Gödel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all but the most trivial axiomatic systems capable of doing arithmetic. The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus giving a negative answer to Hilbert's second problem.

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an "effective procedure" (essentially, a computer program) is capable of proving all truths about the relations of the natural numbers (arithmetic). For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system. The second incompleteness theorem, a corollary of the first, shows that such a system cannot demonstrate its own consistency.

Background

Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a formal proof from a finite set of axioms is valid. This task, known as automatic proof verification, is closely related to automated theorem proving. The difference is that instead of constructing a new proof, the proof verifier simply checks that a provided formal proof (or, in some cases, instructions that can be followed to create a formal proof) is correct. This process is not merely hypothetical; systems such as Isabelle are used today to formalize proofs and then check their validity.

Many theories of interest include an infinite set of axioms, however. To verify a formal proof when the set of axioms is infinite, it must be possible to determine whether a statement that is claimed to be an axiom is actually an axiom. This issue arises in first order theories of arithmetic, such as Peano arithmetic, because the principle of mathematical induction is expressed as an infinite set of axioms (an axiom schema).

A formal theory is said to be ''effectively generated'' if its set of axioms is a recursively enumerable set. This means that there is a computer program that, in principle, could enumerate all the axioms of the theory without listing any statements that are not axioms. This is equivalent to the existence of a program that enumerates all the theorems of the theory without enumerating any statements that are not theorems. Examples of effectively generated theories with infinite sets of axioms include Peano arithmetic and Zermelo–Fraenkel set theory.

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. A set of axioms is complete if, for any statement in the axioms' language, either that statement or its negation is provable from the axioms. 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. 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. Gödel's incompleteness theorems show that in certain cases it is not possible to obtain an effectively generated, complete, consistent theory.

First incompleteness theorem

Gödel's first incompleteness theorem states that: : Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, p. 250).

The true but unprovable statement referred to by the theorem is often referred to as “the Gödel sentence” for the theory. The proof constructs a specific Gödel sentence for each effectively generated theory, but there are infinitely many statements in the language of the theory that share the property of being true but unprovable. For example, the conjunction of the Gödel sentence and any logically valid sentence will have this property.

For each consistent formal theory ''T'' having the required small amount of number theory, the corresponding Gödel sentence ''G'' asserts: “''G'' cannot be proved within the theory ''T''”. This interpretation of ''G'' leads to the following informal analysis. If ''G'' were provable under the axioms and rules of inference of ''T'', then ''T'' would have a theorem, ''G'', which effectively contradicts itself, and thus the theory ''T'' would be inconsistent. This means that if the theory ''T'' is consistent then ''G'' cannot be proved within it, and so the theory ''T'' is incomplete. Moreover, the claim ''G'' makes about its own unprovability is correct. In this sense ''G'' is not only unprovable but true, and provability-within-the-theory-''T'' is not the same as truth. This informal analysis can be formalized to make a rigorous proof of the incompleteness theorem, as described in the section "Proof sketch for the first theorem" below.

Each effectively generated theory has its own Gödel statement. It is possible to define a larger theory ''T’'' that contains the whole of ''T'', plus ''G'' as an additional axiom. This will not result in a complete theory, because Gödel's theorem will also apply to ''T’'', and thus ''T’'' cannot be complete. In this case, ''G'' is indeed a theorem in ''T’'', because it is an axiom. Since ''G'' states only that it is not provable in ''T'', no contradiction is presented by its provability in ''T’''. However, because the incompleteness theorem applies to ''T’'': there will be a new Gödel statement ''G’'' for ''T’'', showing that ''T’'' is also incomplete. ''G’'' will differ from ''G'' in that ''G’'' will refer to ''T’'', rather than ''T''.

To prove the first incompleteness theorem, Gödel represented statements by numbers. Then the theory at hand, which is assumed to prove certain facts about numbers, also proves facts about its own statements, provided that it is effectively generated. Questions about the provability of statements are represented as questions about the properties of numbers, which would be decidable by the theory if it were complete. In these terms, the Gödel sentence states that no natural number exists with a certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under the assumption that the theory is consistent, there is no such number.

Meaning of the first incompleteness theorem

Gödel's first incompleteness theorem shows that any consistent formal system that includes enough of the theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable. Thus no formal system (satisfying the hypotheses of the theorem) that aims to characterize the natural numbers can actually do so, as there will be true number-theoretical statements which that system cannot prove. This fact 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). Some (like Bob Hale and Crispin Wright) argue that it is not a problem for logicism because the incompleteness theorems apply equally to second 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.

The existence of an incomplete formal system is, in itself, not particularly surprising. A system may be incomplete simply because not all the necessary axioms have been discovered. For example, Euclidean geometry without the parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining axioms.

Gödel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent finite list of axioms can ''never'' be created, nor even an infinite list that can be enumerated by a computer program. Each time a new 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.

There ''are'' complete and consistent list of axioms for arithmetic that ''cannot'' be enumerated by a computer program. For example, one might take all true statements about the natural numbers to be axioms (and no false statements), which gives the theory known as "true arithmetic". The difficulty is that there is no mechanical way to decide, given a statement about the natural numbers, whether it is an axiom of this theory, and thus there is no effective way to verify a formal proof in this theory.

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").

Relation to the liar paradox

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 theory ''T'' makes a similar assertion to the liar sentence, but with truth replaced by provability: ''G'' says "''G'' is not provable in the theory ''T''." 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 by Gödel (when he was working on the proof of the incompleteness theorem) and by Alfred Tarski.

Original statements

The first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper ''On Formally Undecidable Propositions in Principia Mathematica and Related Systems I.'' The second incompleteness theorem appeared as "Theorem XI" in the same paper.

Extensions of Gödel's original result

Gödel demonstrated the incompleteness of the theory of ''Principia Mathematica'', a particular theory of arithmetic, but a parallel demonstration could be given for any effective theory 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 theory. 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 theory is not just consistent but ''ω-consistent''. A theory 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 ''n'' the theory proves ~''P''(''n''), and yet the theory also proves that there exists a natural number ''n'' such that ''P''(''n''). That is, the theory says that a number with property ''P'' exists while denying that it has any specific value. The ω-consistency of a theory 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 theory to be consistent, rather than ω-consistent. This is mostly of technical interest, since 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== Gödel's second incompleteness theorem can be stated as follows: : For any formal effectively generated theory ''T'' including basic arithmetical truths and also certain truths about formal provability, ''T'' includes a statement of its own consistency if and only if ''T'' is inconsistent. This strengthens the first incompleteness theorem, because the statement constructed in the first incompleteness theorem does not directly express the consistency of the theory. The proof of the second incompleteness theorem is obtained, essentially, by formalizing the proof of the first incompleteness theorem within the theory itself.

A technical subtlety in the second incompleteness theorem is how to express the consistency of ''T'' as a formula in the language of ''T''. There are many ways to do this, and not all of them lead to the same result. In particular, different formalizations of the claim that ''T'' is consistent may be inequivalent in ''T'', 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 since 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 technically ambiguous, but what is meant here is the largest consistent initial segment of the axioms of PA ordered according to some criteria; for example, by "Gödel numbers", the numbers encoding the axioms as per the scheme used by Gödel mentioned above).

In the case of Peano arithmetic, or any familiar explicitly axiomatized theory ''T'', it is possible to canonically define a formula Con(''T'') expressing the consistency of ''T''; this formula expresses the property that "there does not exist a natural number coding a sequence of formulas, such that each formula is either one of the axioms of ''T'', a logical axiom, or an immediate consequence of preceding formulas according to the rules of inference of first-order logic, and such that the last formula is a contradiction".

The formalization of Con(''T'') depends on two factors: formalizing the notion of a sentence being derivable from a set of sentences and formalizing the notion of being an axiom of ''T''. Formalizing derivability can be done in canonical fashion: given an arithmetical formula A(''x'') defining a set of axioms, one can canonically form a predicate Prov_A(''P'') which expresses that ''P'' is provable from the set of axioms defined by A(''x'').

In addition, the standard proof of the second incompleteness theorem assumes that Prov_A(''P'') satisfies that Hilbert–Bernays provability conditions. Letting #(''P'') represent the Gödel number of a formula ''P'', the derivability conditions say: # If ''T'' proves ''P'', then T proves Prov_A(#(''P'')). # ''T'' proves 1.; that is, ''T'' proves that if ''T'' proves ''P'', then ''T'' proves Prov_A(#(''P'')). In other words, ''T'' proves that Prov_A(#(''P'')) implies Prov_A(#(Prov_A(#(P)))). # ''T'' proves that if ''T'' proves that (''P'' → ''Q'') and ''T'' proves ''P'' then ''T'' proves ''Q''. In other words, ''T'' proves that Prov_A(#(''P'' → ''Q'')) and Prov_A(#(''P'')) imply Prov_A(#(''Q'')).

Implications for consistency proofs

Gödel's second incompleteness theorem also implies that a theory ''T''₁ satisfying the technical conditions outlined above cannot prove the consistency of any theory ''T''₂ which proves the consistency of ''T''₁. This is because such a theory ''T''₁ can prove that if ''T''₂ proves the consistency of ''T''₁, then ''T''₁ is in fact consistent. For the claim that ''T''₁ is consistent has form "for all numbers ''n'', ''n'' has the decidable property of not being a code for a proof of contradiction in ''T''₁". If ''T''₁ were in fact inconsistent, then ''T''₂ would prove for some ''n'' that ''n'' is the code of a contradiction in ''T''₁. But if ''T''₂ also proved that ''T''₁ is consistent (that is, that there is no such n), then it would itself be inconsistent. This reasoning can be formalized in ''T''₁ to show that if ''T''₂ is consistent, then ''T''₁ is consistent. Since, by second incompleteness theorem, ''T''₁ does not prove its consistency, it cannot prove the consistency of ''T''₂ 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 theory the consistency of which is provable in Peano arithmetic. For example, the theory 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.

The corollary also indicates the epistemological relevance of the second incompleteness theorem. It would actually provide no interesting information if a theory ''T'' proved its consistency. This is because inconsistent theories prove everything, including their consistency. Thus a consistency proof of ''T'' in ''T'' would give us no clue as to whether ''T'' really is consistent; no doubts about the consistency of ''T'' would be resolved by such a consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a theory ''T'' in some theory ''T’'' which is in some sense less doubtful than ''T'' itself, for example weaker than ''T''. For many naturally occurring theories ''T'' and ''T’'', such as ''T'' = Zermelo–Fraenkel set theory and ''T’'' = primitive recursive arithmetic, the consistency of ''T’'' is provable in ''T'', and thus ''T’'' can't prove the consistency of ''T'' by the above corollary of the second incompleteness theorem.

The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that could be formalized in the theory that is proved consistent. For example, Gerhard Gentzen proved the consistency of Peano arithmetic (PA) in a different theory which 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

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. The usage of "independent" is also ambiguous, however. Some use it to mean just "not provable", leaving open whether an independent statement might be refuted.

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 proven from ZFC.

In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in standard set theory.

In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the Ramsey theorem, is undecidable in the first-order axiomatization of arithmetic called Peano arithmetic, but can be proven in the larger system of second-order arithmetic. Kirby and Paris later showed Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris-Harrington principle, to be 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 on the basis of a philosophy of mathematics called predicativism. The related but more general graph minor theorem (2003) has consequences for computational complexity theory.

Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's theorem states that for any theory that can represent enough arithmetic, there is an upper bound ''c'' such that no specific number can be proven in that theory 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.

Limitations of Gödel's theorems

The conclusions of Gödel's theorems are only proven for the formal theories that satisfy the necessary hypotheses. Not all axiom systems satisfy these hypotheses, even when these systems have models that include the natural numbers as a subset. For example, there are first-order axiomatizations of Euclidean geometry, of real closed fields, and of arithmetic in which multiplication is not ''provably'' total; none of these meet the hypotheses of Gödel's theorems. The key fact is that these axiomatizations are not expressive enough to define the set of natural numbers or develop basic properties of the natural numbers. Regarding the third example, Dan E. Willard (Willard 2001) has studied many weak systems of arithmetic which do not satisfy the hypotheses of the second incompleteness theorem, and which are consistent and capable of proving their own consistency (see self-verifying theories).

Gödel's theorems only apply to effectively generated (that is, recursively enumerable) theories. If all true statements about natural numbers are taken as axioms for a theory, then this theory is a consistent, complete extension of Peano arithmetic (called true arithmetic) for which none of Gödel's theorems hold, because this theory is not recursively enumerable.

The second incompleteness theorem only shows that the consistency of certain theories cannot be proved from the axioms of those theories themselves. It does not show that the consistency cannot be proved from other (consistent) axioms. For example, the consistency of the Peano arithmetic can be proved in Zermelo–Fraenkel set theory (ZFC), or in theories of arithmetic augmented with transfinite induction, as in Gentzen's consistency proof.

Relationship with computability

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 unsolvable: there is no computer program that can correctly determine, given a program ''P'' as input, whether ''P'' eventually halts when run with some given input. Kleene showed that the existence of a complete effective theory 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 (2005, 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,...,x_k) 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 theory of arithmetic ''T'' will prove this. Moreover, if the theory ''T'' is ω-consistent, then it will never prove that some 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 algorithmically determine 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.

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

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 trick 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 have to somehow contain a reference to ''p'', which could easily give rise to an infinite regress. Gödel's ingenious trick 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 trick 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 HELLO is represented by 72-69-76-76-79 using decimal ASCII, ie the number 7269767679. :* The logical statement x=y => y=x is represented by 120-061-121-032-061-062-032-121-061-120 using octal ASCII, ie 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 which 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 proven (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'').

Now comes the trick: 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.

Diagonalization

The next step in the proof is to obtain a statement that says it is unprovable. 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''), ''p'' is obtained: ''p'' 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 asserts its own unprovability. The proof of the diagonal lemma employs a similar method.

Proof of independence

Now assume that the formal system is ω-consistent. 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 supports construction of 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'', it can be proved that the number 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: it can neither be proved nor disproved within the chosen system. So the chosen system is either inconsistent or incomplete. This logic can be applied to any formal system meeting the criteria. The conclusion is that all formal systems meeting the criteria are either inconsistent or incomplete. It should be noted that ''p'' is not provable (and thus true) in every consistent system. The assumption of ω-consistency is only required for the negation of ''p'' to be not provable. So:

In an ω-consistent formal system, neither ''p'' nor its negation can be proved, and so ''p'' is undecidable.

In a consistent formal system either the same situation occurs, or the negation of ''p'' can be proved; In the later case, a statement ("not ''p''") is false but provable.

Note that if one tries to fix this by "adding the missing axioms" to avoid the undecidability of the system, then one has to add either ''p'' or "not ''p''" as axioms. But this then creates a new formal system₂ (old system + ''p''), to which exactly the same process can be applied, creating a new statement form Bew₂(''x'') for this new system. When the diagonal lemma is applied to this new form Bew₂, a new statement ''p₂'' is obtained; this statement will be different from the previous one, and this new statement will be undecidable in the new system if it is ω-consistent, thus showing that system₂ is equally inconsistent. So adding extra axioms cannot fix the problem.

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).

Formalized proofs

Formalized proofs of versions of the incompleteness theorem have been developed by Natarajan Shankar in 1986 using Nqthm (Shankar 1994) and by R. O'Connor in 2003 using Coq (O'Connor 2005).

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 essentially 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 proven 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 proven in the system.

But this last statement is equivalent to ''p'' itself (and this equivalence can be proven in the system), so ''p'' can be proven 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 formal logic to define their principles. One can paraphrase the first theorem as saying the following:

:An all-encompassing axiomatic system can never be found that is able to prove ''all'' mathematical truths, but no falsehoods.

On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to each formal system.

The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics:

:If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.

Therefore, to establish the consistency of a system S, one needs to use some other ''more powerful'' system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S.

Theories such as Peano arithmetic, for which any computably enumerable consistent extension is incomplete, are called essentially undecidable or essentially incomplete.

Minds and machines

Authors including J. R. Lucas 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."

Paraconsistent logic

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 theory within the language of the theory (Priest 2006:47). Stewart Shapiro (2002) gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism. Carl Hewitt (2008) has proposed that (inconsistent) paraconsistent logics that prove their own Gödel sentences may have applications in software engineering.

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. A number of authors have commented negatively on such extensions and interpretations, including Torkel Franzén (2005); 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.).

The role of self-reference

Torkel Franzén (2005, p. 46) observes:

Gödel's proof of the first incompleteness theorem and Rosser's strengthened version have given many the impression that the theorem can only be proved by constructing self-referential statements [...] or even that only strange self-referential statements are known to be undecidable in elementary arithmetic. To counteract such impressions, we need only introduce a different kind of proof of the first incompleteness theorem.

He then proposes the proofs based on Computability, or on information theory, as described earlier in this article, as examples of proofs that should "counteract such impressions".

History

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 theory 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.

Announcement

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; it was never written.

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 theory 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.

Criticism

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). But Zermelo did not relent and published his criticisms in print with “a rather scathing paragraph on his young competitor” (Grattan-Guinness:513). Gödel decided that to pursue the matter further was pointless, and Carnap agreed (Dawson:77). Much of Zermelo's subsequent work was related to logics stronger than first-order logic, with which he hoped to show both the consistency and categoricity of mathematical theories.

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 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.

Wittgenstein and Gödel

Ludwig Wittgenstein wrote several passages about the incompleteness theorems that were published posthumously in his 1953 ''Remarks on the Foundations of Mathematics''. Gödel was a member of the Vienna Circle during the period in which Wittgenstein's early ideal language philosophy and Tractatus Logico-Philosophicus dominated the circle's thinking; writings of Gödel in his Nachlass express the belief that Wittgenstein willfully misread Gödel's theorems.

Multiple commentators have read Wittgenstein as misunderstanding Gödel (Rodych 2003), although Juliet Floyd and Hilary Putnam (2000) have suggested that the majority of commentary misunderstands Wittgenstein. On their release, Bernays, Dummett, and Kreisel wrote separate reviews on Wittgenstein's remarks, all of which were extremely negative (Berto 2009:208). The unanimity of this criticism caused Wittgenstein's remarks on the incompleteness theorems to have little impact on the logic community. In 1972, Gödel wrote to Karl Menger that Wittgenstein's comments demonstrate a fundamental misunderstanding of the incompleteness theorems.

:"It is clear from the passages you cite that Wittgenstein did "not" understand [the first incompleteness theorem] (or pretended not to understand it). He interpreted it as a kind of logical paradox, while in fact is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics)." (Wang 1996:197)

Since the publication of Wittgenstein's ''Nachlass'' in 2000, a series of papers in philosophy have sought to evaluate whether the original criticism of Wittgenstein's remarks was justified. Floyd and Putnam (2000) argue that Wittgenstein had a more complete understanding of the incompleteness theorem than was previously assumed. They are particularly concerned with the interpretation of a Gödel sentence for an ω-inconsistent theory as actually saying "I am not provable", since the theory has no models in which the provability predicate corresponds to actual provability. Rodych (2003) argues that their interpretation of Wittgenstein is not historically justified, while Bays (2004) argues against Floyd and Putnam's philosophical analysis of the provability predicate. Berto (2009) explores the relationship between Wittgenstein's writing and theories of paraconsistent logic.

See also

Gödel's completeness theorem

Löb's Theorem

Provability logic

Münchhausen Trilemma

Non-standard model of arithmetic

Tarski's undefinability theorem

''Minds, Machines and Gödel''

Third Man Argument

Notes

References

Articles by Gödel

1931, ''Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I.'' ''Monatshefte für Mathematik und Physik 38'': 173-98.

1931, ''Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I.'' and ''On formally undecidable propositions of Principia Mathematica and related systems I'' in Solomon Feferman, ed., 1986. ''Kurt Gödel Collected works, Vol. I''. Oxford University Press: 144-195. The original German with a facing English translation, preceded by a very illuminating introductory note by Kleene.

* Hirzel, Martin, 2000, ''On formally undecidable propositions of Principia Mathematica and related systems I.''. A modern translation by the author.

1951, ''Some basic theorems on the foundations of mathematics and their implications'' in Solomon Feferman, ed., 1995. ''Kurt Gödel Collected works, Vol. III''. Oxford University Press: 304-23.

Translations, during his lifetime, of Gödel’s paper into English

None of the following agree in all translated words and in typography. The typography is a serious matter, because Gödel expressly wished to emphasize “those metamathematical notions that had been defined in their usual sense before . . ."(van Heijenoort 1967:595). Three translations exist. Of the first John Dawson states that: “The Meltzer translation was seriously deficient and received a devastating review in the ''Journal of Symbolic Logic''; ”Gödel also complained about Braithwaite’s commentary (Dawson 1997:216). “Fortunately, the Meltzer translation was soon supplanted by a better one prepared by Elliott Mendelson for Martin Davis’s anthology ''The Undecidable'' . . . he found the translation “not quite so good” as he had expected . . . [but because of time constraints he] agreed to its publication” (ibid). (In a footnote Dawson states that “he would regret his compliance, for the published volume was marred throughout by sloppy typography and numerous misprints” (ibid)). Dawson states that “The translation that Gödel favored was that by Jean van Heijenoort”(ibid). For the serious student another version exists as a set of lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Gödel at to the Institute for Advanced Study during the spring of 1934" (cf commentary by Davis 1965:39 and beginning on p. 41); this version is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication:

B. Meltzer (translation) and R. B. Braithwaite (Introduction), 1962. ''On Formally Undecidable Propositions of Principia Mathematica and Related Systems'', Dover Publications, New York (Dover edition 1992), ISBN 0-486-66980-7 (pbk.) This contains a useful translation of Gödel's German abbreviations on pp. 33–34. As noted above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all its suspect content by

:*Stephen Hawking editor, 2005. ''God Created the Integers: The Mathematical Breakthroughs That Changed History'', Running Press, Philadelphia, ISBN 0-7624-1922-9. Gödel’s paper appears starting on p. 1097, with Hawking’s commentary starting on p. 1089.

Martin Davis editor, 1965. ''The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems and Computable Functions'', Raven Press, New York, no ISBN. Gödel’s paper begins on page 5, preceded by one page of commentary.

Jean van Heijenoort editor, 1967, 3rd edition 1967. ''From Frege to Gödel: A Source Book in Mathematical Logic, 1979-1931'', Harvard University Press, Cambridge Mass., ISBN 0-674-32449-8 (pbk). van Heijenoort did the translation. He states that “Professor Gödel approved the translation, which in many places was accommodated to his wishes.”(p. 595). Gödel’s paper begins on p. 595; van Heijenoort’s commentary begins on p. 592.

Martin Davis editor, 1965, ibid. "On Undecidable Propositions of Formal Mathematical Systems." A copy with Gödel's corrections of errata and Gödel's added notes begins on page 41, preceded by two pages of Davis's commentary. Until Davis included this in his volume this lecture existed only as mimeographed notes.

Articles by others

George Boolos, 1989, "A New Proof of the Gödel Incompleteness Theorem", ''Notices of the American Mathematical Society'' v. 36, pp. 388–390 and p. 676, reprinted in Boolos, 1998, ''Logic, Logic, and Logic'', Harvard Univ. Press. ISBN 0 674 53766 1

Arthur Charlesworth, 1980, "A Proof of Godel's Theorem in Terms of Computer Programs," ''Mathematics Magazine'', v. 54 n. 3, pp. 109–121. JStor

Martin Davis, "The Incompleteness Theorem", in Notices of the AMS vol. 53 no. 4 (April 2006), p. 414.

Jean van Heijenoort, 1963. "Gödel's Theorem" in Edwards, Paul, ed., ''Encyclopedia of Philosophy, Vol. 3''. Macmillan: 348-57.

Geoffrey Hellman, ''How to Gödel a Frege-Russell: Gödel's Incompleteness Theorems and Logicism.'' Noûs, Vol. 15, No. 4, Special Issue on Philosophy of Mathematics. (Nov., 1981), pp. 451–468.

David Hilbert, 1900, "Mathematical Problems." English translation of a lecture delivered before the International Congress of Mathematicians at Paris, containing Hilbert's statement of his Second Problem.

| year=1994 | journal=Notre Dame Journal of Formal Logic | issn=0029-4527 | volume=35 | issue=3 | pages=403–412 | doi=10.1305/ndjfl/1040511346}}

Stephen Cole Kleene, 1943, "Recursive predicates and quantifiers," reprinted from ''Transactions of the American Mathematical Society'', v. 53 n. 1, pp. 41–73 in Martin Davis 1965, ''The Undecidable'' (loc. cit.) pp. 255–287.

John Barkley Rosser, 1936, "Extensions of some theorems of Gödel and Church," reprinted from the ''Journal of Symbolic Logic'' vol. 1 (1936) pp. 87–91, in Martin Davis 1965, ''The Undecidable'' (loc. cit.) pp. 230–235.

John Barkley Rosser, 1939, "An Informal Exposition of proofs of Gödel's Theorem and Church's Theorem", Reprinted from the ''Journal of Symbolic Logic'', vol. 4 (1939) pp. 53–60, in Martin Davis 1965, ''The Undecidable'' (loc. cit.) pp. 223–230

C. Smoryński, "The incompleteness theorems", in J. Barwise, ed., ''Handbook of Mathematical Logic'', North-Holland 1982 ISBN 978-0444863881, pp. 821–866.

Dan E. Willard (2001), "Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection Principles", ''Journal of Symbolic Logic'', v. 66 n. 2, pp. 536–596.

Richard Zach, 2005, "Paper on the incompleteness theorems" in Grattan-Guinness, I., ed., ''Landmark Writings in Western Mathematics''. Elsevier: 917-25.

Books about the theorems

Francesco Berto. ''There's Something about Gödel: The Complete Guide to the Incompleteness Theorem'' John Wiley and Sons. 2010.

Domeisen, Norbert, 1990. ''Logik der Antinomien''. Bern: Peter Lang. 142 S. 1990. ISBN 3-261-04214-1. Zentralblatt MATH

Torkel Franzén, 2005. ''Gödel's Theorem: An Incomplete Guide to its Use and Abuse''. A.K. Peters. ISBN 1568812388 Douglas Hofstadter, 1979. ''Gödel, Escher, Bach: An Eternal Golden Braid''. Vintage Books. ISBN 0465026850. 1999 reprint: ISBN 0465026567. Douglas Hofstadter, 2007. ''I Am a Strange Loop''. Basic Books. ISBN 9780465030781. ISBN 0465030785.

Stanley Jaki, OSB, 2005. ''The drama of the quantities''. Real View Books.

Per Lindström, 1997, ''Aspects of Incompleteness'', Lecture Notes in Logic v. 10.

J.R. Lucas, FBA, 1970. ''The Freedom of the Will''. Clarendon Press, Oxford, 1970.

Ernest Nagel, James Roy Newman, Douglas Hofstadter, 2002 (1958). ''Gödel's Proof'', revised ed. ISBN 0814758169. Rudy Rucker, 1995 (1982). ''Infinity and the Mind: The Science and Philosophy of the Infinite''. Princeton Univ. Press.

Smith, Peter, 2007. ''An Introduction to Gödel's Theorems.'' Cambridge University Press. MathSciNet

N. Shankar, 1994. ''Metamathematics, Machines and Gödel's Proof'', Volume 38 of Cambridge tracts in theoretical computer science. ISBN 0521585333

Raymond Smullyan, 1991. ''Godel's Incompleteness Theorems''. Oxford Univ. Press.

—, 1994. ''Diagonalization and Self-Reference''. Oxford Univ. Press. Hao Wang, 1997. ''A Logical Journey: From Gödel to Philosophy''. MIT Press. ISBN 0262231891

Miscellaneous references

Francesco Berto. "The Gödel Paradox and Wittgenstein's Reasons" ''Philosophia Mathematica'' (III) 17. 2009.

John W. Dawson, Jr., 1997. ''Logical Dilemmas: The Life and Work of Kurt Gödel'', A.K. Peters, Wellesley Mass, ISBN 1-56881-256-6.

Goldstein, Rebecca, 2005, ''Incompleteness: the Proof and Paradox of Kurt Gödel'', W. W. Norton & Company. ISBN 0-393-05169-2

Juliet Floyd and Hilary Putnam, 2000, "A Note on Wittgenstein's 'Notorious Paragraph' About the Gödel Theorem", ''Journal of Philosophy'' v. 97 n. 11, pp. 624–632.

Carl Hewitt, 2008, "Large-scale Organizational Computing requires Unstratified Reflection and Strong Paraconsistency", ''Coordination, Organizations, Institutions, and Norms in Agent Systems III'', Springer-Verlag.

David Hilbert and Paul Bernays, ''Grundlagen der Mathematik'', Springer-Verlag.

John Hopcroft and Jeffrey Ullman 1979, ''Introduction to Automata theory'', Addison-Wesley, ISBN 0-201-02988-X

Stephen Cole Kleene, 1967, ''Mathematical Logic''. Reprinted by Dover, 2002. ISBN 0-486-42533-9

Graham Priest, 2006, ''In Contradiction: A Study of the Transconsistent'', Oxford University Press, ISBN 0-199-26329-9

Graham Priest, 1984, "Logic of Paradox Revisited", ''Journal of Philosophical Logic'', v. 13, n. 2, pp. 153–179

Hilary Putnam, 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.

Russell O'Connor, 2005, "Essential Incompleteness of Arithmetic Verified by Coq", Lecture Notes in Computer Science v. 3603, pp. 245–260.

Victor Rodych, 2003, "Misunderstanding Gödel: New Arguments about Wittgenstein and New Remarks by Wittgenstein", ''Dialectica'' v. 57 n. 3, pp. 279–313. Stewart Shapiro, 2002, "Incompleteness and Inconsistency", ''Mind'', v. 111, pp 817–32.

Alan Sokal and Jean Bricmont, 1999, ''Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science'', Picador. ISBN 0-31-220407-8

Joseph R. Shoenfield (1967), ''Mathematical Logic''. Reprinted by A.K. Peters for the Association of Symbolic Logic, 2001. ISBN 978-156881135-2

Jeremy Stangroom and Ophelia Benson, ''Why Truth Matters'', Continuum. ISBN 0-82-649528-1

George Tourlakis, ''Lectures in Logic and Set Theory, Volume 1, Mathematical Logic'', Cambridge University Press, 2003. ISBN 978-0-52175373-9

Hao Wang, 1996, ''A Logical Journey: From Gödel to Philosophy'', The MIT Press, Cambridge MA, ISBN 0-262-23189-1.

Richard Zach, 2006, "Hilbert's program then and now", in ''Philosophy of Logic'', Dale Jacquette (ed.), Handbook of the Philosophy of Science, v. 5., Elsevier, pp. 411–447.

External links

Stanford Encyclopedia of Philosophy: "Kurt Gödel" -- by Juliette Kennedy.

MacTutor biographies:

*Kurt Gödel.

*Gerhard Gentzen.

*What is Mathematics:Gödel's Theorem and Around by ''Karlis Podnieks''. An online free book.

World's shortest explanation of Gödel's theorem using a printing machine as an example.

Category:Mathematical theorems Category:Mathematical logic Category:Model theory Category:Proof theory Category:Epistemology Category:Metatheorems

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Coordinates	45°30′″N73°40′″N
name	Kurt Gödel
birth date	April 28, 1906
birth place	Brno, Moravia, Austria–Hungary
death date	January 14, 1978
death place	Princeton, New Jersey, United States
field	Mathematics, Mathematical logic
work institutions	Institute for Advanced Study
alma mater	University of Vienna
doctoral advisor	Hans Hahn
known for	Gödel's incompleteness theorems, Gödel's completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel's ontological proof
prizes	Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1974)
religion	Theist
signature	Kurt Gödel signature.svg
footnotes	}}

Kurt Friedrich Gödel (; April 28, 1906 – January 14, 1978) was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for "working mathematicians" to assume the axiom of choice in their proofs. There are many important results whose only known proofs use the axiom of choice, which is equivalent to Zorn's lemma. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Life

Childhood

Gödel was born April 28, 1906, in Brno, Austria-Hungary into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh). At the time of his birth the town had a slight German-speaking majority, and this was the language of his parents. The ancestors of Kurt Gödel were often active in the cultural life of the Brno city. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".

Although he spoke very little Czech himself, Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian empire broke up at the end of World War I. According to his classmate Klepetař, "Gödel considered himself always Austrian and an exile in Czechoslovakia" ("''ein Österreicher im Exil in der Tschechoslowakei''") during this time. He chose to become an Austrian citizen at age 23. When Nazi Germany annexed Austria, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as ''Herr Warum'' ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

Gödel attended the ''Evangelische Volksschule'', a Lutheran school in Brno from 1912 to 1916, and was enrolled in the ''Deutsches Staats-Realgymnasium'' from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's ''Theory of Colours'' and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics. Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's ''Metaphysische Anfangsgründe der Naturwissenschaft'', and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book ''Introduction to Mathematical Philosophy'', he became interested in mathematical logic. According to Gödel mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published ''Grundzüge der theoretischen Logik'' (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: ''Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?''

This was the topic chosen by Gödel for his doctorate work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

The Incompleteness Theorem

In 1931 and while still in Vienna, Gödel published his incompleteness theorems in "Über formal unentscheidbare Sätze der ''Principia Mathematica'' und verwandter Systeme" (called in English "On Formally Undecidable Propositions of ''Principia Mathematica'' and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that: # If the system is consistent, it cannot be complete. # The consistency of the axioms cannot be proven within the system. These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in ''Principia Mathematica'' and Hilbert's formalism, to find a set of axioms sufficient for all mathematics. The incompleteness theorems also imply that not all mathematical questions are computable.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as Gödel numbering.

In his two-page paper "Zum intuitionistischen Aussagenkalkül" (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

The mid 1930s: further work and visits to the US

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a ''Privatdozent'' (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was assassinated by a pro-Nazi student. This triggered "a severe nervous crisis" in Gödel. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend. He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled ''On undecidable propositions of formal mathematical systems''. Stephen Kleene, who had just completed his Ph.D. at Princeton, took notes of these lectures which have been subsequently published.

Gödel would visit the IAS again in the autumn of 1935. The traveling and the hard work had exhausted him, and the next year he took a break to recover from a depression. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.

He married Adele Nimbursky (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, in the Blue Hill Inn at the top of the bay. Gödel was taking a vacation from the IAS.

Gödel was not merely vacationing, and had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

Relocation to Princeton, Einstein and US citizenship

After the Anschluss in 1938, Austria had become a part of Nazi Germany. Germany abolished the title of ''Privatdozent'', so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down. His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the trans-Siberian railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel would accept a position at the Institute for Advanced Study (IAS).

Gödel very quickly resumed his mathematical work. In 1940, he published his work ''Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory'' which is a classic of modern mathematics. In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF). Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.

Albert Einstein was also living at Princeton during this time. Gödel and Einstein subsequently developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely...to have the privilege of walking home with Gödel".

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution, one that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his chances. Fortunately, the judge turned out to be Phillip Forman. Forman knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.

Later years and death

Gödel became a permanent member of the Institute of Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.

In 1951, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. He gave this elaboration to Einstein as a present for his 70th birthday. These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.

During his many years at the Institute, Gödel's interests turned to philosophy and physics. He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, Adele was hospitalized for six months and could no longer prepare Gödel's food. In her absence, he refused to eat, eventually starving to death. He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.

Religious views

Gödel was a convinced theist. He rejected the notion of others like his friend Albert Einstein that God was impersonal.

He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world [Welt] is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is ''theistic'', not pantheistic, following Leibniz rather than Spinoza."

Legacy

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association of Symbolic Logic has invited an annual Kurt Gödel lecture each year since 1990.

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's ''Nachlass'', and the final two include correspondence.

A biography of Gödel was published by John Dawson in 2005. Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled "Dangerous Knowledge".

Douglas Hofstadter wrote a popular book in 1979 called ''Gödel, Escher, Bach: An Eternal Golden Braid'' to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain.

Important publications

In German:

1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." ''Monatshefte für Mathematik und Physik'' 37: 349–60.

1931, "Über formal unentscheidbare Sätze der ''Principia Mathematica'' und verwandter Systeme, I." ''Monatshefte für Mathematik und Physik'' 38: 173–98.

1932, "Zum intuitionistischen Aussagenkalkül", ''Anzeiger Akademie der Wissenschaften Wien'' 69: 65–66.

In English:

1940. ''The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory.'' Princeton University Press.

1947. "What is Cantor's continuum problem?" ''The American Mathematical Monthly 54'': 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). ''Philosophy of Mathematics: Selected Readings''. Cambridge Univ. Press: 470–85.

1950, "Rotating Universes in General Relativity Theory." ''Proceedings of the international Congress of Mathematicians in Cambridge,'' 1: 175–81

In English translation:

Kurt Godel, 1992. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.

Kurt Godel, 2000. ''On Formally Undecidable Propositions Of Principia Mathematica And Related Systems'', tr. Martin Hirzel

Jean van Heijenoort, 1967. ''A Source Book in Mathematical Logic, 1879–1931''. Harvard Univ. Press.

*1930. "The completeness of the axioms of the functional calculus of logic," 582–91.

*1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).

*1931. "On formally undecidable propositions of ''Principia Mathematica'' and related systems," 596–616.

*1931a. "On completeness and consistency," 616–17.

"My philosophical viewpoint", c. 1960, unpublished.

"The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished.

''Collected Works'': Oxford University Press: New York. Editor-in-chief: Solomon Feferman.

*Volume I: Publications 1929–1936 ISBN 978-0-19-503964-1 / Paperback:ISBN 978-0-19-514720-9,

*Volume II: Publications 1938–1974 ISBN 978-0-19-503972-6 / Paperback:ISBN 978-0-19-514721-6,

*Volume III: Unpublished Essays and Lectures ISBN 978-0-19-507255-6 / Paperback:ISBN 978-0-19-514722-3,

*Volume IV: Correspondence, A–G ISBN 978-0-19-850073-5,

*Volume V: Correspondence, H–Z ISBN 978-0-19-850075-9.

See also

Notes

References

Dawson, John W., 1997. ''Logical dilemmas: The life and work of Kurt Gödel''. Wellesley MA: A K Peters.

1911 Encyclopædia Britannica/Brünn. (2007, September 19). In Wikisource, The Free Library. Retrieved 10PM EST March 13, 2008.

Rebecca Goldstein, 2005. ''Incompleteness: The Proof and Paradox of Kurt Gödel''. W. W. Norton & Company, New York. ISBN 0-393-32760-4 pbk.

Further reading

John L. Casti and Werner DePauli, 2000. ''Gödel: A Life of Logic'', Basic Books (Perseus Books Group), Cambridge, MA. ISBN 0-7382-0518-4.

John W. Dawson, Jr. ''Logical Dilemmas: The Life and Work of Kurt Gödel''. AK Peters, Ltd., 1996.

John W. Dawson, Jr, 1999. "Gödel and the Limits of Logic", ''Scientific American'', vol. 280 num. 6, pp. 76–81

Torkel Franzén, 2005. ''Gödel's Theorem: An Incomplete Guide to Its Use and Abuse''. Wellesley, MA: A K Peters.

Ivor Grattan-Guinness, 2000. ''The Search for Mathematical Roots 1870–1940''. Princeton Univ. Press.

Jaakko Hintikka, 2000. ''On Gödel''. Wadsworth.

Douglas Hofstadter, 1980. ''Gödel, Escher, Bach''. Vintage.

Stephen Kleene, 1967. ''Mathematical Logic''. Dover paperback reprint ca. 2001.

Stephen Kleene, 1980. ''Introduction to Metamathematics''. North Holland ISBN 0-7204-2103-9 (Ishi Press paperback. 2009. ISBN 978-0-923891-57-2)

J.R. Lucas, 1970. ''The Freedom of the Will''. Clarendon Press, Oxford.

Ernest Nagel and Newman, James R., 1958. ''Gödel's Proof.'' New York Univ. Press.

Procházka, Jiří, 2006, 2006, 2008, 2008, 2O1O. ''Kurt Gödel: 1906–1978: Genealogie''. ITEM, Brno. Volume I. Brno 2006, ISBN 80-902297-9-4. In Ger., Engl. Volume II. Brno 2006, ISBN 80-903476-0-6. In Germ., Engl. Volume III. Brno 2008, ISBN 80-903476-4-9. In Germ., Engl. Volume IV. Brno, Princeton 2008, ISBN 978-80-903476-5-6. In Germ., Engl.Volume V,Brno,Princeton 2O1O, ISBN 8O-9O3476-9-X.In Germ.,Engl.

Ed Regis, 1987. ''Who Got Einstein's Office?'' Addison-Wesley Publishing Company, Inc.

Raymond Smullyan, 1992. ''Godel's Incompleteness Theorems''. Oxford University Press.

Olga Taussky-Todd, 1983. Remembrances of Kurt Gödel. Engineering & Science, Winter 1988.

Hao Wang, 1987. ''Reflections on Kurt Gödel.'' MIT Press.

Hao Wang, 1996. ''A Logical Journey: From Godel to Philosophy''. MIT Press.

Yourgrau, Palle, 1999. ''Gödel Meets Einstein: Time Travel in the Gödel Universe.'' Chicago: Open Court.

Yourgrau, Palle, 2004. ''A World Without Time: The Forgotten Legacy of Gödel and Einstein.'' Basic Books. Book review by John Stachel in the Notices of the American Mathematical Society (54 (7), p 861–868):

External links

Kennedy, Juliette. "Kurt Gödel." In Stanford Encyclopedia of Philosophy.

Time Bandits – an article about the relationship between Gödel and Einstein by Jim Holt

"Gödel and the limits of logic" by John W Dawson Jr. (June 2006)

Notices of the AMS, April 2006, Volume 53, Number 4 Kurt Gödel Centenary Issue

Paul Davies and Freeman Dyson discuss Kurt Godel

"Gödel and the Nature of Mathematical Truth" Edge: A Talk with Rebecca Goldstein on Kurt Gödel.

It's Not All In The Numbers: Gregory Chaitin Explains Gödel's Mathematical Complexities.

Dangerous Knowledge Google Video of a BBC documentary made by David Malone about the life and work of Kurt Gödel and other revolutionary mathematical thinkers.

Gödel photo g.

Category:1906 births Category:1978 deaths Category:20th-century philosophers Category:20th-century mathematicians Category:American mathematicians Category:American people of Austrian descent Category:Austrian emigrants to the United States Category:Austrian mathematicians Category:Austrian logicians Category:Austrian philosophers Category:Christian philosophers Category:Ontologists Category:Platonists Category:Deaths by starvation Category:Austrian people of Moravian German descent Category:German-language philosophers Category:Institute for Advanced Study faculty Category:National Medal of Science laureates Category:Naturalized citizens of the United States Category:People from Brno Category:Princeton University faculty Category:Set theorists Category:Vienna Circle Category:University of Vienna alumni Category:Burials at Princeton Cemetery Category:Foreign Members of the Royal Society Category:American logicians Category:American philosophers Category:Austrian Lutherans Category:American Lutherans

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Coordinates	45°30′″N73°40′″N
name	The Foundations
background	group_or_band
origin	London, England
genre	Soul
years active	1967–1970
label	Pye, Castle, Uni
past members	1966-1968Eric AllendaleArthur BrownPat BurkeClem CurtisMike ElliottTony GomezTim HarrisPeter MacBethAlan Warner 1968-1970Eric AllendalePat BurkeTony GomezTim HarrisPeter MacBethAlan Warner Colin Young 1970-1971Eric AllendaleSteve BinghamPat BurkeTony GomezTim HarrisAlan Warner Colin Young }}

The Foundations were a British soul band, active from 1967 to 1970. The group, made up of West Indians, White British, and a Sri Lankan, are best known for their two biggest hits, "Baby Now That I've Found You" (a Number One hit in the UK Singles Chart and Canada, and subsequently Top 10 in the US), written by Tony Macaulay and John MacLeod; and "Build Me Up Buttercup" (a number 3 on the Billboard Hot 100 and number 1 in Canada), co-written by Macaulay with Mike d'Abo, at the time the lead vocalist with Manfred Mann. The group was the first multi-racial group to have a number 1 hit in the UK in the 1960s.

The Foundations are notable for being one of the few label acts to successfully imitate what became known as the Motown Sound. In terms of line-up and musical style, they anticipated the sound of the more successful Hot Chocolate. They were in a similar musical vein as Love Affair, who also topped the UK charts in 1968 with their version of Robert Knight's "Everlasting Love". The Foundations signed to Pye, at the time one of only four big UK record companies (the others being EMI with its HMV, Columbia Records, and Parlophone labels; Decca; and Philips who also owned Fontana). the Cabin Boys (led by Tommy Steele's brother, Colin Hicks), and others. Pat Burke, a professional musician, was from the London Music Conservatorium. Eric Allendale had led his own band at one stage as well as having played with Edmundo Ros and being a former member of the Terry Lightfoot and Alex Walsh bands. Alan Warner, the guitarist, was also an experienced musician for his age, having played with numerous semi-pro groups from the age of 16. Bassist Peter Macbeth was a former teacher. Tony Gomez, the keyboard player, was a former clerk, while Clem Curtis had been an interior decorator and professional boxer.

The story of the origins of the Foundations can be somewhat surprising and a bit confusing as to who was responsible for choosing the band's name, and various sources give slightly different accounts of their beginnings. One version is that they were originally called The Ramong Sound, or The Ramongs, and there were two lead singers, Clem Curtis and Psychedelic shock rocker Arthur Brown. Other variations on this are that after they dropped the "Ramong" from their name, they were then called The Foundation Squad or The Foundation Sound. Apparently, Arthur Brown was only a temporary member for about one month, and by the time the Foundations had signed to Pye, he had left the group.

The Foundations actually did come together in Bayswater, London in January 1967. They practiced and played in a basement club called the Butterfly Club, which they ran. While managing the club themselves, they played music nightly, and handled the cooking and cleaning. They would get to bed around 6 or 7 a.m., sleep until 4 p.m., get up and begin again to get ready to open at 8 p.m. Sometimes they barely made enough money to pay the rent, let alone feed themselves. At times, they lived off the leftovers and a couple of pounds of rice. They stayed there until they were eventually forced out by a protection racket gang and then had to move next door to a dingy unused mini-cab office.

Career from 1967

While the biography on Allmusic states that Barry Class was the first to discover them, Alan Warner, the original founding member and lead guitarist of The Foundations and former Ramongs member states that it was actually Fairway who saw them in the dingy office and then introduced them to Class. Either way, both Fairway and Class teamed up to become their managers. Class took care of the business management, while Fairway went out promoting and trying to secure gigs for the group.

When they were at the top spot with "Baby, Now That I've Found You", Fairway commented to ''Melody Maker'' that most management would pull them out of the "bargain priced dates" that were booked for some time. He expressed gratitude to everyone for their support, and said that they would fulfill every engagement for which they had signed. Dopson also noted that Fairway also leaked a story to the media saying that the Foundations had broken up which only served to keep the Foundations name in the news headlines. Fairway and Class eventually introduced them to Pye A&R; man Tony Macaulay. The day Macaulay came to hear them play, he was suffering from what he described as the worst hangover of his life. The band was playing so loud he could not judge how good they were, but he decided to give them a chance.

At first they found progress quite slow, and one of their sax players, Pat Burke, had to drop out of the band and take another job while they went through a rough patch. He did rejoin them again later in 1967.

They did a couple of tours backing The Toys and later Motown Records' Edwin Starr. Curtis doubted if this group called The Toys was the original Toys let alone American. They were noticed by Brian Epstein who added them to the roster of his NEMS Agency but the contract became void when he died.

When "Baby Now That I've Found You" was first released it went nowhere. Luckily the BBC's newly founded BBC Radio 1 were looking to avoid any records being played by the pirate radio stations and they looked back at some recent releases that the pirate stations had missed. "Baby, Now That I've Found You" was one of them. The single then took off and by November was number one in the UK Singles Chart. On the same day PP Arnold was in the studio with Dusty Springfield and Madeline Bell as her backing vocalists.

Around this time after the release of their second single, there were tensions developing between the band and their songwriter/producer, Tony Macaulay. He would not allow them to record any of their own songs. In an interview, the band's organ player, Tony Gomez, told NME Magazine in an interview that he, Peter MacBeth, and Eric Allendale had some ideas that they wanted to put together. Curtis later recalled that Macaulay was a problem. "Tony Macaulay was very talented, but could be difficult to get on with. When we asked to record some of our own material - just as B sides, we weren't after the A side - he called us 'ungrateful' and stormed out of the studio."

Bassist Peter Macbeth left the band in 1969, to join the group Bubastis with Bernie Living, and was replaced by Steve Bingham.

Beginning of 1970 to the breakup in late 1970

After a successful run of hits, The Foundations broke off with their management and a Bill Graham sponsored tour to support The Temptations at the newly opened Copacabana club. This ended up in disaster and the band came back to the UK in low spirits. It had been previously reported in a publicity sheet around early December 1969 that the band had broken away from their manager Barry Class. Jim Dawson who was formerly their agent and Mike Dolan took over the group's affairs. The group's final hits were "Born to Live, Born to Die" which was written by Eric Allendale and Tony Gomez. and "My Little Chickadee", a US only hit which barely made the hot 100. Another member joined the band in 1970. Paul Lockey who had been with Robert Plant in Band Of Joy joined as their bass guitarist.

"My Little Chickadee" proved to be the band's last hit. In spite of releasing "Take A Girl Like You", the title song to the Oliver Reed and Hayley Mills film, and a heavy blues rock song "I'm Gonna Be A Rich Man", the band split in late 1970.

1971 to the end of the 1970s

The last record released in the early 1970s as The Foundations was a single "Stoney Ground" b/w "I'll Give You Love" MCA MCA 5075 1971. By that time the original band had already split up and Alan Warner had been a member of Pluto since joining and helping to form the English progressive rock band in 1970. There would be two more singles released as The Foundations in the mid to late 1970s.

When Curtis returned to the UK, he formed a new version of the group with little success in spite of releasing several singles, but later had a lucrative spell on the 1960s nostalgia circuit. Curtis' re-formed Foundations have on several occasions and among the many musicians to be part of latter day Foundations were Bill and John Springate, the latter becoming a member of The Glitter Band, Derek "Del" Watson, Paul Wilmot (all members of the band "ELEGY") and Roy Carter who later on joined Heatwave.

Also in the 1970s there would be a collaborative attempt between two former members of the Foundations. Original Foundations trombonist Eric Allendale attempted to work with the band's Foundations drummer Tim Harris.

In the mid 1970s, while Clem Curtis and The Foundations were on the road, there was also another Foundations line up that was led by Colin Young who were on the road at the same time, who were playing basically the same material. This eventually led to court action which resulted in Curtis being allowed to bill his group as either The Foundations or Clem Curtis & The Foundations. Young was allowed to bill himself as The New Foundations or Colin Young & The New Foundations. These featured Curtis as the lead singer.

Various sources erroneously state that there was an early 1970s English line up that had nothing, or little to do with, the original Foundations. However, Curtis has been leading a new line up of the Foundations since coming back to the UK and reforming the group in the early 1970s.

1980s to present

In or around the late 1980s Curtis and Alan Warner teamed up to recut "Baby, Now That I've Found You" as well as other hits of The Foundations. Trinidad, West Indies.

Colin Young: lead vocals - b. 12 September 1944, Barbados, West Indies - replaced Clem Curtis in 1968.

Arthur Brown: vocals - b. 24 June 1942 Whitby, Yorkshire, Member for approximately one month in 1967

Alan Warner: lead guitar - b. 21 April 1941, Paddington, west London. Peter Macbeth: bass guitar - b. Peter McGrath, 2 February 1943, Marylebone, North London.

Steve Bingham: bass guitar -

Tim Harris: drums - b. 14 January 1948, St John's Wood, North London - Died ???? Tony Gomez: keyboards - b. 13 December 1938, Colombo, Ceylon - (now Sri Lanka). Pat Burke: tenor saxophone/flute - b. 9 October 1937, Kingston, Jamaica, West Indies. Mike Elliott: tenor saxophone - b. 6 August 1929, Jamaica, West Indies. - Left in 1968 Eric Allandale: trombone - b. Eric Allandale Dubuisson, 4 March 1936, Dominica, West Indies — died 23 August 2001.

The Tweeney: vocals - one guest appearance in Nottingham, 2011

Paul Lockey: bass guitar - joined in 1970 for nine months

Guests

Mike D'Abo :piano - b. Michael David D'Abo, 1 March 1944, Betchworth, Surrey. Co-wrote and guested on "Build Me Up Buttercup" contributing piano.

John Mcleod: piano

Clem Curtis & The Foundations

Clem Curtis: vocals

James Colah: keyboards

Michael J. Parlett: saxophone

Alan Warner: guitar

George Chandler: backing vocals

Valentine Pascal: electric guitar

1970s line-up

Clem Curtis: Vocals

Bill Springate: guitar

John Springate: Vocals

Del Watson: Bass

'''Paul Wilmot : Drums

John Paul: ?

1977 line-up

Clem Curtis: Vocals

Leroy Carter

John Savile

Valentine Pascal

Georges Delanbanque

Discography

;Summary of single releases From the bands beginning to their breakup near the end of 1970, the Foundations released ten singles in the UK including two versions of the same song. A good deal of the songs on the singles were composed by Tony Macaulay and John Macleod. They had four significant hits from these plus a minor hit with one of their own compositions, "Born To Live, Born To Die" and another minor hit with "My Little Chickadee" in the United States. There were other titles announced that were either never recorded or were never released. They were "Our Love Went Thataway", "Tear Jerker, Music-worker, You" which was to be released around the same time as "Better By Far" by Lulu and "No Place On Earth Could Find You" In 1971 the single "Stoney Ground" was released. It is believed that this single was actually by Colin Young and his new backing band Development. It seems quite likely as the Colin Young and Development debut single "Any Time At All" pre-dates "Stoney Ground". In the mid and late seventies there were two more singles released under the Foundations name. They were "Where Were You When I Needed Your Love" and "Closer To Loving You" which featured the Northern Soul classic "Change My Life" as the B side. These last two singles to bear the Foundations name featured Clem Curtis once more as the lead vocalist.

;Summary of album releases During the 1960s the Foundations recorded and released four LPs in the United Kingdom. Before its release it was originally announced, in the October 1967 of ''Beat Instrumental Monthly'' that the debut album's title was to be ''Sound Basis''issue. However, when it was released on Pye, it had the title of ''From The Foundations''. The American version of this album, which was released on the UNI label, was given the title of ''Baby, Now That I've Found You''. This album featured Curtis on lead vocals. The next release was in 1968. It was a live LP called ''Rocking The Foundations'', and also featured Curtis on lead vocals plus two instrumentals "The Look of Love" and "Coming Home Baby". Also in 1968, another LP was released on the Marble Arch label. This featured re-recordings of their previous hits and songs, but with Young on vocals instead of Curtis. It also featured a version of "Build Me Up Buttercup". There was also an American album called ''Build Me Up Buttercup''. This was a compilation of Foundations tracks. Side one consisted of tracks from their ''Rocking The Foundations'' album; and side two consisted of "Build Me Up Buttercup", the B side of that single, plus some earlier Foundations tracks. The groups last LP release was in 1969, which featured their hit "In The Bad Bad Old Days", and the minor US hit "My Little Chickadee". Since then there have been various compilations of the Foundations songs, released on both the Golden Hour and PRT labels.

UK singles

"Baby, Now That I've Found You" / "Come On Back to Me" - Pye 7N 17366 - 1967 - UK #1, US #11

"Back On My Feet Again" / "I Can Take Or Leave Your Loving" - Pye 7N 17417 - 1968 - UK #18

"Any Old Time (You're Lonely And Sad)" / "We Are Happy People" - Pye 7N 17503 -1968 - UK #48

"Build Me Up Buttercup" / "New Direction" Pye 7N 17636 - 1968 - UK #2, US #3

"In the Bad Bad Old Days (Before You Loved Me)" / "Give Me Love" - Pye 7N 17702 - 1969 - UK #8

"Born to Live, Born to Die" / "Why Did You Cry" - Pye 7N 17809 1969 - UK #46

"Baby, I Couldn't See" / "Penny Sir" - Pye 7N 17849 - 1969

"Take A Girl Like You" / "I'm Gonna Be A Rich Man" - Pye - 7N 17904 - 1970

"I'm Gonna be a Rich Man" / "In The Beginning" Pye 7N 17956 - 1970

"Stoney Ground" / "I'll Give You Love" - MCA MKS 5075–1971

"Where Were You When I Needed Your Love" / "Love Me Nice And Easy" - Summit SU 100

"Closer To Loving You" / "Change My Life" - Psycho P 2603–1978

"Baby Now That I've Found You" / "Build Me Up Buttercup" - Old Gold OG9407 - (1979 Re-release)

"Baby Now That I've Found You" / "Build Me Up Buttercup" - Flashback FBS 6 - (1979 Re-release)

"Baby Now That I've Found You" (1989 Re Mix) / "Build Me Up Buttercup" - PRT PYS 24 - 1989

"Build Me Up Buttercup" - CD single - (1998 Re-release) - UK #71

UK albums

''From the Foundations'' - Pye NSPL 18206 - 1967

''Rocking the Foundations'' - Pye NSPL 18227 - 1968

''The Foundations'' - Marble Arch MALS 1157–1968

''Digging the Foundations'' - Pye NSPL 18290 - 1969

''Golden Hour of the Foundations'' (Greatest Hits) - GH 574 - 1973

''Back to the Beat'' - PRT DOW7 - 1983

''Best Of'' - PRT PYL 4003–1987

UK EPs 7"

"It's All Right" - Pye NEP24297 - 1968

"Mini Monster" - Pye PMM.103

UK EPs 12"

"Baby, Now That I've Found You" - Pye Big Deal BD 107 - (4 tracks)

"Baby, Now That I've Found You" - PRT Pyt 24 - 1989 - (3 tracks incl remix)

UK CDs

''Golden Hour Of The Foundations'' - Knight Records KGH CD 104 - 1990

''Strong Foundations'' - The Singles and More - Music Club - MCCD 327 - 1997

''Build Me Up Buttercup'' - Castle Select SELCD 527 - 1998

''Baby, Now That I've Found You'' - Sequel Records - NEECD 300 - 1999

''Build Me Up Buttercup'' (The Complete Pye Collection) [Remastered] - Castle - 2004

US singles

"Baby Now That I've Found You" / "Come On Back To Me" - Uni 55038 - 1967

"Back On My Feet Again" / "I Can Take Or Leave Your Loving" - Uni 55058 - 1968

"Any Old Time (You're Lonely And Sad)" / "We Are Happy People" - Uni 55073 -1968

"Build Me Up Buttercup" / "New Direction" - Uni 55101 - 1968

"In The Bad Bad Old Days (Before You Loved Me)" / "Give Me Love" - Uni 55117 - 1969

"Born To Live, Born To Die" / "Why Did You Cry" - Uni 55162 - 1969 - UK #46

"My Little Chickadee" / "Solomon Grundy" - Uni 55137 - 1969

"Take A Girl Like You" / "I'm Gonna Be A Rich Man" - Uni 55210 - 1970

"Stoney Ground" / "I'll Give You Love" - Uni 55315 - 1971

"Build Me Up Buttercup" / "Baby, Now That I've Found You" - Eric 192 (Re-release)

US albums

''Baby Now That I've Found You'' - Uni 3016 (Mono)/73016 (Stereo) - 1967

''Build Me Up Buttercup'' - Uni 73043—1968

''Digging The Foundations'' - Uni 73058—1969

Canadian singles

"Baby Now That I've Found You" - Pye 827—1968 - #1

"Back On My Feet Again" - Pye 833 - #29

"Build Me Up Buttercup" - Pye 17636 - #1

"In the Bad Bad Old Days (Before You Loved Me)" - Pye 17702 - #23

References

Reference links and other information

The Crazy World Of Arthur Brown http://www.richieunterberger.com/brown.html

Alex Gitlin Pluto page http://www.alexgitlin.com/npp/pluto.htm

Gary Moberley info http://web.archive.org/web/20091027104142/http://geocities.com/SunsetStrip/Diner/2674/moberley_gary_a.htm

Pluto Website http://www.pluto-rockband.com/70141/info.php?p=2&pno;=0

Alan Warner site page http://www.alan-warner.com/10102/info.php?p=4&pno;=0

Ride Your Pony http://rideyourpony-twighlightzone.blogspot.com/2007/09/group-made-up-of-west-indians.html

Mike D'Abo Foundations http://www.mikedabo.com/foundations.htm

Buddah Records @ Disco-Disco.com http://www.disco-disco.com/labels/buddah.shtml

www.javirunner-brightsideofthemoon.blogspot.com Info on Barry Class

www.tamworthbands.com Info on Barry Class

External links

Clem Curtis & The Foundations web site

Clem Curtis web site

Alan Warner site page

Interview with Clem Curtis

Category:British pop music groups Category:British soul musical groups Category:Musical groups established in 1967 Category:Musical groups disestablished in 1970 Category:The Foundations Category:Pye Records artists

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