* A. P. Hazen, " On Gödel's Ontological Proof ", Australasian Journal of Philosophy, Vol.

* Jordan Howard Sobel, " Gödel's Ontological Proof " in On Being and Saying.

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.

* Ernest Nagel and James R. Newman 1958 Gödel's Proof, New York University Press, Card Catalog Number: 58-5610.

* Gödel's Proof by Ernest Nagel and James R. Newman ( 1959 ).

* Proof of the semantic completeness of first order predicate logic ( Gödel's completeness theorem 1930 )

Newman also wrote Gödel's Proof ( 1958 ) with Ernest Nagel, presenting the main results of Gödel's incompleteness theorem and the mathematical work and philosophies leading up to its discovery in a more accessible manner.

This book inspired Douglas Hofstadter to take up the study of mathematical logic, write his famous book Gödel, Escher, Bach, and prepare a second edition of Gödel's Proof, published in 2002.

While Stephen Hawking was initially a believer in the Theory of Everything, after considering Gödel's Incompleteness Theorem, he has concluded that one is not obtainable, and has stated so publicly in his lecture " Gödel and the End of Physics " ( 2002 ).

Likewise, some Catholic theologians have rejected Gödel's revised version.

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

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

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

Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.

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dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.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

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.

* 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.

In 1958, he published with James R. Newman Gödel's proof, a short book explicating Gödel's incompleteness theorems to those not well trained in mathematical logic.

** 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.

George Boolos ( 1989 ) built on a formalized version of Berry's paradox to prove Gödel's Incompleteness Theorem in a new and much simpler way.

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

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

Thus Post in his 1936 was also discounting Kurt Gödel's suggestion to Church in 1934 – 5 that the thesis might be expressed as an axiom or set of axioms.

Hilbert's work had started logic on this course of clarification ; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s.

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

The prototypical example of this abstract notion is the self-referential structure at the core of Gödel's incompleteness theorems.

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

Morgenstern's diary is an important and usually reliable source for Gödel's later years, but the implication of the August 1970 diary entry — that Gödel did not believe in God — is not consistent with the other evidence.

" Wang reports that Gödel's wife, Adele, two days after Gödel's death, told Wang that " Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning.

< p > The Grandjean questionnaire is perhaps the most extended autobiographical item in Gödel's papers.

It is possible that this italicization is Wang's and not Gödel's.