Gödel's incompleteness theorems

Gödel's incompleteness theorems are two theorems of mathematical logic that demonstrate the inherent limitations of every formal axiomatic system capable of modelling basic arithmetic.  These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

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

Employing a diagonal argument, Gödel's incompleteness theorems were the first of several closely related theorems on the limitations of formal systems. They were followed by Tarski's undefinability theorem on the formal undefinability of truth, Church's proof that Hilbert's Entscheidungsproblem is unsolvable, and Turing's theorem that there is no algorithm to solve the halting problem.

Formal systems: completeness, consistency, and effective axiomatization

The incompleteness theorems apply to formal systems that are of sufficient complexity to express the basic arithmetic of the natural numbers and which are consistent, and  effectively axiomatized, these concepts being detailed below. Particularly in the context of first-order logic, formal systems are also called formal theories. In general, a formal system is a deductive apparatus that consists of a particular set of axioms along with rules of symbolic manipulation (or rules of inference) that allow for the derivation of new theorems from the axioms.  One example of such a system is first-order Peano arithmetic, a system in which all variables are intended to denote natural numbers. In other systems, such as set theory, only some sentences of the formal system express statements about the natural numbers. The incompleteness theorems are about formal provability within these systems, rather than about "provability" in an informal sense.

There are several properties that a formal system may have, including completeness, consistency, and the existence of an effective axiomatization. The incompleteness theorems show that systems which contain a sufficient amount of arithmetic cannot possess all three of these properties.

Effective axiomatization

A formal system is said to be effectively axiomatized (also called effectively generated) if its set of theorems is a recursively enumerable set (Franzén 2005, p. 112).

This means that there is a computer program that, in principle, could enumerate all the theorems of the system without listing any statements that are not theorems. Examples of effectively generated theories include Peano arithmetic and Zermelo–Fraenkel set theory (ZFC).

The theory known as true arithmetic consists of all true statements about the standard integers in the language of Peano arithmetic. This theory is consistent, and complete, and contains a sufficient amount of arithmetic. However it does not have a recursively enumerable set of axioms, and thus does not satisfy the hypotheses of the incompleteness theorems.

Completeness

A set of axioms is (syntactically, or negation-) complete if, for any statement in the axioms' language, that statement or its negation is provable from the axioms (Smith 2007, p.  24). This is the notion relevant for Gödel's first Incompleteness theorem. It  is not to be confused with semantic completeness, which means that the set of axioms proves all the semantic tautologies of the given language. In his completeness theorem, Gödel proved that first order logic is semantically complete. But it is not syntactically complete, since there are sentences expressible in the language of first order logic that can be neither proved nor disproved from the axioms of logic alone.

In a mere system of logic it would be absurd to expect syntactic completeness. But in a system of mathematics, thinkers such as Hilbert had believed that it is just a matter of time to find such an axiomatization that would allow one to either prove or disprove (by proving its negation) each and every mathematical formula.

A formal system might be syntactically incomplete by design, as logics generally are. Or it may be incomplete simply because not all the necessary axioms have been discovered or included. For example, Euclidean geometry without the parallel postulate is incomplete, because some statements in the language (such as the parallel postulate itself) can not be proved from the remaining axioms. Similarly, the theory of dense linear orders is not complete, but becomes complete with an extra axiom stating that there are no endpoints in the order. The continuum hypothesis is a statement in the language of ZFC that is not provable within ZFC, so ZFC is not complete. In this case, there is no obvious candidate for a new axiom that resolves the issue.

The theory of first order Peano arithmetic seems to be consistent. Assuming this is indeed the case, note that it has an infinite but recursively enumerable set of axioms, and can encode enough arithmetic for the hypotheses of the incompleteness theorem. Thus by the first incompleteness theorem, Peano Arithmetic is not complete. The theorem gives an explicit example of a statement of arithmetic that is neither provable nor disprovable in Peano's arithmetic. Moreover, this statement is true in the usual model. In addition, no effectively axiomatized, consistent extension of Peano arithmetic can be complete.

Consistency
A set of axioms is (simply) consistent if there is no statement such that both the statement and its negation are provable from the axioms, and inconsistent otherwise.

Peano arithmetic is provably consistent from ZFC, but not from within itself. Similarly, ZFC is not provably consistent from within itself, but ZFC + "there exists an inaccessible cardinal" proves ZFC is consistent because if  is the least such cardinal, then  sitting inside the von Neumann universe is a model of ZFC, and a theory is consistent if and only if it has a model.

If one takes all statements in the language of Peano arithmetic as axioms, then this theory is complete, has a recursively enumerable set of axioms, and can describe addition and multiplication. However, it is not consistent.

Additional examples of inconsistent theories arise from the paradoxes that result when the axiom schema of unrestricted comprehension is assumed in set theory.

Systems which contain arithmetic

The incompleteness theorems apply only to formal systems which are able to prove a sufficient collection of facts about the natural numbers. One sufficient collection is the set of theorems of Robinson arithmetic Q.  Some systems, such as Peano arithmetic, can directly express statements about natural numbers. Others, such as ZFC set theory, are able to interpret statements about natural numbers into their language. Either of these options is appropriate for the incompleteness theorems.

The theory of algebraically closed fields of a given characteristic is complete, consistent, and has an infinite but recursively enumerable set of axioms. However it is not possible to encode the integers into this theory, and the theory cannot describe arithmetic of integers.  A similar example is the theory of real closed fields, which is essentially equivalent to Tarski's axioms for Euclidean geometry. So Euclidean geometry itself (in Tarski's formulation) is an example of a complete, consistent, effectively axiomatized theory.

The system of Presburger arithmetic consists of a set of axioms for the natural numbers with just the addition operation (multiplication is omitted). Presburger arithmetic is complete, consistent, and recursively enumerable and can encode addition but not multiplication of natural numbers, showing that for Gödel's theorems one needs the theory to encode not just addition but also multiplication.

Dan Willard (2001) has studied some weak families of arithmetic systems which allow enough arithmetic as relations to formalise Gödel numbering, but which are not strong enough to have multiplication as a function, and so fail to prove the second incompleteness theorem; these systems are consistent and capable of proving their own consistency (see self-verifying theories).

Conflicting goals
In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any incorrect results. For example, we could imagine a set of true axioms which allow us to prove every true arithmetical claim about the natural numbers (Smith 2007, p 2). In the standard system of first-order logic, an inconsistent set of axioms will prove every statement in its language (this is sometimes called the principle of explosion), and is thus automatically complete. A set of axioms that is both complete and consistent, however, proves a maximal set of non-contradictory theorems (Hinman 2005, p. 143).

The pattern illustrated in the previous sections with Peano arithmetic, ZFC, and ZFC + "there exists an inaccessible cardinal" cannot generally be broken. Here ZFC + "there exists an inaccessible cardinal" cannot from itself, be proved consistent. It is also not complete, as illustrated by the in ZFC + "there exists an inaccessible cardinal" theory unresolved continuum hypothesis.

The first incompleteness theorem shows that, in formal systems that can express basic arithmetic, a complete and consistent finite list of axioms can never be created: each time an additional, consistent statement is added as an axiom, there are other true statements that still cannot be proved, even with the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system inconsistent. It is not even possible for an infinite list of axioms to be complete, consistent, and effectively axiomatized.

First incompleteness theorem

Gödel's first incompleteness theorem first appeared as "Theorem VI" in Gödel's 1931 paper "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I".  The hypotheses of the theorem were improved shortly thereafter by J. Barkley Rosser (1936) using Rosser's trick. The resulting theorem (incorporating Rosser's improvement) may be paraphrased in English as follows, where "formal system" includes the assumption that the system is effectively generated.

First Incompleteness Theorem: "Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F." (Raatikainen 2015)

The unprovable statement GF referred to by the theorem is often referred to as "the Gödel sentence"  for the system F. The proof constructs a particular Gödel sentence for the system F, but there are infinitely many statements in the language of the system that share the same properties, such as the conjunction of the Gödel sentence and any logically valid sentence.

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

Syntactic form of the Gödel sentence

The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in F. However, the sequence of steps is such that the constructed sentence turns out to be GF itself. In this way, the Gödel sentence GF indirectly states its own unprovability within F (Smith 2007, p. 135).

To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.

Thus, although the Gödel sentence refers indirectly to sentences of the system F, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation (Smith 2007, p. 141).   As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form. In particular, it can be expressed as a formula in the language of arithmetic consisting of a number of leading universal quantifiers followed by a quantifier-free body (these formulas are at level  of the arithmetical hierarchy).  Via the MRDP theorem, the Gödel sentence can be re-written as a statement that a particular polynomial in many variables with integer coefficients never takes the value zero when integers are substituted for its variables (Franzén 2005, p. 71).

Truth of the Gödel sentence

The first incompleteness theorem shows that the Gödel sentence  GF of an appropriate formal theory F is unprovable in F. Because, when interpreted as a statement about arithmetic, this unprovability is exactly what the sentence (indirectly) asserts, the Gödel sentence is, in fact, true (Smoryński 1977 p. 825; also see Franzén 2005 pp. 28–33). For this reason, the sentence GF is often said to be "true but unprovable." (Raatikainen 2015).  However, since the Gödel sentence cannot itself formally specify its intended interpretation, the truth of the sentence GF may only be arrived at via a meta-analysis from outside the system. In general, this meta-analysis can be carried out within the weak formal system known as primitive recursive arithmetic, which proves the implication Con(F)→GF, where Con(F) is a canonical sentence asserting the consistency of F (Smoryński 1977 p. 840, Kikuchi and Tanaka 1994 p. 403).

Although the Gödel sentence of a consistent theory is true as a statement about the intended interpretation of arithmetic, the Gödel sentence will be false in some nonstandard models of arithmetic, as a consequence of Gödel's completeness theorem (Franzén 2005, p. 135).  That theorem shows that, when a sentence is independent of a theory, the theory will have models in which the sentence is true and models in which the sentence is false.  As described earlier, the Gödel sentence of a system F is an arithmetical statement which claims that no number exists with a particular property. The incompleteness theorem shows that this claim will be independent of the system F, and the truth of the Gödel sentence follows from the fact that no standard natural number has the property in question. Any model in which the Gödel sentence is false must contain some element which satisfies the property within that model. Such a model must be "nonstandard" – it must contain elements that do not correspond to any standard natural number (Raatikainen 2015, Franzén 2005, p. 135).

Relationship with the liar paradox

Gödel specifically cites Richard's paradox and the liar paradox as semantical analogues to his syntactical incompleteness result in the introductory section of "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I".  The liar paradox is the sentence "This sentence is false."  An analysis of the liar sentence shows that it cannot be true (for then, as it asserts, it is false), nor can it be false (for then, it is true). A Gödel sentence G for a system F makes a similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the system F."  The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar sentence.

It is not possible to replace "not provable" with "false" in a Gödel sentence because the predicate "Q is the Gödel number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's undefinability theorem, was discovered independently both by Gödel, when he was working on the proof of the incompleteness theorem, and by the theorem's namesake, Alfred Tarski.

Extensions of Gödel's original result

Compared to the theorems stated in Gödel's 1931 paper, many contemporary statements of the incompleteness theorems are more general in two ways. These generalized statements are phrased to apply to a broader class of systems, and they are phrased to incorporate weaker consistency assumptions.

Gödel demonstrated the incompleteness of the system of Principia Mathematica, a particular system of arithmetic, but a parallel demonstration could be given for any effective system of a certain expressiveness. Gödel commented on this fact in the introduction to his paper, but restricted the proof to one system for concreteness.  In modern statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for the incompleteness theorem, so that it is not limited to any particular formal system. The terminology used to state these conditions was not yet developed in 1931 when Gödel published his results.

Gödel's original statement and proof of the incompleteness theorem requires the assumption that the system is not just consistent but ω-consistent. A system is ω-consistent if it is not ω-inconsistent, and is ω-inconsistent if there is a predicate P such that for every specific natural number m the system proves ~P(m), and yet the system also proves that there exists a natural number n such that P(n). That is, the system says that a number with property P exists while denying that it has any specific value. The ω-consistency of a system implies its consistency, but consistency does not imply ω-consistency. J. Barkley Rosser (1936) strengthened the incompleteness theorem by finding a variation of the proof (Rosser's trick) that only requires the system to be consistent, rather than ω-consistent. This is mostly of technical interest, because all true formal theories of arithmetic (theories whose axioms are all true statements about natural numbers) are ω-consistent, and thus Gödel's theorem as originally stated applies to them. The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel–Rosser theorem.

Second incompleteness theorem

For each formal system F containing basic arithmetic, it is possible to canonically define a formula Cons(F) expressing the consistency of F. This formula expresses the property that "there does not exist a natural number coding a formal derivation within the system F whose conclusion is a syntactic contradiction."   The syntactic contradiction is often taken to be "0=1", in which case Cons(F) states "there is no natural number that codes a derivation of '0=1' from the axioms of F."

Gödel's second incompleteness theorem shows that, under general assumptions, this canonical consistency statement Cons(F) will not be provable in F. The theorem first appeared as "Theorem XI" in Gödel's 1931 paper "On Formally Undecidable Propositions in Principia Mathematica and Related Systems I". In the following statement, the term "formalized system" also includes an assumption that F is effectively axiomatized.
Second Incompleteness Theorem: "Assume F is a consistent formalized system which contains elementary arithmetic. Then ." (Raatikainen 2015)

This theorem is stronger than the first incompleteness theorem because the statement constructed in the first incompleteness theorem does not directly express the consistency of the system. The proof of the second incompleteness theorem is obtained by formalizing the proof of the first incompleteness theorem within the system F itself.

Expressing consistency

There is a technical subtlety in the second incompleteness theorem regarding the method of expressing the consistency of F as a formula in the language of F. There are many ways to express the consistency of a system, and not all of them lead to the same result.  The formula Cons(F) from the second incompleteness theorem is a particular expression of consistency.

Other formalizations of the claim that F is consistent may be inequivalent in F, and some may even be provable. For example, first-order Peano arithmetic (PA) can prove that "the largest consistent subset of PA" is consistent. But, because PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest consistent subset of PA" is meant here to be the largest consistent initial segment of the axioms of PA under some particular effective enumeration.)

The Hilbert–Bernays conditions

The standard proof of the second incompleteness theorem assumes that the provability predicate ProvA(P) satisfies the Hilbert–Bernays provability conditions. Letting #(P) represent the Gödel number of a formula P, the derivability conditions say:

 If F proves P, then F proves ProvA(#(P)).
 F proves 1.; that is, F proves that if F proves P, then F proves ProvA(#(P)). In other words, F proves that ProvA(#(P)) implies ProvA(#(ProvA(#(P)))).
 F proves that if F proves that (P → Q) and F proves P then F proves Q. In other words, F proves that ProvA(#(P → Q)) and ProvA(#(P)) imply ProvA(#(Q)).

There are systems, such as Robinson arithmetic, which are strong enough to meet the assumptions of the first incompleteness theorem, but which do not prove the Hilbert—Bernays conditions. Peano arithmetic, however, is strong enough to verify these conditions, as are all theories stronger than Peano arithmetic.

Implications for consistency proofs

Gödel's second incompleteness theorem also implies that a system F1 satisfying the technical conditions outlined above cannot prove the consistency of any system F2 that proves the consistency of F1. This is because such a system F1 can prove that if F2 proves the consistency of F1, then F1 is in fact consistent. For the claim that F1 is consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction in F1". If F1 were in fact inconsistent, then F2 would prove for some n that n is the code of a contradiction in F1. But if F2 also proved that F1 is consistent (that is, that there is no such n), then it would itself be inconsistent. This reasoning can be formalized in F1 to show that if F2 is consistent, then F1 is consistent. Since, by second incompleteness theorem, F1 does not prove its consistency, it cannot prove the consistency of F2 either.

This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the consistency of Peano arithmetic using any finitistic means that can be formalized in a system the consistency of which is provable in Peano arithmetic (PA). For example, the system of primitive recursive arithmetic (PRA), which is widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA cannot prove the consistency of PA. This fact is generally seen to imply that Hilbert's program, which aimed to justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out (Franzén 2005, p. 106).

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

The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that can be formalized in the system that is proved consistent. For example, Gerhard Gentzen proved the consistency of Peano arithmetic in a different system that includes an axiom asserting that the ordinal called ε0 is wellfounded; see Gentzen's consistency proof. Gentzen's theorem spurred the development of ordinal analysis in proof theory.

Examples of undecidable statements

There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is the proof-theoretic sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified deductive system.  The second sense, which will not be discussed here, is used in relation to computability theory and applies not to statements but to decision problems, which are countably infinite sets of questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable function that correctly answers every question in the problem set (see undecidable problem).

Because of the two meanings of the word undecidable, the term independent is sometimes used instead of undecidable for the "neither provable nor refutable" sense.

Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of whether the truth value of the statement is well-defined, or whether it can be determined by other means. Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be known or is ill-specified, is a controversial point in the philosophy of mathematics.

The combined work of Gödel and Paul Cohen has given two concrete examples of undecidable statements (in the first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC axioms except the axiom of choice).  These results do not require the incompleteness theorem.   Gödel proved in 1940 that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither is provable from ZF, and the continuum hypothesis cannot be proved from ZFC.

In 1973, Saharon Shelah showed that the Whitehead problem in group theory is undecidable, in the first sense of the term, in standard set theory.

Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another incompleteness theorem in that setting. Chaitin's incompleteness theorem states that for any system that can represent enough arithmetic, there is an upper bound c such that no specific number can be proved in that system to have Kolmogorov complexity greater than c. While Gödel's theorem is related to the liar paradox, Chaitin's result is related to Berry's paradox.

Undecidable statements provable in larger systems

These are natural mathematical equivalents of the Gödel "true but undecidable" sentence. They can be proved in a larger system which is generally accepted as a valid form of reasoning, but are undecidable in a more limited system such as Peano Arithmetic.

In 1977, Paris and Harrington proved that the Paris–Harrington principle, a version of the infinite Ramsey theorem, is undecidable in (first-order) Peano arithmetic, but can be proved in the stronger system of second-order arithmetic. Kirby and Paris later showed that Goodstein's theorem, a statement about sequences of natural numbers somewhat simpler than the Paris–Harrington principle, is also undecidable in Peano arithmetic.

Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system codifying the principles acceptable based on a philosophy of mathematics called predicativism.  The related but more general graph minor theorem (2003) has consequences for computational complexity theory.

Relationship with computability

The incompleteness theorem is closely related to several results about undecidable sets in recursion theory.

Stephen Cole Kleene (1943) presented a proof of Gödel's incompleteness theorem using basic results of computability theory. One such result shows that the halting problem is undecidable: there is no computer program that can correctly determine, given any program P as input, whether P eventually halts when run with a particular given input. Kleene showed that the existence of a complete effective system of arithmetic with certain consistency properties would force the halting problem to be decidable, a contradiction. This method of proof has also been presented by Shoenfield (1967, p. 132); Charlesworth (1980); and Hopcroft and Ullman (1979).

Franzén (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,...,xk) with integer coefficients, determines whether there is an integer solution to the equation p = 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language of arithmetic, if a multivariate integer polynomial equation p = 0 does have a solution in the integers then any sufficiently strong system of arithmetic T will prove this. Moreover, if the system T is ω-consistent, then it will never prove that a particular polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were complete and ω-consistent, it would be possible to determine algorithmically whether a polynomial equation has a solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in contradiction to Matiyasevich's theorem. Moreover, for each consistent effectively generated system T, it is possible to effectively generate a multivariate polynomial p over the integers such that the equation p = 0 has no solutions over the integers, but the lack of solutions cannot be proved in T (Davis 2006:416, Jones 1980).

Smorynski (1977, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially undecidable (see Kleene 1967, p. 274).

Chaitin's incompleteness theorem gives a different method of producing independent sentences, based on Kolmogorov complexity. Like the proof presented by Kleene that was mentioned above, Chaitin's theorem only applies to theories with the additional property that all their axioms are true in the standard model of the natural numbers. Gödel's incompleteness theorem is distinguished by its applicability to consistent theories that nonetheless include statements that are false in the standard model; these theories are known as ω-inconsistent.

Proof sketch for the first theorem

The proof by contradiction has three essential parts. To begin, choose a formal system that meets the proposed criteria:

 Statements in the system can be represented by natural numbers (known as Gödel numbers). The significance of this is that properties of statements—such as their truth and falsehood—will be equivalent to determining whether their Gödel numbers have certain properties, and that properties of the statements can therefore be demonstrated by examining their Gödel numbers. This part culminates in the construction of a formula expressing the idea that "statement S is provable in the system" (which can be applied to any statement "S" in the system).
 In the formal system it is possible to construct a number whose matching statement, when interpreted, is self-referential and essentially says that it (i.e. the statement itself) is unprovable. This is done using a technique called "diagonalization" (so-called because of its origins as Cantor's diagonal argument).
 Within the formal system this statement permits a demonstration that it is neither provable nor disprovable in the system, and therefore the system cannot in fact be ω-consistent. Hence the original assumption that the proposed system met the criteria is false.

Arithmetization of syntax

The main problem in fleshing out the proof described above is that it seems at first that to construct a statement p that is equivalent to "p cannot be proved", p would somehow have to contain a reference to p, which could easily give rise to an infinite regress. Gödel's ingenious technique is to show that statements can be matched with numbers (often called the arithmetization of syntax) in such a way that "proving a statement" can be replaced with "testing whether a number has a given property". This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. The same technique was later used by Alan Turing in his work on the Entscheidungsproblem.

In simple terms, a method can be devised so that every formula or statement that can be formulated in the system gets a unique number, called its Gödel number, in such a way that it is possible to mechanically convert back and forth between formulas and Gödel numbers.  The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that such numbers can be constructed. A simple example is the way in which English is stored as a sequence of numbers in computers using ASCII or Unicode:
 The word HELLO is represented by 72-69-76-76-79 using decimal ASCII, i.e. 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, i.e. the number 120061121032061062032121061120.

In principle, proving a statement true or false can be shown to be equivalent to proving that the number matching the statement does or doesn't have a given property. Because the formal system is strong enough to support reasoning about numbers in general, it can support reasoning about numbers that represent formulae and statements as well. Crucially, because the system can support reasoning about properties of numbers, the results are equivalent to reasoning about provability of their equivalent statements.

Construction of a statement about "provability"

Having shown that in principle the system can indirectly make statements about provability, by analyzing properties of those numbers representing statements it is now possible to show how to create a statement that actually does this.

A formula F(x) that contains exactly one free variable x is called a statement form or class-sign. As soon as x is replaced by a specific number, the statement form turns into a bona fide statement, and it is then either provable in the system, or not. For certain formulas one can show that for every natural number n, F(n) is true if and only if it can be proved (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as "2×3=6".

Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form F(x) can be assigned a Gödel number denoted by G(F). The choice of the free variable used in the form F(x) is not relevant to the assignment of the Gödel number G(F).

The notion of provability itself can also be encoded by Gödel numbers, in the following way: since a proof is a list of statements which obey certain rules, the Gödel number of a proof can be defined. Now, for every statement p, one may ask whether a number x is the Gödel number of its proof. The relation between the Gödel number of p and x, the potential Gödel number of its proof, is an arithmetical relation between two numbers.  Therefore, there is a statement form Bew(y) that uses this arithmetical relation to state that a Gödel number of a proof of y exists:
Bew(y) = ∃ x ( y is the Gödel number of a formula and x is the Gödel number of a proof of the formula encoded by y).
The name Bew is short for beweisbar, the German word for "provable"; this name was originally used by Gödel to denote the provability formula just described.  Note that "Bew(y)" is merely an abbreviation that represents a particular, very long, formula in the original language of T; the string "Bew" itself is not claimed to be part of this language.

An important feature of the formula Bew(y) is that if a statement p is provable in the system then Bew(G(p)) is also provable. This is because any proof of p would have a corresponding Gödel number, the existence of which causes Bew(G(p)) to be satisfied.

Diagonalization

The next step in the proof is to obtain a statement which, indirectly, asserts its own unprovability. Although Gödel constructed this statement directly, the existence of at least one such statement follows from the diagonal lemma, which says that for any sufficiently strong formal system and any statement form F there is a statement p such that the system proves
p ↔ F(G(p)).
By letting F be the negation of Bew(x), we obtain the theorem
p ↔ ~Bew(G(p))
and the p defined by this roughly states that its own Gödel number is the Gödel number of an unprovable formula.

The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the resulting Gödel number will be that of an unprovable statement. But when this calculation is performed, the resulting Gödel number turns out to be the Gödel number of p itself. This is similar to the following sentence in English:
", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable.
This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is obtained as a result, and thus this sentence indirectly asserts its own unprovability. The proof of the diagonal lemma employs a similar method.

Now, assume that the axiomatic system is ω-consistent, and let p be the statement obtained in the previous section.

If p were provable, then Bew(G(p)) would be provable, as argued above. But p asserts the negation of Bew(G(p)). Thus the system would be inconsistent, proving both a statement and its negation.  This contradiction shows that p cannot be provable.

If the negation of p were provable, then Bew(G(p)) would be provable (because p was constructed to be equivalent to the negation of Bew(G(p))). However, for each specific number x, x cannot be the Gödel number of the proof of p, because p is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with a certain property (that it is the Gödel number of the proof of p), but on the other hand, for every specific number x, we can prove that it does not have this property. This is impossible in an ω-consistent system. Thus the negation of p is not provable.

Thus the statement p is undecidable in our axiomatic system: it can neither be proved nor disproved within the system.

In fact, to show that p is not provable only requires the assumption that the system is consistent. The stronger assumption of ω-consistency is required to show that the negation of p is not provable. Thus, if p is constructed for a particular system:
If the system is ω-consistent, it can prove neither p nor its negation, and so p is undecidable.
If the system is consistent, it may have the same situation, or it may prove the negation of p. In the later case, we have a statement ("not p") which is false but provable, and the system is not ω-consistent.

If one tries to "add the missing axioms" to avoid the incompleteness of the system, then one has to add either p or "not p" as axioms. But then the definition of "being a Gödel number of a proof" of a statement changes. which means that the formula Bew(x) is now different. Thus when we apply the diagonal lemma to this new Bew, we obtain a new statement p, different from the previous one, which will be undecidable in the new system if it is ω-consistent.

Proof via Berry's paradox
George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently discovered by Saul Kripke (Boolos 1998, p. 383).  Boolos's proof proceeds by constructing, for any computably enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a "different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388).

Computer verified proofs

The incompleteness theorems are among a relatively small number of nontrivial theorems that have been transformed into formalized theorems that can be completely verified by proof assistant software. Gödel's original proofs of the incompleteness theorems, like most mathematical proofs, were written in natural language intended for human readers.

Computer-verified proofs of versions of the first incompleteness theorem were announced by Natarajan Shankar in 1986 using Nqthm (Shankar 1994), by Russell O'Connor in 2003 using Coq (O'Connor 2005) and by John Harrison in 2009 using HOL Light (Harrison 2009). A computer-verified proof of both incompleteness theorems was announced by Lawrence Paulson in 2013 using Isabelle (Paulson 2014).

Proof sketch for the second theorem

The main difficulty in proving the second incompleteness theorem is to show that various facts about provability used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate for provability. Once this is done, the second incompleteness theorem follows by formalizing the entire proof of the first incompleteness theorem within the system itself.

Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be proved from within the system itself. The demonstration above shows that if the system is consistent, then p is not provable. The proof of this implication can be formalized within the system, and therefore the statement "p is not provable", or "not P(p)" can be proved in the system.

But this last statement is equivalent to p itself (and this equivalence can be proved in the system), so p can be proved in the system. This contradiction shows that the system must be inconsistent.

Discussion and implications

The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a single system of formal logic to define their principles.

Consequences for logicism and Hilbert's second problem

The incompleteness theorem is sometimes thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451–468). Bob Hale and Crispin Wright argue that it is not a problem for logicism because the incompleteness theorems apply equally to first order logic as they do to arithmetic. They argue that only those who believe that the natural numbers are to be defined in terms of first order logic have this problem.

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the status of Hilbert's second problem is not yet decided (see "Modern viewpoints on the status of the problem").

Minds and machines

Authors including the philosopher J. R. Lucas and physicist Roger Penrose have debated what, if anything, Gödel's incompleteness theorems imply about human intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the Church–Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Gödel's incompleteness theorems would apply to it.

Hilary Putnam (1960) suggested that while Gödel's theorems cannot be applied to humans, since they make mistakes and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. Assuming that it is consistent, either its consistency cannot be proved or it cannot be represented by a Turing machine.

Avi Wigderson (2010) has proposed that the concept of mathematical "knowability" should be based on computational complexity rather than logical decidability.  He writes that "when knowability is interpreted by modern standards, namely via computational complexity, the Gödel phenomena are very much with us."

Douglas Hofstadter, in his books Gödel, Escher, Bach and I Am a Strange Loop, cites Gödel's theorems as an example of what he calls a strange loop, a hierarchical, self-referential structure existing within an axiomatic formal system. He argues that this is the same kind of structure which gives rise to consciousness, the sense of "I", in the human mind. While the self-reference in Gödel's theorem comes from the Gödel sentence asserting its own unprovability within the formal system of Principia Mathematica, the self-reference in the human mind comes from the way in which the brain abstracts and categorises stimuli into "symbols", or groups of neurons which respond to concepts, in what is effectively also a formal system, eventually giving rise to symbols modelling the concept of the very entity doing the perception.
Hofstadter argues that a strange loop in a sufficiently complex formal system can give rise to a "downward" or "upside-down" causality, a situation in which the normal hierarchy of cause-and-effect is flipped upside-down. In the case of Gödel's theorem, this manifests, in short, as the following:

"Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false." (I Am a Strange Loop.)

In the case of the mind, a far more complex formal system, this "downward causality" manifests, in Hofstadter's view, as the ineffable human instinct that the causality of our minds lies on the high level of desires, concepts, personalities, thoughts and ideas, rather than on the low level of interactions between neurons or even fundamental particles, even though according to physics the latter seems to possess the causal power.

"There is thus a curious upside-downness to our normal human way of perceiving the world: we are built to perceive “big stuff” rather than “small stuff”, even though the domain of the tiny seems to be where the actual motors driving reality reside." (I Am a Strange Loop.)

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 system within the language of the system (Priest 2006:47). Stewart Shapiro (2002) gives a more mixed appraisal of the applications of Gödel's theorems to dialetheism.

Appeals to the incompleteness theorems in other fields

Appeals and analogies are sometimes made to the incompleteness theorems in support of arguments that go beyond mathematics and logic. Several authors have commented negatively on such extensions and interpretations, including Torkel Franzén (2005); Panu Raatikainen (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.).

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 system of arithmetic without any axioms of induction. By 1928, Ackermann had communicated a modified proof to Bernays; this modified proof led Hilbert to announce his belief in 1929 that the consistency of arithmetic had been demonstrated and that a consistency proof of analysis would likely soon follow. After the publication of the incompleteness theorems showed that Ackermann's modified proof must be erroneous, von Neumann produced a concrete example showing that its main technique was unsound (Zach 2006, p. 418, Zach 2003, p. 33).

In the course of his research, Gödel discovered that although a sentence which asserts its own falsehood leads to paradox, a sentence that asserts its own non-provability does not. In particular, Gödel was aware of the result now called Tarski's indefinability theorem, although he never published it. Gödel announced his first incompleteness theorem to Carnap, Feigel and Waismann on August 26, 1930; all four would attend the Second Conference on the Epistemology of the Exact Sciences, 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,

This speech quickly became known as a summary of Hilbert's beliefs on mathematics (its final six words, "Wir müssen wissen. Wir werden wissen!", were used as Hilbert's epitaph in 1943). Although Gödel was likely in attendance for Hilbert's address, the two never met face to face (Dawson 1996, p. 72).

Gödel announced his first incompleteness theorem at a roundtable discussion session on the third day of the conference. The announcement drew little attention apart from that of von Neumann, who pulled Gödel aside for conversation. Later that year, working independently with knowledge of the first incompleteness theorem, von Neumann obtained a proof of the second incompleteness theorem, which he announced to Gödel in a letter dated November 20, 1930 (Dawson 1996, p. 70). Gödel had independently obtained the second incompleteness theorem and included it in his submitted manuscript, which was received by Monatshefte für Mathematik on November 17, 1930.

Gödel's paper was published in the Monatshefte in 1931 under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" ("On Formally Undecidable Propositions in Principia Mathematica and Related Systems I"). As the title implies, Gödel originally planned to publish a second part of the paper in the next volume of the Monatshefte; the prompt acceptance of the first paper was one reason he changed his plans (van Heijenoort 1967:328, footnote 68a).

Generalization and acceptance

Gödel gave a series of lectures on his theorems at Princeton in 1933–1934 to an audience that included Church, Kleene, and Rosser. By this time, Gödel had grasped that the key property his theorems required is that the system must be effective (at the time, the term "general recursive" was used). Rosser proved in 1936 that the hypothesis of ω-consistency, which was an integral part of Gödel's original proof, could be replaced by simple consistency, if the Gödel sentence was changed in an appropriate way. These developments left the incompleteness theorems in essentially their modern form.

Gentzen published his consistency proof for first-order arithmetic in 1936. Hilbert accepted this proof as "finitary" although (as Gödel's theorem had already shown) it cannot be formalized within the system of arithmetic that is being proved consistent.

The impact of the incompleteness theorems on Hilbert's program was quickly realized. Bernays included a full proof of the incompleteness theorems in the second volume of Grundlagen der Mathematik (1939), along with additional results of Ackermann on the ε-substitution method and Gentzen's consistency proof of arithmetic. This was the first full published proof of the second incompleteness theorem.

Criticisms

Finsler

Paul Finsler (1926) used a version of Richard's paradox to construct an expression that was false but unprovable in a particular, informal framework he had developed. Gödel was unaware of this paper when he proved the incompleteness theorems (Collected Works Vol. IV., p. 9). Finsler wrote to Gödel in 1931 to inform him about this paper, which Finsler felt had priority for an incompleteness theorem. Finsler's methods did not rely on formalized provability, and had only a superficial resemblance to Gödel's work (van Heijenoort 1967:328). Gödel read the paper but found it deeply flawed, and his response to Finsler laid out concerns about the lack of formalization (Dawson:89). Finsler continued to argue for his philosophy of mathematics, which eschewed formalization, for the remainder of his career.

Zermelo

In September 1931, Ernst Zermelo wrote to Gödel to announce what he described as an "essential gap" in Gödel's argument (Dawson:76). In October, Gödel replied with a 10-page letter (Dawson:76, Grattan-Guinness:512-513), where he pointed out that Zermelo mistakenly assumed that the notion of truth in a system is definable in that system (which is not true in general by Tarski's undefinability theorem). 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.

Wittgenstein

Ludwig Wittgenstein wrote several passages about the incompleteness theorems that were published posthumously in his 1953 Remarks on the Foundations of Mathematics, in particular one section sometimes called the "notorious paragraph" where he seems to confuse the notions of "true" and "provable" in Russell's system. 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.  There has been some controversy about whether Wittgenstein misunderstood the incompleteness theorem or just expressed himself unclearly. Writings in Gödel's Nachlass express the belief that Wittgenstein misread his ideas.

Multiple commentators have read Wittgenstein as misunderstanding Gödel (Rodych 2003), although Juliet Floyd and Hilary Putnam (2000), as well as Graham Priest (2004) have provided textual readings arguing that most 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 stated: "Has Wittgenstein lost his mind? Does he mean it seriously? He intentionally utters trivially nonsensical statements" (Wang 1996:179), and wrote to Karl Menger  that Wittgenstein's comments demonstrate a misunderstanding of the incompleteness theorems writing:

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 system as actually saying "I am not provable", since the system 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 machine
 Gödel's completeness theorem
 Gödel's speed-up theorem
 Gödel, Escher, Bach
 Löb's Theorem
 Minds, Machines and Gödel
 Münchhausen trilemma
 Non-standard model of arithmetic
 Proof theory
 Provability logic
 Quining
 Tarski's undefinability theorem
 Theory of everything#Gödel's incompleteness theorem
 Third Man Argument

References

Citations

Articles by Gödel
 Kurt Gödel, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", Monatshefte für Mathematik und Physik, v. 38 n. 1, pp. 173–198.
 —, 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I", in Solomon Feferman, ed., 1986. Kurt Gödel Collected works, Vol. I.  Oxford University Press, pp. 144–195. . The original German with a facing English translation, preceded by an introductory note by Stephen Cole Kleene.
 —, 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, pp. 304–323. .

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),  (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, . 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, 1879-1931, Harvard University Press, Cambridge Mass.,  (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.
 Bernd Buldt, 2014, "The Scope of Gödel's First Incompleteness Theorem", Logica Universalis, v. 8, pp. 499–552.
 Arthur Charlesworth, 1980, "A Proof of Godel's Theorem in Terms of Computer Programs", Mathematics Magazine, v. 54 n. 3, pp. 109–121.
 Martin Davis, 2006, "The Incompleteness Theorem", Notices of the AMS, v. 53 n. 4, pp. 414.
 Jean van Heijenoort, 1963, "Gödel's Theorem" in Edwards, Paul, ed., Encyclopedia of Philosophy, v. 3. Macmillan, pp. 348–57.
 Geoffrey Hellman, 1981, "How to Gödel a Frege-Russell: Gödel's Incompleteness Theorems and Logicism", Noûs, v. 15 n. 4, Special Issue on Philosophy of Mathematics, 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.
 Martin Hirzel, 2000, "On formally undecidable propositions of Principia Mathematica and related systems I.." An English translation of Gödel's paper. Archived from the original. Sept. 16, 2004.
 Makoto Kikuchi and Kazayuki Tanaka, 1994, "On formalization of model-theoretic proofs of Gödel's theorems", Notre Dame Journal of Formal Logic, v. 5 n. 3, pp. 403–412.
 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.
 Panu Raatikainen, 2015, "Gödel's Incompleteness Theorems", Stanford Encyclopedia of Philosophy. Accessed April 3, 2015.
 Panu Raatikainen, 2005, "On the philosophical relevance of Gödel's incompleteness theorems", Revue Internationale de Philosophie 59 (4):513-534.
 John Barkley Rosser, 1936, "Extensions of some theorems of Gödel and Church", reprinted from the Journal of Symbolic Logic, v. 1 (1936) pp. 87–91, in Martin Davis 1965, The Undecidable (loc. cit.) pp. 230–235.
 —, 1939, "An Informal Exposition of proofs of Gödel's Theorem and Church's Theorem", Reprinted from the Journal of Symbolic Logic, v. 4 (1939) pp. 53–60, in Martin Davis 1965, The Undecidable (loc. cit.) pp. 223–230
 C. Smoryński, 1982, "The incompleteness theorems", in Jon Barwise, ed., Handbook of Mathematical Logic, North-Holland, 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, 2003. "The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program" Synthese, v. 137 n. 1, pp. 211–259.
 —, 2005, "Kurt Gödel, Paper on the incompleteness theorems" in Ivor Grattan-Guinness, ed. Landmark Writings in Western Mathematics, Elsevier, pp. 917–925.

Books about the theorems
 Francesco Berto. There's Something about Gödel:  The Complete Guide to the Incompleteness Theorem John Wiley and Sons. 2010.
 Norbert Domeisen, 1990. Logik der Antinomien. Bern: Peter Lang. 142 S. 1990. . .
 Torkel Franzén, 2005. Gödel's Theorem: An Incomplete Guide to its Use and Abuse. A.K. Peters.
 Douglas Hofstadter, 1979. Gödel, Escher, Bach: An Eternal Golden Braid. Vintage Books. . 1999 reprint: .
 —, 2007. I Am a Strange Loop. Basic Books. . .
 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.  .
 Rudy Rucker, 1995 (1982). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton Univ. Press.
 Peter Smith, 2007. An Introduction to Gödel's Theorems. Cambridge University Press.
 N. Shankar, 1994.  Metamathematics, Machines and Gödel's Proof, Volume 38 of Cambridge tracts in theoretical computer science.
 Raymond Smullyan, 1987. Forever Undecided  - puzzles based on undecidability in formal systems
 —, 1991. Godel's Incompleteness Theorems. Oxford Univ. Press.
 —, 1994. Diagonalization and Self-Reference. Oxford Univ. Press.
 —, 2013. The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs. Courier Corporation. .
 Hao Wang, 1997. A Logical Journey: From Gödel to Philosophy. MIT Press.

Miscellaneous references
 Francesco Berto, 2009, "The Gödel Paradox and Wittgenstein's Reasons" Philosophia Mathematica (III) 17.
 John W. Dawson, Jr., 1997. Logical Dilemmas: The Life and Work of Kurt Gödel, A. K. Peters, Wellesley Mass, .
 Rebecca Goldstein, 2005, Incompleteness: the Proof and Paradox of Kurt Gödel, W. W. Norton & Company.
 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.
 John Harrison, 2009, "Handbook of Practical Logic and Automated Reasoning", Cambridge University Press,
 David Hilbert and Paul Bernays, Grundlagen der Mathematik, Springer-Verlag.
 John Hopcroft and Jeffrey Ullman 1979, Introduction to Automata Theory, Languages, and Computation, Addison-Wesley,
 James P. Jones, Undecidable Diophantine Equations, Bulletin of the American Mathematical Society, v. 3 n. 2, 1980, pp. 859–862.
 Stephen Cole Kleene, 1967, Mathematical Logic. Reprinted by Dover, 2002.
 Russell O'Connor, 2005, "Essential Incompleteness of Arithmetic Verified by Coq", Lecture Notes in Computer Science v. 3603, pp. 245–260.
  Lawrence Paulson, 2013, "A machine-assisted proof of Gödel's incompleteness theorems for the theory of hereditarily finite sets", Review of Symbolic Logic, v. 7 n. 3, 484–498.
 Graham Priest, 1984, "Logic of Paradox Revisited", Journal of Philosophical Logic, v. 13,` n. 2,  pp. 153–179.
 —, 2004, Wittgenstein's Remarks on Gödel's Theorem in  Max Kölbel, ed., Wittgenstein's lasting significance, Psychology Press, pp. 207–227.
 —, 2006, In Contradiction: A Study of the Transconsistent, Oxford University Press,
 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.
 Wolfgang Rautenberg, 2010, A Concise Introduction to Mathematical Logic, 3rd. ed., Springer,
 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.
 Joseph R. Shoenfield (1967), Mathematical Logic. Reprinted by A.K. Peters for the Association for Symbolic Logic, 2001.
 Jeremy Stangroom and Ophelia Benson, Why Truth Matters, Continuum.
 George Tourlakis, Lectures in Logic and Set Theory, Volume 1, Mathematical Logic, Cambridge University Press, 2003.
 Avi Wigderson, 2010, "The Gödel Phenomena in Mathematics: A Modern View", in Kurt Gödel and the Foundations of Mathematics: Horizons of Truth, Cambridge University Press.
 Hao Wang, 1996, A Logical Journey: From Gödel to Philosophy, The MIT Press, Cambridge MA, .
 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

 .
 .
 Paraconsistent Logic § Arithmetic and Gödel's Theorem entry in the Stanford Encyclopedia of Philosophy.
 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.
 October 2011 RadioLab episode about/including Gödel's Incompleteness theorem

Category:Theorems in the foundations of mathematics
Category:Mathematical logic
Category:Model theory
Category:Proof theory
Category:Epistemology
Category:Metatheorems
Incompleteness theorems


Andreu Vivó

Andreu Vivó Tomás ( in Manresa - 30 December 2012 in San Juan de Torruella) was a Spanish male artistic gymnast, representing his nation at international competitions.  He participated at the 2000 Summer Olympics. He also competed at the 2002, 2003 and 2005 World Artistic Gymnastics Championships. He won the gold medal in the team all-around event at the 2005 Mediterranean Games together with Victor Cano, Manuel Carballo, Rafael Martinez and Iván San Miguel.

Vivo died of a heart attack when he was mountain climbing in the morning on 30 December 2012. He began to feel ill after climbing the Collbaix peak in the Catalan district with a friend and collapsed soon after beginning the descent. When emergency management personnel arrived they were unable to revive him. His funeral was in Manresa, his hometown, at the Church of Christ the King.

References

External links

Category:1978 births
Category:2012 deaths
Category:Spanish male artistic gymnasts
Category:People from Manresa
Category:Gymnasts at the 2000 Summer Olympics
Category:Olympic gymnasts of Spain
Category:Universiade medalists in gymnastics
Category:Mediterranean Games silver medalists for Spain
Category:Mediterranean Games medalists in gymnastics
Category:Competitors at the 2005 Mediterranean Games
Category:Universiade bronze medalists for Spain


Trapp Filling Station

The Trapp Filling Station, at 252-256 W. Capitol Drive in Hartland, Wisconsin, was built in 1922.  It was listed on the National Register of Historic Places in 1986.

Originally, the structure was built and used as a house.  It was located on the major highway into Hartland, though, and in the 1920s Cornelius Trapp began to operate it as a filling station.  Apparently "its location was obviously not prominent enough to succeed in sales. The site was soon used by others, twice as a grocery store by the Elmer Hornburg family and the Ferdi Dombroski family. More recently it has seen use as a handcraft outlet store and a health food market. Members of the Trapp family remained residents of the upper residential areas even when changing commercial
agents operated out of the lower area."

References

Category:Gas stations on the National Register of Historic Places in Wisconsin
Category:National Register of Historic Places in Waukesha County, Wisconsin
Category:Tudor Revival architecture in the United States
Category:Buildings and structures completed in 1922


Michael Carr (composer)

Michael Carr (11 March 1905 – 16 September 1968), real name Maurice Alfred Cohen, was a British popular music composer and lyricist born in Leeds. He is best remembered for the song "South of the Border (Down Mexico Way)", written with Jimmy Kennedy for the 1939 film of the same name.

Life and work

Born the son of a cabinet maker and boxer, Morris "Cockney" Cohen, and English mother Gertrude J Beresford, he was brought up in Dublin where his father opened a restaurant. In his teens, he ran away to sea, and eventually became a journalist in the USA. Under the name of Michael Carr, he played a number of small roles in Hollywood films.

In 1934 he settled in London where he worked for a music company. He wrote and co-wrote several songs for such performers as Gracie Fields. One of his best-known songs, written with lyricist Jimmy Kennedy, was "We're Going to Hang out the Washing on the Siegfried Line" (1939). During World War II he served in the army. In 1942 he composed "Freedom Concerto".

Among Carr's other compositions were "Lonely Ballerina" (1954) for the Mantovani Orchestra, and two instrumental pieces which were hits for The Shadows: "Man of Mystery" (UK #5 hit in December 1960) and "Kon-Tiki" (UK #1 hit in October 1961). His "Man of Mystery" was the theme music for the early 1960s film series "Edgar Wallace Mysteries" based on the prodigious output of the writer Edgar Wallace known as the "man of mystery". In recent years this instrumental rock tune has featured in the repertoire of the British alternative rock band Muse.

With Ben Nisbet, Carr co-wrote the theme song to the popular 1960s children's TV series "The White Horses". The song was released as the single "White Horses" sung by "Jacky" (Jackie Lee) which became a UK #10 hit in April 1968. In 2003 it was named in The Penguin Television Companion as the greatest theme song in television history. Carr and Nisbet collaborated again for another song for Jacky, "Off and Running".

Among many other songs, Carr also wrote/co-wrote:

 Ol' Faithful" (1934) (with Kennedy)
 "Dinner for One, Please James" (1935)
 "The General's Fast Asleep" (1935) (with Kennedy)
 "Getting Around and About" (1935) (with Lewis Ilda)
 "Old Timer" (1935) (with Lewis Ilda)
 "Girl with the Dreamy Eyes" (1935) (with Eddie Pola)
 "Orchids to My Lady" (1935)
 "Misty Islands of the Highlands" (1935) (with Kennedy)
 "The Wheel of the Wagon Is Broken" (1935) (with Elton Box & Desmond Cox)
 "Did Your Mother Come from Ireland" (1936) (with Kennedy)
 "The Sunset Trail" (1936) (with Kennedy)
 "Why Did She Fall for the Leader of the Band?" (1936) (with Kennedy)
 "Cowboy" (1937)
 "Home Town" (1937) (with Kennedy)
 "The Little Boy that Santa Claus Forgot" (1937) (with Tommie Connor & Jimmy Leach)
 "Cinderella, Stay in My Arms" (1938)
 "South of the Border" (1939) (with Kennedy)
 "On the Outside Looking in" (1939)
 "Somewhere in France with You" (1939)
 "(We're Gonna Hang out the) Washing on the Siegfried Line" (1939) (with Kennedy)
 "A Handsome Territorial" (1939) (with Kennedy)
 "The First Lullaby" (1941) (with Popplewell)
 "A Pair Of Silver Wings" (1941) (with Eric Maschwitz)
 "I Love To Sing" (1943) (with Paul Misrake & Tommie Connor)
 "When You Talk About Old Ireland" (1951) (with Tommie Connor)

Carr also composed "Ole Faithful", which became the Hull Rugby League Football Club's supporter's anthem.

He collaborated with many other British songwriters such as Jack Popplewell (also from Leeds), Jack Strachey and Eric Maschwitz and with Norrie Paramor for another Shadows' tune "The Miracle".

He died in London.

References

Category:1905 births
Category:1968 deaths
Category:English songwriters
Category:English composers
Category:Musicians from Leeds
Category:20th-century classical musicians
Category:20th-century English musicians


Battle Damage: Code Red

Battle Damage: Code Red is a 1982 expansion for Star Fleet Battles published by Task Force Games.

Gameplay
Battle Damage: Code Red is a fast damage allocation system for Star Fleet Battles, and uses a card-dealing system, with each card denoting a particular ship system.

Reception
Andy Davis reviewed Battle Damage: Code Red in The Space Gamer No. 60. Davis commented that "I can only recommend Battle Damage: Code Red to those players who want a slight change of pace and have [the money] to spare. For the price I paid, t was less than satisfactory."

References

Category:Star Fleet Battles


1994 World Artistic Gymnastics Championships

The 1994 World Artistic Gymnastics Championships were held in Brisbane, Queensland, Australia from 19–24 April 1994.

Only the all-around and apparatus events were contested at this meet. A team World Championships meet was held in Dortmund, Germany in November 1994. 1994 was the only year in which the Worlds were split into two separate competitions in this fashion.

Results

Men

All-around

Floor Exercise

Pommel Horse

Still Rings

Vault

Parallel Bars

Horizontal Bar

Women

All-around

Vault

Uneven Bars

Balance Beam

Floor Exercise

NB: For this competition, tie-breaker policies were used. When scores were identical, the gymnast with the higher score in the preliminary round was awarded the higher placement in finals.

Category:World Artistic Gymnastics Championships
Category:International gymnastics competitions hosted by Australia
G
G
Category:1994 in gymnastics
Category:1990s in Brisbane
Category:April 1994 sports events in Oceania


Benson Building (Baltimore, Maryland)

Benson Building, also known as the IPC Building, is a historic retail and office located at Baltimore, Maryland, United States. It is on the corner of East Franklin and North Charles Streets. The main side is on Charles Street and has seven bays with store windows and entrances on the first floor, and office windows on the upper floors. The recessed storefronts feature bronzed aluminum infill panels above and below the glass panes.  It was constructed in 1911 and the principal original occupant was C.J. Benson and Company, a local interior decorating and furniture establishment.

The Benson Building was listed on the National Register of Historic Places in 1980.

References

External links
, including undated photo, at Maryland Historical Trust

Category:Buildings and structures in Baltimore
Category:Commercial buildings on the National Register of Historic Places in Baltimore
Category:Commercial buildings completed in 1911
Category:Downtown Baltimore
Category:1911 establishments in Maryland


Cieplinki

Cieplinki  is a village in the administrative district of Gmina Izbica Kujawska, within Włocławek County, Kuyavian-Pomeranian Voivodeship, in north-central Poland. It lies approximately  south-east of Izbica Kujawska,  south-west of Włocławek, and  south of Toruń.

The village has a population of 60.

References

Cieplinki


2005 Malaysian haze

Haze is caused by "hotspots" (zones with high temperature levels as seen via satellite imagery) in Malaysia and Indonesia. Lingering smoke from forest fires on the Indonesian island of Sumatra are identified as the primary cause. Farmers regularly burn scrub and forest to clear land during the dry season for agricultural purposes, but this had been the worst haze to hit Malaysia since the 1997 haze.

On 10 August 2005, air quality in the Malaysian capital city of Kuala Lumpur was so poor that health officials advised citizens to stay at home with doors closed. Some schools were closed to keep children from being exposed to the haze.

State of emergency in Port Klang and Kuala Selangor
On 11 August 2005 a state of emergency was announced for the world's 12th largest port, Port Klang and the district of Kuala Selangor after air pollution there reached dangerous levels (defined as a value greater than 500 on the Air Pollution Index or API). This was the first time the state of emergency had been imposed in Malaysia since the September 1997 haze, when Sarawak was placed in a state of emergency due to similar reasons.

The state of emergency in the two affected areas meant that schools, government officials, the port, and offices were closed. Shops carrying necessities, however, such as supermarkets and pharmacies remained open.

After the API levels dropped to acceptable levels, the state of emergency was lifted on 13 August.

Resolution
By 13 August 2005, air quality and visibility returned to normal in Kuala Lumpur, as the haze moved northwards to the states of Perlis, Kedah and Penang, according to the Department of Environment (JAS) API reading. On 16 August, air quality throughout Malaysia had returned to normal according to JAS statistics, as the haze was blown further northwards into Thailand.

Kuala Lumpur International Airport was unaffected during the hazy period but flights for non-ILS equipped planes from Subang airport were suspended until conditions cleared on 13 August.

Cloud seeding operations using RMAF C-130s were used  throughout Malaysia, although the haze had moved north of Malaysia into Thailand and Cambodia.

Malaysia's Environment Minister, Adenan Satem, and Commodities Minister, Peter Chin, met with Indonesia's forestry minister and officials from its environment ministry in Medan. Malaysia had sent 125 firefighters while Australia had sent 12 bushfire experts to fight Indonesia's estimated more than a thousand forest and scrub fires (estimate reached by counting hotspots greater than 1 km on satellite imagery, example of such an image shown below).

Other nations pledged aid to Indonesia, including an offer from Singapore of planes for cloud seeding. Japan, South Korea and Thailand also offered help.

On 22 August, Malaysia extended its cloud seeding operations to include Indonesia, as requested by Indonesia.

The haze made a brief return on 11 September.

API readings
The air pollutant index readings (API) were released for the first time by the Malaysian government since 1997. Up to date readings are available for the 51 sites monitored automatically and continuously via the Department of Environment, Malaysia.

The following are API readings recorded during the onset of haze on 10 August 2005 (refer to this Related article from the New Straits Times):
 Port Klang 424
 Kuala Selangor 418
 Shah Alam 349
 Putrajaya 337
 Petaling Jaya 304
 Kuala Lumpur 276
 Tanjung Malim 205
 Gombak 200
 Country Heights, Kajang 185
 Nilai 174
 Seremban 137
 Jerantut 131
 Bukit Rambai, Malacca 107

References

 Malaysia seeks talks with Indonesia over choking haze (CRI)
 Malaysia chokes on Indonesia's smoke (7news)
 Malaysia haze worsens; schools close as air becomes hazardous (SP Times)
 Haze in Malaysia closes schools (BBC)
 Malaysians told to pray for rain (BBC)

External links

 Eyewitness photographs of the haze
 National pollutant inventory - Particulate matter fact sheet
 Forest fires result from government failure in Indonesia
 Smoky haze chokes Southeast Asia

Category:Smog events
Haze
Haze
Category:Air pollution in Malaysia
Category:Fires in Indonesia
Category:2005 in the environment


Miroslav Krobot

Miroslav Krobot (born 12 November 1951) is a Czech theatre director and film actor. He starred in the film The Man from London, which was entered into the 2007 Cannes Film Festival.

Selected filmography
 Wrong Side Up (2005)
 The Man from London (2007)
 3 Seasons in Hell (2009)
 Alois Nebel (2011)
 The House (2011)
 Leaving (2011)
 In the Shadow (2012)
 Revival (2013)
 Nowhere in Moravia (2014)
 Spoor (2017)

References

External links

Category:1951 births
Category:Living people
Category:Czech theatre directors
Category:Czech male film actors
Category:People from Šumperk
Category:Sun in a Net Awards winners


Germans in the Philippines

German settlement in the Philippines began during the Colonial era when the German Empire attempted to acquire the Philippines. This article also refers to the choice of Filipino citizenship and/or settlement in the Philippines by persons of either pure or mixed German descent who continued to reside in the country for a significant number of years or decades.

History

Spanish Rule
The first Germans to arrive to the Philippines were colonists. Otto von Bismarck’s Germany was one of the United States' rivals in replacing Spanish rule in the archipelago.  From 1890 to the outbreak of the Spanish–American War in 1898, there was a lull in Germany’s colonial campaigns. Like other colonialist nations, Germany sought to protect its overseas nationals and trade interests to the extent of safeguarding free access to markets. A German squadron arrived in Manila and engaged in maneuvers which Commodore George Dewey seeing this as obstruction of his blockade, offered war — after which the Germans backed down.

First Philippine Republic
The Battle of Manila Bay took place on 1 May 1898, following the outbreak of the Spanish–American War. The German Emperor expected an American defeat, with Spain left in a sufficiently weak position for the revolutionaries to capture Manila—leaving the Philippines ripe for German picking. Following the American victory in the war, the Philippines and the Far East were brought to the attention of the world and Germany recognized the great potentialities of the islands as a major commercial market.

On 12 June, the day the Philippines declared its independence from Spain, Vice-Admiral Otto von Diederichs arrived in Manila Bay. The number of German war vessels in Philippine waters increased to three. Earlier, on 6 and 9 May, respectively, the German ships Irene and the Cormoran arrived in the bay with a separate instruction from the German government, mainly to protect German nationals in Manila. German’s interest in the Philippines was cut short with the signing of the Treaty of Paris on 10 December 1898. The Philippines was finally annexed by the United States in 1899.

American period
The Philippines was part of the United States between 1898 and 1946. During the era of the Philippine Commonwealth, 1935–1946, Jewish refugees including German Jews from Europe sought a safe haven in Manila. The migration of Jews escaping Europe between 1935 and 1941 was the last major immigration of Jews to the Philippines. The first German Jews to arrive in Manila actually came from the Jewish community in Shanghai. With the occupation of Peking by the Japanese in 1937, the four million inhabitants of Shanghai were endangered. Germany's shift of alliance from China to Japan at this time alarmed German Jews in Shanghai, fearing German pressure on Japan to adopt Nazi anti-Jewish policies. Fearing for them as well, the Jewish Community in Manila, led by the Frieder Brothers of Cincinnati, organized the Jewish Refugee Committee of Manila (JRC) with the intention of rescuing German members of the Shanghai Jewish community.

Modern Era
In recent years, several German businesses have set up shops in the Philippines, and a number of Germans have chosen the Philippines as their new residence. In the Philippines, since its formation in January 1906, the German Club has provided a place of respite and interaction for Germans and Filipinos alike. In the past century, it has stood witness to the country’s unfolding history and today enjoys the regular patronage of members and guests at its current location in Legaspi Village, Makati City.

Notable Filipinos of German descent

 Jose Rizal - Philippine National Hero, doctor
 Christopher de Leon - Actor
 Michael de Mesa - Actor
 Krista K - Model
 Carla Abellana - Actress
 Ivan Dorschner - Actor, model
 Paolo Duterte - Businessman
 William Beier -  Figure ice skating
 Jestoni Alarcon - Actor
 Panchito Alba - Actor active from the 1950s through the 1990s, FAMAS Award recipient
 Arjo Atayde - Actor
 Ria Atayde - Actress
 Rita Avila - Actress
 Kurt Bachmann - Basketball player, Philippine National Team
 Charles Borck - Basketball player
 Aleck Bovick - Actress
 Michele Bumgarner - Racing driver
 Felix Brawner Jr. - Army brigadier general
 Romeo A. Brawner - Former Commissioner, COMELEC
 Sara Duterte-Carpio - Davao City mayor
 Alan Peter Cayetano - Senator, Secretary of Foreign Affairs
 Lino Cayetano - Director, producer
 Pia Cayetano - Senator, Representative
 Will Devaughn - Actor
 Etang Discher - Actress during 1930s through 1950s
 Lilia Dizon - Actress
 Mathew Dumba - Ice hockey player
 Andi Eigenmann - Actress
 Gabby Eigenmann - Actor
 Geoff Eigenmann - Actor
 Ryan Eigenmann - Actor
 Maike Evers - Model, VJ
 E. J. Feihl - Basketball player
 Gerphil Flores - Soprano
 Amalia Fuentes - Actress
 Baron Geisler - Actor
 Donald Geisler - Businessman, former athlete
 Melody Gersbach -  Beauty pageant contestant
 Cherie Gil - Actress
 Enrique Gil - Actor, commercial model, dancer, performer
 Mark Gil - Actor
 Nikki Gil - Actress, songstress
 Roxanne Guinoo - Star Circle Teen Quest contestant, model and actress
 Edward Hagedorn - Puerto Princesa City mayor
 Christine Jacob - TV personality
 Jaclyn Jose - Actress
 Bianca King - Actress
 Krista Kleiner - Beauty pageant contestant
 Doug Kramer - PBA basketball player, model
 Helga Krapf - Actress
 Juan Karlos Labajo - Actor, singer
 Renaldo Lapuz - Musician, singer-songwriter
 Laura Lehmann - Beauty pageant contestant
 Katharina Lehnert - Tennis player
 Gabriel de Leon - Actor
 Malaya Lewandowski - Actress, songstress
 Sid Lucero - Actor, model
 Eddie Mesa - Filipino actor from 1950s - 1990s
 Colby Miller - MTV Asia VJ
 Ara Mina - Actress
 Jewel Mische - Actress
 Julia Montes - ABS-CBN Actress
 Aga Muhlach - Actor, model and product endorser
 Niño Muhlach - Former actor and former child actor
 Diether Ocampo  - Filipino actor and model
 Paraluman - Actress, FAMAS Award recipient
 Gina Pareño - Filipina actress
 Piolo Pascual - Filipino model, film and television actor, musician, and producer
 Polo Ravales - Actor
 Delia Razon - Actress
 John Regala - Actor, christian minister and environmentalist
 Cristine Reyes - Actress
 Alden Richards - Actor
 Christopher Ricketts - Martial arts instructor and actor
 Ronnie Ricketts - Actor, fight director, martial arts instructor, film producer and director
 Ross Rival - Actor
 Rosanna Roces - Actress
 Lana Roi - Singer, model
 Jen Rosendahl - Actress
 Lou Salvador - former basketball player turned film producer
 Lou Salvador, Jr. - former teen star during 1950s
 Nadine Samonte - Starstruck Season 1 contestant, model and actress
 Empress Schuck - Actress
 Imelda Schweighart- Beauty titleholder
 Sandra Seifert - Beauty pageant contestant
 Mary Walter - Actress from the 1920s through the 1990s
 Freddie Webb - Actor, former politician, former athlete
 Pinky Webb - Broadcasting jounalistic, newscaster currently employer of CNN Philippines
 Valerie Weigmann - Filipina actress, TV Host, fashion model, and beauty pageant titleholder
 Pia Wurtzbach - Filipina actress, model and Miss Universe 2015
 The Zobel de Ayala family - A successful Filipino business family
 Zaldy Zshornack - Actor
  James Hampel - Chess Boxer and elite high school walker

See also
 History of the Jews in the Philippines
 Filipinos in Germany
 Germany–Philippines relations

References

German
Category:European diaspora in the Philippines
Philippines
Category:Germany–Philippines relations


Lamborghini 400 GT

Lamborghini 400 GT is the name given to two grand tourers produced by Italian manufacturer Lamborghini.

History
The first 400 GT, commonly referred to as simply the 400 GT or 400 GT Interim, was essentially the older 350 GT featuring an enlarged, 3,929 cc V12 engine, with a power output of . Twenty-three of these cars were built, with three featuring aluminium bodywork.

The second 400 GT, commonly known as the 400 GT 2+2, had a different roofline, and minor sheetmetal changes compared to the 350 GT and first 400 GT. It was first presented at the 1966 Geneva Auto Show. The larger body shape enabled the +2 seating to be installed in the rear, where the 350 GT only had room for luggage or +1 seating. The bodywork was designed by Carrozzeria Touring. The 400 GT 2+2 also had a Lamborghini designed gearbox, with Porsche style synchromesh on all gears, which greatly improved the drivetrain. When leaving the factory the 400 GT originally fitted Pirelli Cinturato 205VR15 tyres (CN72).

A total of 23 units of the 400 GT Interim and 224 units of the 400 GT 2+2 were built from 1966 to 1968, when it was replaced with the Islero.

The 400 GT 2+2, 400 GT Interim and the 350 GT all shared the same  wheelbase.

400 GT Monza

The Lamborghini 400 GT Monza was a one-off two-seater sports car based on the 400 GT "Interim", featuring unique bodywork by the shop of Neri and Bonacini, who were already previously known for their work on the "Nembo" series of Ferraris.

Giorgio Neri and Luciano Bonacini were initially hired by Ferruccio Lamborghini in 1963 to construct the chassis of and assemble the first prototype Lamborghini, the 350 GTV. Following this, they supplied some early production chassis, before turning that job over to Marchesi once series production of the 350 GT was well underway.

Through this pre-existing relationship with Lamborghini, Neri and Bonacini were commissioned to create a one-off two-seater sports car based on a 350 GT chassis (number 01030) and a 400 GT V-12 engine. Possibly built for an unknown American client to race at the 24 Hours of Le Mans, the car was completed in 1966 and named the 400 GT Monza. The original name was the "400GT Neri and Bonacini," but the final name Monza was eventually chosen due to its brevity and evocation of racing history. The aluminum body was hand built in the Neri and Bonacini shop and went through many revisions during the fabrication process. The final result was a distinctive fastback, with a long hood and Kamm tail. The designers integrated a roll bar into the extremely thick C-pillar, possibly increasing roll-over safety but compromising rearward visibility. Other details included a very low and raked windshield, prominent but non-functional air vent grilles behind the front wheels and stylized "400 Monza" badging. Overall, the design shows visual similarities to other contemporary Italian sports cars such as the Bizzarrini 5300 GT and the Lamborghini Miura.

Homologation problems prevented the car from ever racing and the American customer who commissioned it apparently never received the car. It was instead displayed at the 1967 Barcelona Motor Show on the Lamborghini importer Amato's stand. A wealthy Spaniard purchased it at the show and the 400GT Monza remained in his family after his death in the early 1990s. In 1996, auction house Brooks (now Bonhams) discovered the car in storage when they were contacted by the family of the now deceased owner to appraise some other cars. After nine years of negotiations, the car was finally sold at Bonhams' December 2005 auction in London for £177,500 GBP.

References

External links

 lp112.com a register of over 100 400 GTs
 www.400gt.com additional historical site
 November 2010 Octane Magazine article on the 400 GT Monza
 January 2011 Octane Magazine article on the 400 GT Monza

400 GT 2+2
Category:Cars introduced in 1966
Category:Coupés
Category:Rear-wheel-drive vehicles
Category:Group 4 cars


Harzungen

Harzungen is a village and a former municipality in the district of Nordhausen, in Thuringia, Germany. Since July 2018, it is part of the municipality Harztor.

During World War II a concentration camp with 4000 inmates was built in this city.
It was a subcamp of Mittelbau-Dora who was itself a subcamp of the Buchenwald concentration camp.

References

Category:Former municipalities in Thuringia
Category:Nordhausen (district)
Category:Bezirk Erfurt