Gödel's completeness theorem is a fundamental A theorem is a statement which can be proven true within some logical framework. Proving theorems is a central activity of mathematics. Note that theorem is distinct from theory. A theorem generally has a set-up - a number of conditions, which may be listed in the theorem or described beforehand... theorem in 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. Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more... mathematical logic proved by Kurt Gödel [ kurt gøːdl], ( April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher of mathematics. He was born in Brünn in Moravia, Austria-Hungary (now Brno in the Czech Republic), became a Czechoslovak citizen at age 12 when the... Kurt Gödel in 1929 was a common year starting on Tuesday (link will take you to calendar). Events January January 2 - Canada and the United States agree on a plan to preserve Niagara Falls. January 9 - The Seeing Eye is established with the mission to train dogs to assist the blind ( Nashville, Tennessee... 1929. It states, in its most familiar form, that in 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... or for any X, it is the case that..., where X is an element of a set called... first-order predicate calculus every universally valid formula can be proved.

The word "proved" above means, in effect: proved by a method whose validity can be checked Flowcharts are often used to represent algorithms. An algorithm is a finite set of well-defined instructions for accomplishing some task which, given an initial state, will result in a corresponding recognisable end-state (contrast with heuristic). Algorithms can be implemented by computer programs, although often in restricted forms; an... algorithmically, for example, by a The tower of a personal computer (specifically a Power Mac G5). A computer is a device or machine for making calculations or controlling operations that are expressible in numerical or logical terms. Computers are constructed from components that perform simple well-defined functions. The complex interactions of these components endow... computer (although no such machines existed in 1929).

A logical formula is called universally valid if it is true in every possible domain and with every possible interpretation, inside that domain, of non-constant symbols used in the formula. To say that it can be proved means that there exists a formal proof of that formula which uses only the logical axioms and rules of inference adopted in some particular In logic, mathematics, and computer science, a formal system is a formal grammar used for modelling purposes. Formalization is the act of creating a formal system, in an attempt to capture the essential features of a real-world or conceptual system in formal language. For example, in some colleges the... formalisation of 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... or for any X, it is the case that..., where X is an element of a set called... first-order predicate calculus.

The theorem can be seen as a justification of the logical axioms and inference rules of first-order logic. The rules are "complete" in the sense that they are strong enough to prove every universally valid statement. A converse to completeness is A logical argument is sound if and only if the argument is valid all of its premises are true. A proof procedure (e.g. natural deduction) for a logic is sound if it proves only valid formulas (also tautologies). Sound Arguments Suppose we have a sound argument (in this case... soundness, i.e., the fact that only universally valid statements can be proven in first-order logic. (In fact, the axioms of first-order logic are chosen in such a way that soundness is more or less obvious.)

To cleanly state Gödel's completeness theorem, one has to refer to an underlying Set theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed. For more information on set theory in Wikipedia, see: Set gives a basic introduction to elementary... set theory in order to clarify what the word "domain" in the definition of "universally valid" means.

The branch of mathematical logic that deals with what is true in different domains and under different interpretations is In mathematics, model theory is the study of the representation of mathematical concepts in terms of set theory, or the study of the models which underlie mathematical systems. It assumes that there are some pre-existing mathematical objects out there, and asks questions regarding how or what can be proven... model theory; the branch that deals with what can be formally proved is Proof theory, studied as a branch of mathematical logic, represents proofs as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures, such as plain lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of... proof theory. The completeness theorem, therefore, establishes a fundamental connection between what can be proved and what is (universally) true; between model theory and proof theory; between semantics and syntax in mathematical logic. It should not, however, be misinterpreted as obliterating the difference between these two concepts; in fact, another celebrated result by the same author, In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. Somewhat simplified, the first theorem states: In any consistent formal system that is sufficiently strong to axiomatize the natural numbers – that is, sufficiently strong to define the operations... Gödel's incompleteness theorem, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

In mathematical logic, Gödels incompleteness theorems are two celebrated theorems proved by Kurt Gödel in 1931. Somewhat simplified, the first theorem states: In any consistent formal system that is sufficiently strong to axiomatize the natural numbers – that is, sufficiently strong to define the operations... Gödel's incompleteness theorem was the death knell of an attempt by the renowned mathematicians Alfred North Whitehead Alfred North Whitehead ( February 15, 1861, Ramsgate, Kent, UK – December 30, 1947, Cambridge, MA) was a British-American philosopher, physicist and mathematician who worked in logic, mathematics, philosophy of science and metaphysics. His best known work in mathematics is the Principia Mathematica which he wrote with... Alfred North Whitehead and Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell ( May 18, 1872 – February 2, 1970) was one of the most influential mathematicians, philosophers, and logicians of the modern age, working mostly in the 20th century. A prolific writer, Russell was also a populariser of philosophy and a commentator on... Bertrand Russell to patch up set theories developed by Friedrich Ludwig Gottlob Frege Friedrich Ludwig Gottlob Frege (November 8, 1848 - July 26, 1925) was a German mathematician, logician, and philosopher who is regarded as a founder of both modern mathematical logic and analytic philosophy. Freges life Frege was born in Wismar. He started studying at the University of... Gottlob Frege and Georg Ferdinand Ludwig Philipp Cantor (March 3, 1845 – January 6, 1918) was a mathematician who was born in Russia and lived in Germany for most of his life. He is best known as the creator of modern set theory. He is recognized by mathematicians for having extended set theory... Georg Cantor, which attempted to avoid certain paradoxes derived more or less from an ancient one, The Epimenides paradox is a problem in logic. This problem is named after the Cretan philosopher Epimenides of Knossos (flourished circa 600 BC), who stated Κρητες αει ψευσται, Cretans, always liars. There is no single statement of... Epimenides paradox, which, itself is resolvable as usually stated. The Whitehead-Russell proposal was, in very simple terms, to classify sets of sets at a higher level, or type, than the underlying sets of which they were composed. At the broadest level, type theory is the branch of mathematics and logic that concerns itself with classifying entities into sets called types. In this sense, it is related to the metaphysical notion of type. Modern type theory was invented partly in response to Russells paradox, and features prominently... Type theory is a surviving descendant of their work, but their basic aim of allowing the creation of certain axiomatic systems was defeated by Gödel.

Proofs

For an explanation of Gödel's original proof of the theorem, see The proof of Gödels completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 (and a rewritten version of the dissertation, published as an article in 1930) is not easy to read today; it uses concepts and formalism that are outdated and terminology that is... original proof.

In modern logic texts, Gödel's completeness theorem is usually proved with Leon Henkin is a logician, currently Emeritus Professor at Berkeley. He is principally known for the Henkin Completeness Proof: his version of the proof of the semantic completeness of standard systems of first-order logic. Henkins result was not novel — it had first been proved by Kurt G... Henkin's proof rather than with Gödel's original proof.

Further reading

• Kurt Gödel, "Über die Vollständigkeit des Logikkalküls", doctoral dissertation, University Of Vienna, 1929. This dissertation is the original source of the proof of the completeness theorem.
• Kurt Gödel, "Die Vollständigkeit der Axiome des logischen Funktionen-kalküls", Monatshefte für Mathematik und Physik 37 (1930), 349-360. This article contains the same material as the doctoral dissertation, in a rewritten and shortened form. The proofs are more brief, the explanations more succinct, and the lengthy introduction has been omitted.

External link

• Vilnis Detlovs and Karlis Podnieks, "Introduction to mathematical logic", http://www.ltn.lv/~podnieks/