But it is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone: for example, "the flower is pretty".
In a mere system of logic it would be absurd to expect syntactic completeness. But in a system of mathematics, thinkers such as Hilbert had believed that it is just a matter of time to find such an axiomatization that would allow one to either prove or disprove (by proving its negation) each and every mathematical formula.
A formal system might be syntactically incomplete by design, such as logics generally are. Or it may be incomplete simply because not all the necessary axioms have been discovered or included. For example, Euclidean geometry without the parallel postulate is incomplete, because some statements in the language (such as the parallel postulate itself) can not be proved from the remaining axioms. Similarly, the theory of dense linear orders is not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order. The continuum hypothesis is a statement in the language of ZFC that is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue.
The theory of first-order Peano arithmetic is consistent, has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus, by the first incompleteness theorem, Peano Arithmetic is not complete. The theorem gives an explicit example of a statement of arithmetic that is neither provable nor disprovable in Peano's arithmetics. Moreover, this statement is true in the usual model. Moreover, no effectively axiomatized, consistent extension of Peano arithmetic can be complete.
A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms, and inconsistent otherwise.
Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an inaccessible cardinal" proves ZFC is consistent because if κ is the least such cardinal, then Vκ sitting inside the von Neumann universe is a model of ZFC, and a theory is consistent if and only if it has a model.
If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent.
Additional examples of inconsistent theories arise from the paradoxes that result when the axiom schema of unrestricted comprehension is assumed in set theory.
Systems which contain arithmetic
The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of Robinson arithmeticQ. Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems.
The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers. A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.
The system of Presburger arithmetic consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication.
Dan Willard (2001) has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; these systems are consistent and capable of proving their own consistency (see self-verifying theories).
In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers (Smith 2007, p 2). In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called the principle of explosion), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a maximal set of non-contradictory theorems (Hinman 2005, p. 143).
The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the in ZFC + "there exists an inaccessible cardinal" theory unresolved continuum hypothesis.
The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.
First incompleteness theorem
Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I". The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (1936) using Rosser's trick. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated.
First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015)
The unprovable statement GF referred to by the theorem is often referred to as "the Gödel sentence" for the system F. The proof constructs a particular Gödel sentence for the system F, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any logically valid sentence.
Each effectively generated system has its own Gödel sentence. It is possible to define a larger system F’ that contains the whole of F plus GF as an additional axiom. This will not result in a complete system, because Gödel's theorem will also apply to F’, and thus F’ also cannot be complete. In this case, GF is indeed a theorem in F’, because it is an axiom. Because GF states only that it is not provable in F, no contradiction is presented by its provability within F’. However, because the incompleteness theorem applies to F’, there will be a new Gödel statement GF′ for F’, showing that F’ is also incomplete. GF′ will differ from GF in that GF′ will refer to F’, rather than F.
Syntactic form of the Gödel sentence
The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in F. However, the sequence of steps is such that the constructed sentence turns out to be GF itself. In this way, the Gödel sentence GF indirectly states its own unprovability within F (Smith 2007, p. 135).
To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.
Thus, although the Gödel sentence refers indirectly to sentences of the system F, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation (Smith 2007, p. 141). As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at level of the arithmetical hierarchy). Via the MRDP theorem, the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables (Franzén 2005, p. 71).
Truth of the Gödel sentence
The first incompleteness theorem shows that the Gödel sentence GF of an appropriate formal theory F is unprovable in F. Because this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (Smoryński 1977 p. 825; also see Franzén 2004 pp. 28–33). For this reason, the sentence GF is often said to be "true but unprovable." (Raatikainen 2015). The truth of the sentence GF may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as primitive recursive arithmetic, which proves the implication Con(F)→GF, where Con(F) is a canonical sentence asserting the consistency of F (Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403).
Although the Gödel sentence of a consistent theory is true as a statement about the intended interpretation of arithmetic, the Gödel sentence will be false in some nonstandard models of arithmetic, as a consequence of Gödel's completeness theorem (Franzén 2005, p. 135). That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false. As described earlier, the Gödel sentence of a system F is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system F, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (Raatikainen 2015, Franzén 2005, p. 135).
Relationship with the liar paradox
Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F." The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.
It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski.
Extensions of Gödel's original result
Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions.
Gödel demonstrated the incompleteness of the system of Principia Mathematica, a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results.
Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but ω-consistent. A system is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate P such that for every specific natural number m the system proves ~P(m), and yet the system also proves that there exists a natural number n such that P(n). That is, the system says that a number with property P exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. J. Barkley Rosser (1936) strengthened the incompleteness theorem by finding a variation of the proof (Rosser's trick) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.
Second incompleteness theorem
For each formal system F containing basic arithmetic, it is possible to canonically define a formula Cons(F) expressing the consistency of F. This formula expresses the property that "there does not exist a natural number coding a formal derivation within the system F whose conclusion is a syntactic contradiction." The syntactic contradiction is often taken to be "0=1", in which case Cons(F) states "there is no natural number that codes a derivation of '0=1' from the axioms of F."
Gödel's second incompleteness theorem shows that, under general assumptions, this canonical consistency statement Cons(F) will not be provable in F. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". In the following statement, the term "formalized system" also includes an assumption that F is effectively axiomatized.
Second Incompleteness Theorem: "Assume F is a consistent formalized system which contains elementary arithmetic. Then ." (Raatikainen 2015)
This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the system F itself.
There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of F as a formula in the language of F. There are many ways to express the consistency of a system, and not all of them lead to the same result. The formula Cons(F) from the second incompleteness theorem is a particular expression of consistency.
Other formalizations of the claim that F is consistent may be inequivalent in F, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent subset of PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration).
The Hilbert–Bernays conditions
The standard proof of the second incompleteness theorem assumes that the provability predicate ProvA(P) satisfies the Hilbert–Bernays provability conditions. Letting #(P) represent the Gödel number of a formula P, the derivability conditions say:
- If F proves P, then F proves ProvA(#(P)).
- F proves 1.; that is, F proves that if F proves P, then F proves ProvA(#(P)). In other words, F proves that ProvA(#(P)) implies ProvA(#(ProvA(#(P)))).
- F proves that if F proves that (P → Q) and F proves P then F proves Q. In other words, F proves that ProvA(#(P → Q)) and ProvA(#(P)) imply ProvA(#(Q)).
There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert—Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic.
Implications for consistency proofs
Gödel's second incompleteness theorem also implies that a system F1 satisfying the technical conditions outlined above cannot prove the consistency of any system F2 that proves the consistency of F1. This is because such a system F1 can prove that if F2 proves the consistency of F1, then F1 is in fact consistent. For the claim that F1 is consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction in F1". If F1 were in fact inconsistent, then F2 would prove for some n that n is the code of a contradiction in F1. But if F2 also proved that F1 is consistent (that is, that there is no such n), then it would itself be inconsistent. This reasoning can be formalized in F1 to show that if F2 is consistent, then F1 is consistent. Since, by second incompleteness theorem, F1 does not prove its consistency, it cannot prove the consistency of F2 either.
This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic (PA). For example, the system of primitive recursive arithmetic (PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA cannot prove the consistency of PA. This fact is generally seen to imply that Hilbert's program, which aimed to justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out (Franzén 2004, p. 106).
The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually provide no interesting information if a system F proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of F in F would give us no clue as to whether F really is consistent; no doubts about the consistency of F would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a system F in some system F’ that is in some sense less doubtful than F itself, for example weaker than F. For many naturally occurring theories F and F’, such as F = Zermelo–Fraenkel set theory and F’ = primitive recursive arithmetic, the consistency of F’ is provable in F, and thus F’ cannot prove the consistency of F by the above corollary of the second incompleteness theorem.
The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that can be formalized in the system that is proved consistent. For example, Gerhard Gentzen proved the consistency of Peano arithmetic in a different system that includes an axiom asserting that the ordinal called ε0 is wellfounded; see Gentzen's consistency proof. Gentzen's theorem spurred the development of ordinal analysis in proof theory.
Examples of undecidable statements
See also: List of statements independent of ZFC
There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).
Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense.
Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the philosophy of mathematics.
The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice). These results do not require the incompleteness theorem. Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC.
In 1973, Saharon Shelah showed that the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory.
Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound c such that no specific number can be proved in that system to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.
Undecidable statements provable in larger systems
These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.
In 1977, Paris and Harrington proved that the Paris–Harrington principle, a version of the infinite Ramsey theorem, is undecidable in (first-order) Peano arithmetic, but can be proved in the stronger system of second-order arithmetic. Kirby and Paris later showed that Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic.
Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism. The related but more general graph minor theorem (2003) has consequences for computational complexity theory.
Relationship with computability
See also: Halting problem § Relationship with Gödel's incompleteness theorems
The incompleteness theorem is closely related to several results about undecidable sets in recursion theory.
Stephen Cole Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that the halting problem is undecidable: there is no computer program that can correctly determine, given any program P as input, whether P eventually halts when run with a particular given input. Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. This method of proof has also been presented by Shoenfield (1967, p. 132); Charlesworth (1980); and Hopcroft and Ullman (1979).
Franzén (2004, p. 73) explains how Matiyasevich's solution to Hilbert's 10th problem can be used to obtain a proof to Gödel's first incompleteness theorem. Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x1, x2,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong system of arithmetic T will prove this. Moreover, if the system T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem. Moreover, for each consistent effectively generated system T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980).
Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274).
Chaitin's incompleteness theorem gives a different method of producing independent sentences, based on Kolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include statements that are false in the standard model; these theories are known as ω-inconsistent.
Proof sketch for the first theorem
Main article: Proof sketch for Gödel's first incompleteness theorem
The proof by contradiction has three essential parts. To begin, choose a formal system that meets the proposed criteria:
- Statements in the system can be represented by natural numbers (known as Gödel numbers). The significance of this is that properties of statements—such as their truth and falsehood—will be equivalent to determining whether their Gödel numbers have certain properties, and that properties of the statements can therefore be demonstrated by examining their Gödel numbers. This part culminates in the construction of a formula expressing the idea that "statement S is provable in the system" (which can be applied to any statement "S" in the system).
- In the formal system it is possible to construct a number whose matching statement, when interpreted, is self-referential and essentially says that it (i.e. the statement itself) is unprovable. This is done using a technique called "diagonalization" (so-called because of its origins as Cantor's diagonal argument).
- Within the formal system this statement permits a demonstration that it is neither provable nor disprovable in the system, and therefore the system cannot in fact be ω-consistent. Hence the original assumption that the proposed system met the criteria is false.
Arithmetization of syntax
The main problem in fleshing out the proof described above is that it seems at first that to construct a statement p that is equivalent to "p cannot be proved", p would somehow have to contain a reference to p, which could easily give rise to an infinite regress. Gödel's ingenious technique is to show that statements can be matched with numbers (often called the arithmetization of syntax) in such a way that "proving a statement" can be replaced with "testing whether a number has a given property". This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. The same technique was later used by Alan Turing in his work on the Entscheidungsproblem.
In simple terms, a method can be devised so that every formula or statement that can be formulated in the system gets a unique number, called its Gödel number, in such a way that it is possible to mechanically convert back and forth between formulas and Gödel numbers. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is the way in which English is stored as a sequence of numbers in computers using ASCII or Unicode:
- The word is represented by 72-69-76-76-79 using decimal ASCII, i.e. the number 7269767679.
- The logical statement is represented by 120-061-121-032-061-062-032-121-061-120 using octal ASCII, i.e. the number 120061121032061062032121061120.
In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the statement does or doesn't have a given property. Because the formal system is strong enough to support reasoning about numbers in general, it can support reasoning about numbers that represent formulae and statements as well. Crucially, because the system can support reasoning about properties of numbers, the results are equivalent to reasoning about provability of their equivalent statements.
Construction of a statement about "provability"
Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this.
A formula F(x) that contains exactly one free variable x is called a statement form or class-sign. As soon as x is replaced by a specific number, the statement form turns into a bona fide statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number n, F(n) is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as "2×3=6".
Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form F(x) can be assigned a Gödel number denoted by G(F). The choice of the free variable used in the form F(x) is not relevant to the assignment of the Gödel number G(F).
The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement p, one may ask whether a number x is the Gödel number of its proof. The relation between the Gödel number of p and x, the potential Gödel number of its proof, is an arithmetical relation between two numbers. Therefore, there is a statement form Bew(y) that uses this arithmetical relation to state that a Gödel number of a proof of y exists:
- Bew(y) = ∃ x ( y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y).
The name Bew is short for beweisbar, the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described. Note that "Bew(y)" is merely an abbreviation that represents a particular, very long, formula in the original language of T; the string "Bew" itself is not claimed to be part of this language.
An important feature of the formula Bew(y) is that if a statement p is provable in the system then Bew(G(p)) is also provable. This is because any proof of p would have a corresponding Gödel number, the existence of which causes Bew(G(p)) to be satisfied.
The next step in the proof is to obtain a statement which, indirectly, asserts its own unprovability. Although Gödel constructed this statement directly, the existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement form F there is a statement p such that the system proves
- p ↔ F(G(p)).
By letting F be the negation of Bew(x), we obtain the theorem
- p ↔ ~Bew(G(p))
and the p defined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula.
The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number of p itself. This is similar to the following sentence in English:
- ", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable.
This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence indirectly asserts its own unprovability. The proof of the diagonal lemma employs a similar method.
Now, assume that the axiomatic system is ω-consistent, and let p be the statement obtained in the previous section.
If p were provable, then Bew(G(p)) would be provable, as argued above. But p asserts the negation of Bew(G(p)). Thus the system would be inconsistent, proving both a statement and its negation. This contradiction shows that p cannot be provable.
If the negation of p were provable, then Bew(G(p)) would be provable (because p was constructed to be equivalent to the negation of Bew(G(p))). However, for each specific number x, x cannot be the Gödel number of the proof of p, because p is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof of p), but on the other hand, for every specific number x, we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation of p is not provable.
Thus the statement p is undecidable in our axiomatic system: it can neither be proved nor disproved within the system.
In fact, to show that p is not provable only requires the assumption that the system is consistent. The stronger assumption of ω-consistency is required to show that the negation of p is not provable. Thus, if p is constructed for a particular system:
- If the system is ω-consistent, it can prove neither p nor its negation, and so p is undecidable.
- If the system is consistent, it may have the same situation, or it may prove the negation of p. In the later case, we have a statement ("not p") which is false but provable, and the system is not ω-consistent.
If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either p or "not p" as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formula Bew(x) is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement p, different from the previous one, which will be undecidable in the new system if it is ω-consistent.
Proof via Berry's paradox
George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently discovered by Saul Kripke (Boolos 1998, p. 383). Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388).
Computer verified proofs
See also: Automated theorem proving
The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural language intended for human readers.
Computer-verified proofs of versions of the first incompleteness theorem were announced by Natarajan Shankar in 1986 using Nqthm (Shankar 1994), by Russell O'Connor in 2003 using Coq (O'Connor 2005) and by John Harrison in 2009 using HOL Light (Harrison 2009). A computer-verified proof of both incompleteness theorems was announced by Lawrence Paulson in 2013 using Isabelle (Paulson 2014).
Proof sketch for the second theorem
The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system itself.
Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be proved from within the system itself. The demonstration above shows that if the system is consistent, then p is not provable. The proof of this implication can be formalized within the system, and therefore the statement "p is not provable", or "not P(p)" can be proved in the system.
But this last statement is equivalent to p itself (and this equivalence can be proved in the system), so p can be proved in the system. This contradiction shows that the system must be inconsistent.
Discussion and implications
The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system of formal logic to define their principles.
Consequences for logicism and Hilbert's second problem
The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "Modern viewpoints on the status of the problem").
Minds and machines
Main article: Mechanism (philosophy) § Gödelian arguments
Authors including the philosopher J. R. Lucas and physicist Roger Penrose have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.
Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.
Avi Wigderson (2010) has proposed that the concept of mathematical "knowability" should be based on computational complexity rather than logical decidability. He writes that "when knowability is interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."
Although Gödel's theorems are usually studied in the context of classical logic, they also have a role in the study of paraconsistent logic and of inherently contradictory statements (dialetheia). Graham Priest (1984, 2006) argues that replacing the notion of formal proof in Gödel's theorem with the usual notion of informal proof can be used to show that naive mathematics is inconsistent, and uses this as evidence for dialetheism. The cause of this inconsistency is the inclusion of a truth predicate for a system within the language of the system (Priest 2006:47). Stewart Shapiro (2002) gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.
Appeals to the incompleteness theorems in other fields
Appeals and analogies are sometimes made to the incompleteness theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including Torkel Franzén (2004); Alan Sokal and Jean Bricmont (1999); and Ophelia Benson and Jeremy Stangroom (2006). Bricmont and Stangroom (2006, p. 10), for example, quote from Rebecca Goldstein's comments on the disparity between Gödel's avowed Platonism and the anti-realist uses to which his ideas are sometimes put. Sokal and Bricmont (1999, p. 187) criticize Régis Debray's invocation of the theorem in the context of sociology; Debray has defended this use as metaphorical (ibid.).
After Gödel published his proof of the completeness theorem as his doctoral thesis in 1929, he turned to a second problem for his habilitation. His original goal was to obtain a positive solution to Hilbert's second problem (Dawson 1997, p. 63). At the time, theories of the natural numbers and real numbers similar to second-order arithmetic were known as "analysis", while theories of the natural numbers alone were known as "arithmetic".
Gödel was not the only person working on the consistency problem. Ackermann had published a flawed consistency proof for analysis in 1925, in which he attempted to use the method of ε-substitution originally developed by Hilbert. Later that year, von Neumann was able to correct the proof for a system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistency proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound (Zach 2006, p. 418, Zach 2003, p. 33).
In the course of his research, Gödel discovered that although a sentence which asserts its own falsehood leads to paradox, a sentence that asserts its own non-provability does not. In particular, Gödel was aware of the result now called Tarski's indefinability theorem, although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel and Waismann on August 26, 1930; all four would attend a key conference in Königsberg the following week.
The 1930 Königsberg conference was a joint meeting of three academic societies, with many of the key logicians of the time in attendance. Carnap, Heyting, and von Neumann delivered one-hour addresses on the mathematical philosophies of logicism, intuitionism, and formalism, respectively (Dawson 1996, p. 69). The conference also included Hilbert's retirement address, as he was leaving his position at the University of Göttingen. Hilbert used the speech to argue his belief that all mathematical problems can be solved. He ended his address by saying,
- For the mathematician there is no Ignorabimus, and, in my opinion, not at all for natural science either. ... The true reason why [no one] has succeeded in finding an unsolvable problem is, in my opinion, that there is no unsolvable problem. In contrast to the foolish Ignoramibus, our credo avers: We must know. We shall know!
This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "Wir müssen wissen. Wir werden wissen!", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face (Dawson 1996, p. 72).
Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930 (Dawson 1996, p. 70). Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by Monatshefte für Mathematik on November 17, 1930.
Gödel's paper was published in the Monatshefte in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions in Principia Mathematica and Related Systems I"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the Monatshefte; the prompt acceptance of the first paper was one reason he changed his plans (van Heijenoort 1967:328, footnote 68a).
Generalization and acceptance
Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency, if the Gödel sentence was changed in an appropriate way. These developments left the incompleteness theorems in essentially their modern form.
Gentzen published his consistency proof for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent.
The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of Grundlagen der Mathematik (1939), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem.
Paul Finsler (1926) used a version of Richard's paradox to construct an expression that was false but unprovable in a particular, informal framework he had developed. Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability, and had only a superficial resemblance to Gödel's work (van Heijenoort 1967:328). Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization (Dawson:89). Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career.
In September 1931, Ernst Zermelo wrote Gödel to announce what he described as an "essential gap" in Gödel's argument (Dawson:76). In October, Gödel replied with a 10-page letter (Dawson:76, Grattan-Guinness:512-513), where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system (which is not true in general by Tarski's undefinability theorem
"Mechanistic materialism" redirects here. It is not to be confused with Mechanistic Materialism.
Mechanism is the belief that natural wholes (principally living things) are like complicated machines or artifacts, composed of parts lacking any intrinsic relationship to each other. Thus, the source of an apparent thing's activities is not the whole itself, but its parts or an external influence on the parts.
The doctrine of mechanism in philosophy comes in two different flavors. They are both doctrines of metaphysics, but they are different in scope and ambitions: the first is a global doctrine about nature; the second is a local doctrine about humans and their minds, which is hotly contested. For clarity, we might distinguish these two doctrines as universal mechanism and anthropic mechanism.
There is no constant meaning in the history of philosophy for the word Mechanism. Originally, the term meant that cosmological theory which ascribes the motion and changes of the world to some external force. In this view material things are purely passive, while according to the opposite theory (i. e., Dynamism), they possess certain internal sources of energy which account for the activity of each and for its influence on the course of events; These meanings, however, soon underwent modification. The question as to whether motion is an inherent property of bodies, or has been communicated to them by some external agency, was very often ignored. With a large number of cosmologists the essential feature of Mechanism is the attempt to reduce all the qualities and activities of bodies to quantitative realities, i. e. to mass and motion. But a further modification soon followed. Living bodies, as is well known, present at first sight certain characteristic properties which have no counterpart in lifeless matter. Mechanism aims to go beyond these appearances. It seeks to explain all "vital" phenomena as physical and chemical facts; whether or not these facts are in turn reducible to mass and motion becomes a secondary question, although Mechanists are generally inclined to favour such reduction. The theory opposed to this biological mechanism is no longer Dynamism, but Vitalism or Neo-vitalism, which maintains that vital activities cannot be explained, and never will be explained, by the laws which govern lifeless matter.
— "Mechanism" in Catholic Encyclopedia (1913)
The older doctrine, here called universal mechanism, is the ancient philosophies closely linked with materialism and reductionism, especially that of the atomists and to a large extent, stoic physics. They held that the universe is reducible to completely mechanical principles—that is, the motion and collision of matter. Later mechanists believed the achievements of the scientific revolution had shown that all phenomena could eventually be explained in terms of 'mechanical' laws, natural laws governing the motion and collision of matter that implied a thorough going determinism: if all phenomena could be explained entirely through the motion of matter under the laws of classical physics, then even more surely than the gears of a clock determine that it must strike 2:00 an hour after striking 1:00, all phenomena must be completely determined: whether past, present or future. (One of the philosophical implications of modern quantum mechanics is to cast doubt on this view of determinism.)
The French mechanist and determinist Pierre Simon de Laplace formulated the sweeping implications of this thesis by saying:
We may regard the present state of the universe as the effect of the past and the cause of the future. An intellect which at any given moment knew all of the forces that animate nature and the mutual positions of the beings that compose it, if this intellect were vast enough to submit the data to analysis, could condense into a single formula the movement of the greatest bodies of the universe and that of the lightest atom; for such an intellect nothing could be uncertain and the future just like the past would be present before its eyes.
— Pierre Simon Laplace, A Philosophical Essay on Probabilities
One of the first and most famous expositions of universal mechanism is found in the opening passages of Leviathan by Thomas Hobbes (1651). What is less frequently appreciated is that René Descartes was a staunch mechanist, though today, in Philosophy of Mind, he is remembered for introducing the mind–body problem in terms of dualism and physicalism.
Descartes was a substance dualist, and argued that reality was composed of two radically different types of substance: extended matter, on the one hand, and immaterial mind, on the other. Descartes argued that one cannot explain the conscious mind in terms of the spatial dynamics of mechanistic bits of matter cannoning off each other. Nevertheless, his understanding of biology was thoroughly mechanistic in nature:
- "I should like you to consider that these functions (including passion, memory, and imagination) follow from the mere arrangement of the machine’s organs every bit as naturally as the movements of a clock or other automaton follow from the arrangement of its counter-weights and wheels." (Descartes, Treatise on Man, p.108)
His scientific work was based on the traditional mechanistic understanding that animals and humans are completely mechanistic automata. Descartes' dualism was motivated by the seeming impossibility that mechanical dynamics could yield mental experiences.
Isaac Newton ushered in a much weaker acceptation of mechanism that tolerated the antithetical, and as yet inexplicable, action at a distance of gravity. However, his work seemed to successfully predict the motion of both celestial and terrestrial bodies according to that principle, and the generation of philosophers who were inspired by Newton's example carried the mechanist banner nonetheless. Chief among them were French philosophers such as Julien Offray de La Mettrie and Denis Diderot (see also: French materialism).
The thesis in anthropic mechanism is not that everything can be completely explained in mechanical terms (although some anthropic mechanists may also believe that), but rather that everything about human beings can be completely explained in mechanical terms, as surely as can everything about clocks or the internal combustion engine.
One of the chief obstacles that all mechanistic theories have faced is providing a mechanistic explanation of the human mind; Descartes, for one, endorsed dualism in spite of endorsing a completely mechanistic conception of the material world because he argued that mechanism and the notion of a mind were logically incompatible. Hobbes, on the other hand, conceived of the mind and the will as purely mechanistic, completely explicable in terms of the effects of perception and the pursuit of desire, which in turn he held to be completely explicable in terms of the materialistic operations of the nervous system. Following Hobbes, other mechanists argued for a thoroughly mechanistic explanation of the mind, with one of the most influential and controversial expositions of the doctrine being offered by Julien Offray de La Mettrie in his Man a Machine (1748).
Today, as in the past, the main points of debate between anthropic mechanists and anti-mechanists are mainly occupied with two topics: the mind — and consciousness, in particular — and free will. Anti-mechanists argue that anthropic mechanism is incompatible with our commonsense intuitions: in philosophy of mind they argue that unconscious matter cannot completely explain the phenomenon of consciousness, and in metaphysics they argue that anthropic mechanism implies determinism about human action, which (they argue) is incompatible with our understanding of ourselves as creatures with free will. Contemporary philosophers who have argued for this position include Norman Malcolm and David Chalmers.
Anthropic mechanists typically respond in one of two ways. In the first, they agree with anti-mechanists that mechanism conflicts with some of our commonsense intuitions, but go on to argue that our commonsense intuitions are simply mistaken and need to be revised. Down this path lies eliminative materialism in philosophy of mind, and hard determinism on the question of free will. This option is popular with some scientists, but it is rejected by most philosophers[who?], although not by its most well-known advocate, the eliminative materialist philosopher Paul Churchland. Some have questioned how eliminative materialism is compatible with the freedom of will apparently required for anyone (including its adherents) to make truth claims. The second option, common amongst philosophers who adopt anthropic mechanism, is to argue that the arguments given for incompatibility are specious: whatever it is we mean by "consciousness" and "free will," they urge, it is fully compatible with a mechanistic understanding of the human mind and will. As a result, they tend to argue for one or another non-eliminativist physicalist theories of mind, and for compatibilism on the question of free will. Contemporary philosophers who have argued for this sort of account include J. J. C. Smart and Daniel Dennett.
Some scholars have debated over what, if anything, Gödel's incompleteness theorems imply about anthropic mechanism. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church-Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.
Gödelian arguments claim that a system of human mathematicians (or some idealization of human mathematicians) is both consistent and powerful enough to recognize its own consistency. Since this is impossible for a Turing machine, the Gödelian concludes that human reasoning must be non-mechanical.
However, the modern consensus in the scientific and mathematical community is that actual human reasoning is inconsistent; that any consistent "idealized version" H of human reasoning would logically be forced to adopt a healthy but counter-intuitive open-minded skepticism about the consistency of H (otherwise H is provably inconsistent); and that Gödel's theorems do not lead to any valid argument against mechanism. This consensus that Gödelian anti-mechanist arguments are doomed to failure is laid out strongly in Artificial Intelligence: "any attempt to utilize (Gödel's incompleteness results) to attack the computationalist thesis is bound to be illegitimate, since these results are quite consistent with the computationalist thesis."
One of the earliest attempts to use incompleteness to reason about human intelligence was by Gödel himself in his 1951 Gibbs lecture entitled "Some basic theorems on the foundations of mathematics and their philosophical implications". In this lecture, Gödel uses the incompleteness theorem to arrive at the following disjunction: (a) the human mind is not a consistent finite machine, or (b) there exist Diophantine equations for which it cannot decide whether solutions exist. Gödel finds (b) implausible, and thus seems to have believed the human mind was not equivalent to a finite machine, i.e., its power exceeded that of any finite machine. He recognized that this was only a conjecture, since one could never disprove (b). Yet he considered the disjunctive conclusion to be a "certain fact".
In subsequent years, more direct anti-mechanist lines of reasoning were apparently floating around the intellectual atmosphere. In 1960, Hilary Putnam published a paper entitled "Minds and Machines," in which he points out the flaws of a typical anti-mechanist argument. Informally, this is the argument that the (alleged) difference between "what can be mechanically proven" and "what can be seen to be true by humans" shows that human intelligence is not mechanical in nature. Or, as Putnam puts it:
Let T be a Turing machine which "represents" me in the sense that T can prove just the mathematical statements I prove. Then using Gödel's technique I can discover a proposition that T cannot prove, and moreover I can prove this proposition. This refutes the assumption that T "represents" me, hence I am not a Turing machine.
Hilary Putnam objects that this argument ignores the issue of consistency. Gödel's technique can only be applied to consistent systems. It is conceivable, argues Putnam, that the human mind is inconsistent. If one is to use Gödel's technique to prove the proposition that T cannot prove, one must first prove (the mathematical statement representing) the consistency of T, a daunting and perhaps impossible task. Later Putnam suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. If we are to believe that it is consistent, then either we cannot prove its consistency, or it cannot be represented by a Turing machine.
J. R. Lucas in Minds, Machines and Gödel (1961), and later in his book The Freedom of the Will (1970), lays out an anti-mechanist argument closely following the one described by Putnam, including reasons for why the human mind can be considered consistent. Lucas admits that, by Gödel's second theorem, a human mind cannot formally prove its own consistency, and even says (perhaps facetiously) that women and politicians are inconsistent. Nevertheless, he sets out arguments for why a male non-politician can be considered consistent. These arguments are philosophical in nature and are the subject of much debate; Lucas provides references to responses on his own website.
Another work was done by Judson Webb in his 1968 paper "Metamathematics and the Philosophy of Mind". Webb claims that previous attempts have glossed over whether one truly can see that the Gödelian statement p pertaining to oneself, is true. Using a different formulation of Gödel's theorems, namely, that of Raymond Smullyan and Emil Post, Webb shows one can derive convincing arguments for oneself of both the truth and falsity of p. He furthermore argues that all arguments about the philosophical implications of Gödel's theorems are really arguments about whether the Church-Turing thesis is true.
Later, Roger Penrose entered the fray, providing somewhat novel anti-mechanist arguments in his books, The Emperor's New Mind (1989) [ENM] and Shadows of the Mind (1994) [SM]. These books have proved highly controversial. Martin Davis responded to ENM in his paper "Is Mathematical Insight Algorithmic?" (ps), where he argues that Penrose ignores the issue of consistency. Solomon Feferman gives a critical examination of SM in his paper "Penrose's Gödelian argument" (pdf). The response of the scientific community to Penrose's arguments has been negative, with one group of scholars calling Penrose's repeated attempts to form a persuasive Gödelian argument "a kind of intellectual shell game, in which a precisely defined notion to which a mathematical result applies... is switched for a vaguer notion".
A Gödel-based anti-mechanism argument can be found in Douglas Hofstadter's book Gödel, Escher, Bach: An Eternal Golden Braid, though Hofstadter is widely viewed as a known skeptic of such arguments:
Looked at this way, Gödel's proof suggests – though by no means does it prove! – that there could be some high-level way of viewing the mind/brain, involving concepts which do not appear on lower levels, and that this level might have explanatory power that does not exist – not even in principle – on lower levels. It would mean that some facts could be explained on the high level quite easily, but not on lower levels at all. No matter how long and cumbersome a low-level statement were made, it would not explain the phenomena in question. It is analogous to the fact that, if you make derivation after derivation in Peano arithmetic, no matter how long and cumbersome you make them, you will never come up with one for G – despite the fact that on a higher level, you can see that the Gödel sentence is true.
What might such high-level concepts be? It has been proposed for eons, by various holistically or "soulistically" inclined scientists and humanists that consciousness is a phenomenon that escapes explanation in terms of brain components; so here is a candidate at least. There is also the ever-puzzling notion of free will. So perhaps these qualities could be "emergent" in the sense of requiring explanations which cannot be furnished by the physiology alone (Gödel, Escher, Bach, p. 708).
- ^"Mechanism". Catholic Encyclopedia. 1913.
- ^Hans Jonas, The Phenomenon of Life: Toward a Philosophical Biology (Evanston, Illinois: Northwestern University Press, 2001/1966), p. 175.
- ^Graham Oppy (20 January 2015). "Gödel's Incompleteness Theorems". Stanford Encyclopedia of Philosophy. Retrieved 27 April 2016.
- ^Stuart J. Russell; Peter Norvig (2010). "26.1.2: Philosophical Foundations/Weak AI: Can Machines Act Intelligently?/The mathematical objection". Artificial Intelligence: A Modern Approach (3rd ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-604259-7.
- ^Mark Colyvan. An introduction to the philosophy of mathematics. Cambridge University Press, 2012. From 2.2.2, 'Philosophical significance of Gödel's incompleteness results': "The accepted wisdom (with which I concur) is that the Lucas-Penrose arguments fail."
- ^ abLaForte, G., Hayes, P. J., Ford, K. M. 1998. Why Gödel's theorem cannot refute computationalism. Artificial Intelligence, 104:265-286, 1998.
- ^Gödel, Kurt, 1951, Some basic theorems on the foundations of mathematics and their implications in Solomon Feferman, ed., 1995. Collected works / Kurt Gödel, Vol. III. Oxford University Press: 304-23.
- ^Putnam, Hilary, 1960, Minds and Machines in Sidney Hook, ed., Dimensions of Mind: A Symposium. New York University Press. Reprinted in Anderson, A. R., ed., 1964. Minds and Machines. Prentice-Hall: 77.
- ^The Gödel Theorem and Human Nature, a talk given by Hilary Putnam in the Gödel centenary 2006 
- ^Lucas, J. R., 1961, "Minds, Machines, and Gödel." Philosophy 36:112-27.
- ^Webb, Judson, 1968, "Metamathematics and the Philosophy of Mind," Philosophy of Science 35: 156-78.