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In mathematical logic, Gödel's incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ...
1931 is a common year starting on Thursday. ...
First theorem
The first theorem is one of the most famous outside of mathematics, and one of the most misunderstood. It is a theorem in formal logic, and as such is easy to misinterpret. There are many statements that sound similar to Gödel's first incompleteness theorem, but are in fact not true, see misconceptions about Gödel's theorems below. Somewhat simplified, this theorem can be paraphrased as: - In any consistent formal system that is sufficiently strong to axiomatize the natural numbers – that is, sufficiently strong to define the operations that collectively define the natural numbers – one can construct a true statement that can neither be proved nor disproved within the system itself.
This simplified formulation however omits some critical qualifications that make it quite misleading. A more informative (but still omitting some important technical details) paraphrase is: In mathematical logic, a formal system is said to be consistent if it doesnt contain a contradiction, or, more precisely, for no proposition are both and provable. ...
In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ...
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
- If a formal system is sufficiently strong to axiomatize the natural numbers and is limited to finitistic induction and is reinterpreted as its own proof system, it is possible to construct a true statement that can only be proven if inconsistency is allowed between the theory and its reintepretation as a proof theory
The key qualifications of the theorem are the limitation to finitistic induction, the reinterpretation of the system as a proof system for itself, and that completeness can only be proven at the expense of consistency between the theory and its reinterpretation. In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. ...
In epistemology, an axiom is a self-evident truth upon which other knowledge must rest, from which other knowledge is built up. ...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
These assumptions for the theorem correspond to those of Hilbert's program for a reductionist approach to the foundations of mathematics, and the theorem was designed to show that such a reductionist approach is not successful. Hilberts Program was to formalize all existing theories to finite real complete set of axioms, and provide a proof that these axioms were consistent. ...
The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
David Hilbert hoped that the completeness and consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the completeness and consistency of all of mathematics could be reduced to that of basic arithmetic, and these might be proven using itself as its own proof system, thus sidestepping difficult issues concerning the definition of truth, which were later addressed by Tarski. David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
Second theorem Gödel's second incompleteness theorem is motivated by the question whether the first theorem's result depends on the limitation to finitistic induction.
The question is relevant because if the first theorem only applies to systems restricted to finitistic induction, a weaker form of Hilbert's program where non finitistic induction is allowed, but still relies on the device of using a theory as its own proof theory, can then be successful. Hilberts Program was to formalize all existing theories to finite real complete set of axioms, and provide a proof that these axioms were consistent. ...
The second theorem, which is proved by formalizing part of the proof of the first within the system itself, is often paraphrased as:
- No consistent system can be used to prove its own consistency.
This popular formulation also omits some critically important qualifications and is misleading. A more informative (but still omitting some significant technical details) paraphrase is: - Any formal system of sufficient power to express arithmetic, if used as its own proof theory, cannot be used to prove both its own consistency and completeness, no matter which power of induction is allowed within the system.
This is the same as the first theorem, but generalizes it to formal systems using induction of any order, and shows that Hilbert's program is not feasible even if one allows non finitistic induction. Hilberts Program was to formalize all existing theories to finite real complete set of axioms, and provide a proof that these axioms were consistent. ...
Both of Gödel's theorems are narrow, negative, technical results designed to show that Hilbert's program, both in its strong and weak forms, is unfeasible and another approach is needed which does not sidestep the difficult issue of defining logical truth (which was eventually developed by Tarski) by reducing it to self-referential provability. Hilberts Program was to formalize all existing theories to finite real complete set of axioms, and provide a proof that these axioms were consistent. ...
The publication of Gödel's theorems then prompted naturally the question: if it is impossible to prove both consistency and completeness of a theory of arithmetic based on finitistic induction reinterpreting it self-referentially as its own proof theory (first theorem) or any theory of arithmetic (second theorem), and both the strong and weak forms of Hilbert's approach are unfeasible, under which conditions, if any, is it then possible?
Gentzen's theorem The answer to this questions is a far more important and significant theorem by Gerhard Gentzen which soon followed Gödel's theorems, and it can be paraphrased as: Gerhard Gentzen (November 24, 1909 – August 4, 1945) was a German mathematician and logician. ...
- It is possible to prove both the consistency and completeness of a formal system, but only in a proof theory with induction of strictly greater order
Gentzen's theorem implies both of Gödel's theorems, and was the natural consequence of wondering whether Gödel's narrow and negative results depended on the assumption of using a theory self referentially as its own proof theory.
As Gentzen's theorem states, this is not quite the case: any attempt to prove a theory as both complete and consistent using any proof theory (itself being just a special case) with induction of an equivalent order is not possible, but it is indeed possible with a proof theory using induction of higher order (which proof theory of course cannot then be the theory itself used self-referentially).
After Gentzen's theorem it was then shown that it is indeed possible to have a theory of arithmetic based on finitistic induction that can be proven to be both consistent and complete, but this can be and was proven only by using a proof theory with non-finitistic induction to epsilon-0.
Note that this means that (if one accepts the use of non finistic induction [at least in the proof theory] which Hilbert did not) there is then no regress to infinity, because there is no need to prove that proof theory itself to be complete and consistent, which would require, per Gentzen's theorem, another theory with an higher order of induction, and so on. There is no need because the proof theory itself only has to be proven consistent, and this can be done within itself, stopping the recursion.
It is also important to observe that neither Gödel's nor Gentzen's theorems state that it is impossible to prove a theory to be complete using itself as its own proof theory. It is indeed indeed possible to do so, at the cost of being unable to prove its consistency.
But most logicians care much more about the ability to prove consistency than completeness, therefore the option to self-referentially prove completeness at the expense of proving consistency is often not even considered, and this is why Gödel's theorems are usually called incomplete theorems, not more properly as incompleteness-aut-inconsistency theorems.
There is an even more subtle aspect of Gödel's theorems and it is that neither asserts that a theory of arithmetic is inconsistent or incomplete, only that it cannot be proven to be so if (but not iff) meta-interpretation is used to make it into its own proof theory; this is because it is indeed possible to prove that the theory of arithmetic is both consistent and complete by using a proof theory with higher-order induction, per Gentzen's theorem.
In the self referential case it cannot be proven to be both because this would introduce a contradiction between the theory of arithmetic and its reinterpretation as its own proof theory, which is what Hilbert wanted to avoid; the proof of Gödel's theorems does not, critically, involve creating any contradiction in the theory of arithmetic itself).
Meaning of Gödel's theorems Gödel's theorems are theorems in first-order logic, and must ultimately be understood in that context. In formal logic, both mathematical statements and proofs are written in a symbolic language, one where we can mechanically check the validity of proofs so that there can be no doubt that a theorem follows from our starting list of axioms. In theory, such a proof can be checked by a computer, and in fact there are computer programs that will check the validity of proofs (this is called automated reasoning). First-order predicate calculus or first-order logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as there is at least one X such that. ...
Automated theorem proving (currently the most important subfield of automated reasoning) is the proving of mathematical theorems by a computer program. ...
To be able to perform this process, we need to know what our axioms are. We could start with a finite set of axioms, such as in Euclidean geometry, or more generally we could allow an infinite list of axioms, with the requirement that we can mechanically check for any given statement if it is an axiom from that set or not (an axiom schema). In computer science, this is known as having a recursive set of axioms. While an infinite list of axioms may sound strange, this is exactly what's used in the usual axioms for the natural numbers, the Peano axioms: the inductive axiom is in fact an axiom schema - it states that if zero has any property and the successor of any natural number has that property, all natural numbers have that property - it does not specify which property and the only way to say in first-order logic that this is true of all properties is to have an infinite number of statements. In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...
In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether a given element belongs to the set or not. ...
In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...
Gödel's first incompleteness theorem shows that any such system that allows you to define the natural numbers is necessarily incomplete: it contains statements that are neither provably true nor provably false.
The existence of an incomplete system is in itself not particularly surprising. For example, if you take Euclidean geometry and you drop the parallel postulate, you get an incomplete system (in the sense that system does not contain all the true statements). An incomplete system can mean simply that you haven't discovered all the necessary axioms. In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...
In geometry, the parallel postulate, also called Euclids fifth postulate since it is the fifth postulate in Euclids Elements, is a distinctive axiom in what is now called Euclidean geometry. ...
What Gödel showed is that in most cases, such as in number theory or real analysis, you can never discover the complete list of axioms. Each time you add a statement as an axiom, there will always be another statement out of reach. Traditionally, number theory is that branch of pure mathematics concerned with the properties of integers. ...
Real analysis is that branch of mathematical analysis dealing with the set of real numbers and functions of real numbers. ...
You can add an infinite number of axioms; for example, you can add all true statements about the natural numbers to your list of axioms, but such a list will not be a recursive set. Given a random statement, there will be no way to know if it is an axiom of your system. If I give you a proof, in general there will be no way for you to check if that proof is valid.
Gödel's theorem has another interpretation in the language of computer science. In first-order logic, theorems are recursively enumerable: you can write a computer program that will eventually generate any valid proof. You can ask if they satisfy the stronger property of being recursive: can you write a computer program to definitively determine if a statement is true or false? Gödel's theorem says that in general you cannot. In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts if it...
In computability theory a countable set is called recursive, computable or decidable if we can construct an algorithm which terminates after a finite amount of time and decides whether a given element belongs to the set or not. ...
Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's program towards a universal mathematical formalism. The generally agreed upon stance is that the second theorem is what specifically dealt this blow. However some believe it was the first, and others believe that neither did. David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Examples of undecidable statements The existence of an undecidable statement within a formal system is not in itself a surprising phenomenon.
The subsequent combined work of Gödel and Paul Cohen has given concrete examples of undecidable statements (statements which can be neither proven nor disproven): both the axiom of choice and the continuum hypothesis are undecidable in the standard axiomatization of set theory. These results do not require the incompleteness theorem. Paul Joseph Cohen (born April 2, 1934) is an American mathematician. ...
In mathematics, the axiom of choice is an axiom of set theory. ...
In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
In 1936, Alan Turing proved that the halting problem—the question of whether or not a Turing machine halts on a given program—is undecidable. This result was later generalised in the field of recursive functions to Rice's theorem which shows that all non-trivial decision problems are undecidable in a system that is Turing-complete. 1936 was a leap year starting on Wednesday (link will take you to calendar). ...
Alan Turing was a respected scientist and war hero who was later prosecuted for being a homosexual. ...
In computability theory the halting problem is a decision problem which can be informally stated as follows: Given a description of an algorithm and its initial input, determine whether the algorithm, when executed on this input, ever halts (completes). ...
The Turing machine is an abstract machine introduced in 1936 by Alan Turing to give a mathematically precise definition of algorithm or mechanical procedure. As such it is still widely used in theoretical computer science, especially in complexity theory and the theory of computation. ...
In mathematical logic and computer science, the recursive functions are a class of functions from natural numbers to natural numbers which are computable in some intuitive sense. ...
Rices theorem (also known as The Rice-Myhill-Shapiro theorem) is an important result in the theory of recursive functions. ...
In computability theory a programming language or any other logical system is called Turing-complete if it has a computational power equivalent to a universal Turing machine. ...
In 1973, the Whitehead problem in group theory was shown to be undecidable in standard set theory. In 1977, Kirby, Paris and Harrington proved that a statement in combinatorics, a version of the Ramsey theorem, is undecidable in the axiomatization of arithmetic given by the Peano axioms but can be proven to be true in the larger system of set theory. Kruskal's tree theorem, which has applications in computer science, is also undecidable from the Peano axioms but provable in set theory. Goodstein's theorem is a relatively simple statement about natural numbers that is undecidable in Peano arithmetic. 1973 was a common year starting on Monday. ...
In group theory, the Whitehead problem is the following question: Is every abelian group A with Ext1(A, Z) = 0 a free abelian group? The condition Ext1(A, Z) = 0 can be equivalently formulated as follows: whenever B is an abelian group and f : B → A is a surjective group...
Group theory is that branch of mathematics concerned with the study of groups. ...
For the album by Ash, see 1977 (album). ...
Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria, and is in particular concerned with counting the objects in those collections (enumerative combinatorics) and with deciding whether certain optimal objects exist (extremal combinatorics). ...
This article goes into technical details quite quickly. ...
In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...
In mathematical logic, Goodsteins theorem is a statement about the natural numbers that is undecidable in Peano arithmetic but can be proven to be true using the stronger axiom system of set theory, in particular using the axiom of infinity. ...
Gregory Chaitin produced undecidable statements in algorithmic information theory and in fact proved his own incompleteness theorem in that setting. Gregory J. Chaitin (born 1947) is an American contemporary mathematician and computer scientist. ...
Algorithmic information theory is a field of study which attempts to capture the concept of complexity by using tools from theoretical computer science. ...
One of the first problems suspected to be undecidable was the word problem for groups, first posed by Max Dehn in 1911, which states that there is a finitely presented group that has no algorithm to state whether two words are equivalent. It was not proven to be undecidable until 1952. In abstract algebra, the word problem for groups is the problem of deciding whether two given words of a presentation of a group represent the same element. ...
Max Dehn (November 13, 1878 – June 27, 1952) was a German mathematician. ...
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In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
1952 - Wikipedia /**/ @import /skins/monobook/IE50Fixes. ...
Misconceptions about Gödel's theorems Since Gödel's first incompleteness theorem is so famous, it has given rise to many misconceptions. They are refuted here: - The theorem does not imply that every interesting axiom system is incomplete. For example, Euclidean geometry can be axiomatized so that it is a complete system. (In fact, Euclid's original axioms are pretty close to being a complete axiomatization. The missing axioms express properties that seem so obvious that it took the emergence of the idea of a formal proof before their absence was noticed.)
- The theorem only applies to systems that allow you to define the natural numbers as a set. It is not sufficient that the system contain the natural numbers. You must also be able to express the concept "x is a natural number" using your axioms and first-order logic. There are plenty of systems that contain the natural numbers and are complete. For example both the real numbers and complex numbers have complete axiomatizations.
In mathematics, Euclidean geometry is the familiar kind of geometry on the plane or in three dimensions. ...
Euclid of Alexandria (Greek: ) (circa 365–275 BC) was a Greek mathematician, now known as the father of geometry. His most famous work is Elements, widely considered to be historys most successful textbook. ...
Please refer to Real vs. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
Discussion and implications The incompleteness results affect the philosophy of mathematics, particularly viewpoints like formalism, which uses formal logic to define its principles. One can paraphrase the first theorem as saying that "we can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but no falsehoods." Philosophy of mathematics is that branch of philosophy which attempts to answer questions such as: why is mathematics useful in describing nature?, in which sense, if any, do mathematical entities such as numbers exist? and why and how are mathematical statements true?. The various approaches to answering these questions will...
Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics. ...
On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to each formal system.
The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics: The term foundations of mathematics is sometimes used for certain fields of mathematics itself, namely for mathematical logic, axiomatic set theory, proof theory, model theory, and recursion theory. ...
- If an axiomatic system can be proven to be consistent from within itself, then it is inconsistent.
Therefore, in order to establish the consistency of a system S, one needs to utilize some other system T, but a proof in T is not completely convincing unless T's consistency has already been established without using S. The consistency of the Peano axioms for natural numbers for example can be proven in set theory, but not in the theory of natural numbers alone. This provides a negative answer to problem number 2 on David Hilbert's famous list of important open questions in mathematics (called Hilbert's problems). In mathematics, the Peano axioms (or Peano postulates) are a set of first-order axioms proposed by Giuseppe Peano which determine the theory of Peano arithmetic (also known as first-order arithmetic). ...
Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...
Hilberts problems are a list of 23 problems in mathematics put forth by German mathematician David Hilbert in the Paris conference of the International Congress of Mathematicians in 1900. ...
In principle, Gödel's theorems still leave some hope: it might be possible to produce a general algorithm that for a given statement determines whether it is undecidable or not, thus allowing mathematicians to bypass the undecidable statements altogether. However, the negative answer to the Entscheidungsproblem shows that no such algorithm exists. Flowcharts are often used to represent algorithms. ...
The Entscheidungsproblem (German: decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ...
There are some who hold that a statement that is unprovable within a deductive system may be quite provable in a metalanguage. And what cannot be proven in that metalanguage can likely be proven in a meta-metalanguage, recursively, ad infinitum, in principle. By invoking a sort of super Theory of Types with an axiom of Reducibility -- which by an inductive assumption applies to the entire stack of languages -- one may, for all practical purposes, overcome the obstacle of incompleteness. Ad infinitum is a Latin phrase meaning to infinity. ...
Note that Gödel's theorems only apply to sufficiently strong axiomatic systems.
"Sufficiently strong" means that the theory contains enough arithmetic to carry out the coding constructions needed for the proof of the first incompleteness theorem. Essentially, all that is required are some basic facts about addition and multiplication as formalized, e.g., in Robinson arithmetic Q. There are even weaker axiomatic systems that are consistent and complete, for instance Presburger arithmetic which proves every true first-order statement involving only addition. Presburger arithmetic is the first-order theory of the natural numbers with addition. ...
The axiomatic system may consist of infinitely many axioms (as first-order Peano arithmetic does), but for Gödel's theorem to apply, there has to be an effective algorithm which is able to check proofs for correctness. For instance, one might take the set of all first-order sentences which are true in the standard model of the natural numbers. This system is complete; Gödel's theorem does not apply because there is no effective procedure that decides if a given sentence is an axiom. In fact, that this is so is a consequence of Gödel's first incompleteness theorem. Natural number can mean either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). Natural numbers have two main purposes: they can be used for counting (there are 3 apples on the table), or they can be used for ordering (this is...
Another example of a specification of a theory to which Gödel's first theorem does not apply can be constructed as follows: order all possible statements about natural numbers first by length and then lexicographically, start with an axiomatic system initially equal to the Peano axioms, go through your list of statements one by one, and, if the current statement cannot be proven nor disproven from the current axiom system, add it to that system. This creates a system which is complete, consistent, and sufficiently powerful, but not recursively enumerable. In mathematics, the lexicographical order, or dictionary order, is a natural order structure of the cartesian product of two ordered sets. ...
In computability theory, often less suggestively called recursion theory, a countable set S is called recursively enumerable, computably enumerable, semi-decidable or provable if There is an algorithm that, when given an input — typically an integer or a tuple of integers or a sequence of characters — eventually halts if it...
Gödel himself only proved a technically slightly weaker version of the above theorems; the first proof for the versions stated above was given by J. Barkley Rosser in 1936. John Barkley Rosser Sr. ...
In essence, the proof of the first theorem consists of constructing a statement p within a formal axiomatic system that can be given a meta-mathematical interpretation of:
- p = "This statement cannot be proven"
As such, it can be seen as a modern variant of the Liar paradox. Unlike the Liar sentence, p does not directly refer to itself; the above interpretation can only be "seen" from outside the formal system. In philosophy and logic, the liar paradox encompasses paradoxical statements such as: or To avoid having a sentence directly refer to its own truth value, one can also construct the paradox as follows: Eubulides of Miletus words The oldest version of the liar paradox is attributed to the Greek philosopher...
If the axiomatic system is consistent, Gödel's proof shows that p (and its negation) cannot be proven in the system. Therefore p is "true" (p claims to be not provable, and it is not provable) yet it cannot be formally proved in the system. Note that adding p to the axioms of the system would not solve the problem: there would be another Gödel sentence for the enlarged theory.
An easy solution for provability in and of itself is to insist that in a truly consistent proof-theorem system each true theorem should actually explicitly contain its own proof. It is obvious that this Prior's solution does not injure completeness.
Roger Penrose claims that this (alleged) difference between "what can be mechanically proven" and "what can be seen to be true by humans" shows that human intelligence is not mechanical in nature. This claim is also addressed by JR Lucas in Minds, Machines and Gödel (http://users.ox.ac.uk/~jrlucas/mmg.html). Sir Roger Penrose OM (born August 8, 1931) is an English mathematical physicist. ...
John Randolph Lucas (born 18 June 1929) is a philosopher. ...
This view is not widely accepted, because as stated by Marvin Minsky, human intelligence is capable of error and of understanding statements which are in fact inconsistent or false. However, Marvin Minsky has reported that Kurt Gödel told him personally that he believed that human beings had an intuitive, not just computational, way of arriving at truth and that therefore his theorem did not limit what can be known to be true by humans. Marvin Minsky Marvin Lee Minsky (born August 9, 1927), sometimes affectionately known as Old Man Minsky, is an American scientist in the field of artificial intelligence (AI), co-founder of MITs AI laboratory, and author of several texts on AI and philosophy. ...
Kurt Gödel Kurt Gödel [kurt gøːdl], (April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. ...
The position that the theorem shows humans to have an ability that transcends formal logic can also be criticized as follows: We do not know whether the sentence p is true or not, because we do not (and can not) know whether the system is consistent. So in fact we do not know any truth outside of the system. All we know is the following statement:
- Either p is unprovable within the system, or the system is inconsistent.
This statement is easily proved within the system. In fact, such a proof will now be given.
Proof sketch for the first theorem The main problem in fleshing out the above mentioned proof idea is the following: in order to construct a statement p that is equivalent to "p cannot be proved", p would have to somehow contain a reference to p, which could easily give rise to an infinite regress. Gödel's ingenious trick, which was later used by Alan Turing to solve the Entscheidungsproblem, will be described below. Alan Turing was a respected scientist and war hero who was later prosecuted for being a homosexual. ...
The Entscheidungsproblem (German: decision problem) is the challenge in symbolic logic to find a general algorithm which decides for given first-order statements whether they are universally valid or not. ...
To begin with, every formula or statement that can be formulated in our system gets a unique number, called its Gödel number. This is done in such a way that it is easy to mechanically convert back and forth between formulas and Gödel numbers. Because our system is strong enough to reason about numbers, it is now also possible to reason about formulas. In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ...
A formula F(x) that contains exactly one free variable x is called a statement form. 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. Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement form F(x) has a Gödel number which we will denote 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). In law, good faith (in Latin, bona fides) is the mental and moral state of honest, even if objectively unfounded, conviction as to the truth or falsehood of a proposition or body of opinion, or as to the rectitude or depravity of a line of conduct. ...
By carefully analyzing the axioms and rules of the system, one can then write down a statement form P(x) which embodies the idea that x is the Gödel number of a statement which can be proved in our system. Formally: P(x) can be proved if x is the Gödel number of a provable statement, and its negation ~P(x) can be proved if it isn't. (While this is good enough for this proof sketch, it is technically not completely accurate. See Gödel's paper for the problem and Rosser's paper for the resolution. The key word is "omega-consistency".)
Now comes the trick: a statement form F(x) is called self-unprovable if the form F, applied to its own Gödel number, is not provable. This concept can be defined formally, and we can construct a statement form SU(z) whose interpretation is that z is the Gödel number of a self-unprovable statement form. Formally, SU(z) is defined as: z = G(F) for some particular form F(x), and y is the Gödel number of the statement F(G(F)), and ~P(y). Now the desired statement p that was mentioned above can be defined as:
- p = SU(G(SU)).
Intuitively, when asking whether p is true, we ask: "Is the property of being self-unprovable itself self-unprovable?" This is very reminiscent of the Barber paradox about the barber who shaves precisely those people who don't shave themselves: does he shave himself? The Barber paradox is a paradox that relates to mathematical logic and set theory. ...
We will now assume that our axiomatic system is consistent.
If p were provable, then SU(G(SU)) would be true, and by definition of SU, z = G(SU) would be the Gödel number of a self-unprovable statement form. Hence SU would be self-unprovable, which by definition of self-unprovable means that SU(G(SU)) is not provable, but this was our p: p is not provable. This contradiction shows that p cannot be provable.
If the negation of p= SU(G(SU)) were provable, then by definition of SU this would mean that z = G(SU) is not the Gödel number of a self-unprovable form, which implies that SU is not self-unprovable. By definition of self-unprovable, we conclude that SU(G(SU)) is provable, hence p is provable. Again a contradiction. This one shows that the negation of p cannot be provable either.
So the statement p can neither be proved nor disproved within our system.
Proof sketch for the second theorem Let p stand for the undecidable sentence constructed above, and let's assume that the consistency of the system can be proven from within the system itself. We have seen above that if the system is consistent, then p is not provable. The proof of this implication can be formalized in the system itself, and therefore the statement "p is not provable", or "not P(p)" can be proven in the system.
But this last statement is equivalent to p itself (and this equivalence can be proven in the system), so p can be proven in the system. This contradiction shows that the system must be inconsistent.
See also Consistency has three technical meanings: In mathematics and logic, as well as in theoretical physics, it refers to the proposition that a formal theory or a physical theory contains no contradictions. ...
A self-reference occurs when an object refers to itself. ...
Logicism is one of the schools of thought in the Philosophy of mathematics. ...
Minds, Machines and Gödel is the title of a philosophical paper published in 1961 by J. R. Lucas. ...
In mathematical logic, Löbs theorem states that in a theory with Peano arithmetic, for any formula P, if it is provable that if P is provable then P, then P is provable. ...
References
Scientific articles - K. Gödel: Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I. (http://home.ddc.net/ygg/etext/godel/) Monatshefte für Mathematik und Physik, 38 (1931), pp. 173-198. Translated in van Heijenoort: From Frege to Gödel. Harvard University Press, 1971.
- B. Rosser: Extensions of some theorems of Gödel and Church. Journal of Symbolic Logic, 1 (1936), N1, pp. 87-91
Books - Hao Wang: A Logical Journey: From Gödel to Philosophy Bradford Books (January 10, 1997) ISBN 0262231891
- Kārlis Podnieks: Around Goedel's Theorem, http://www.ltn.lv/~podnieks/gt.html
- D. Hofstadter: Gödel, Escher, Bach: An Eternal Golden Braid, 1979, ISBN 0465026850. (1999 reprint: ISBN 0465026567).
- Ernest Nagel, James Roy Newman, Douglas R. Hofstadter: Gödel's Proof, revised edition (2002). ISBN 0814758169.
- Hilbert's second problem (http://aleph0.clarku.edu/~djoyce/hilbert/problems.html#prob2) (English translation)
- Norbert Domeisen, Logik der Antinomien. Bern etc.: Peter Lang. 142 S. 1990. (ISBN 3-261-04214-1), Zentralblatt MATH (http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?first=1&maxdocs=3&type=html&an=0724.03003&format=complete)
GEB cover Gödel, Escher, Bach: an Eternal Golden Braid is a Pulitzer Prize-winning book by Douglas Hofstadter, first published in 1979 by Basic Books. ...
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