Gödel's original formulation is deduced by taking the particular case of a theory without any axiom.

In modern logic texts, Gödel's completeness theorem is usually proved with Henkin's proof, rather than with Gödel's original proof.

Gödel's original statement and proof of the incompleteness theorem requires the assumption that the theory is not just consistent but ω-consistent.

Although not a translation of the original paper, a very useful 4th version exists that " cover ground quite similar to that covered by Godel's original 1931 paper on undecidability " ( Davis 1952: 39 ), as well as Gödel's own extensions of and commentary on the topic.

In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.

* Gödel's ontological proof

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel.

The first version of the ontological proof in Gödel's papers is dated " around 1941 ".

* Original proof of Gödel's completeness theorem

The proof of Gödel's 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the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

Depending on the particular formalism adopted for the calculus, it may be seen as a simple application of a " functional substitution " rule of inference, as in Gödel's paper, or it may be proved by considering the formal proof of, replacing in it all occurrences of Q by some other formula with the same free variables, and noting that all logical axioms in the formal proof remain logical axioms after the substitution, and all rules of inference still apply in the same way.

Gödel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's incompleteness theorems and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

Hofstadter claims this happens in the proof of Gödel's Incompleteness Theorem:

Hofstadter points to Bach's Canon per Tonos, M. C. Escher's drawings Waterfall, Drawing Hands, Ascending and Descending, and the liar paradox as examples that illustrate the idea of strange loops, which is expressed fully in the proof of Gödel's incompleteness theorem.

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.

* Gödel's ontological proof

# REDIRECT Gödel's ontological proof

Gödel's second incompleteness theorem shows that it is not possible for any proof that Peano Arithmetic is consistent to be carried out within Peano arithmetic itself.

Kreisel ( 1976 ) states that although Gödel's results imply that no finitistic syntactic consistency proof can be obtained, semantic ( in particular, second-order ) arguments can be used to give convincing consistency proofs.

Detlefsen ( 1990: p. 65 ) argues that Gödel's theorem does not prevent a consistency proof because its hypotheses might not apply to all the systems in which a consistency proof could be carried out.

** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.

Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

The notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

This incompleteness result is similar to Gödel's incompleteness theorem in that it shows that no consistent formal theory for arithmetic can be complete.

Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself.

( This doesn't violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.

The work of both authors was heavily influenced by Kurt Gödel's earlier work on his incompleteness theorem, especially by the method of assigning numbers ( a Gödel numbering ) to logical formulas in order to reduce logic to arithmetic.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

Gödel's completeness theorem says that a deductive system of first-order predicate calculus is " complete " in the sense that no additional inference rules are required to prove all the logically valid formulas.

Gödel's incompleteness theorem, another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics.

In it, he established the completeness of the first-order predicate calculus ( Gödel's completeness theorem ).

Examples of early theorems from classical model theory include Gödel's completeness theorem, the upward and downward Löwenheim – Skolem theorems, Vaught's two-cardinal theorem, Scott's isomorphism theorem, the omitting types theorem, and the Ryll-Nardzewski theorem.

* Gödel's Proof ( 2002 revised edition ) by Ernest Nagel and James R. Newman, edited by Hofstadter ( ISBN 0-8147-5816-9 ).

"< ref > Gödel's answer to a special questionnaire sent him by the sociologist Burke Grandjean.

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course.

Gödel, Escher, Bach by Douglas Hofstadter, published in 1979, discusses the ideas of self-reference and strange loops, drawing on a wide range of artistic and scientific work, including the art of M. C. Escher and the music of J. S. Bach, to illustrate ideas behind Gödel's incompleteness theorems.

He explicitly includes universe representations describable by non-halting programs whose output bits converge after finite time, although the convergence time itself may not be predictable by a halting program, due to Kurt Gödel's limitations.

The most famous result is Gödel's incompleteness theorem ; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.

For a philosophy of mathematics that attempts to overcome some of the shortcomings of Quine and Gödel's approaches by taking aspects of each see Penelope Maddy's Realism in Mathematics.

Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency.

For instance, there is a phonograph that destroys itself by playing a record titled " I Cannot Be Played on Record Player X " ( an analogy to Gödel's incompleteness theorems ), an examination of canon form in music, and a discussion of Escher's lithograph of two hands drawing each other.

The analysis of logical concepts and the machinery of formalization that is essential to Principia Mathematica ( 3 vols., 1910 – 1913 ) ( by Bertrand Russell, 1872 – 1970, and Alfred North Whitehead, 1861 – 1947 ), to Russell's theory of descriptions, to Kurt Gödel's ( 1906 – 1978 ) incompleteness theorems, and to Alfred Tarski's ( 1901 – 1983 ) theory of truth, is ultimately due to Frege.

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

** What is Mathematics: Gödel's Theorem and Around by Karlis Podnieks.

Post's solution to the problem is described in the demonstration An Example of a Successful Absolute Proof of Consistency offered by Ernest Nagel and James R. Newman in their 1958 Gödel's Proof.

* a set in Kurt Gödel's universe L, which may be constructed by transfinite application of certain constructions in set theory ; see constructible universe.

Earlier statements of this type had either been, except for Gentzen, extremely complicated, ad-hoc constructions ( such as the statements generated by the construction given in Gödel's incompleteness theorem ) or concerned metamathematics or combinatorial results.