* Gödel's ontological proof

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

* Gödel's ontological proof

# REDIRECT Gödel's ontological proof

* Gödel's ontological proof

* Gödel's ontological proof

* Gödel's ontological proof of the existence of God is published posthumously.

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

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.

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

* 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the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.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 original statement and proof of the incompleteness theorem requires the assumption that the theory is not just consistent but ω-consistent.

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.

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.

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.

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

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.

< 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.

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

By Gödel's incompleteness theorem, Peano arithmetic is incomplete and its consistency is not internally provable.

In 1930, Gödel's completeness theorem showed that propositional logic itself was complete in a much weaker sense — that is, any sentence that is unprovable from a given set of axioms must actually be false in some model of the axioms.

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 ).

Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.

A number of scholars claim that Gödel's incompleteness theorem proves that any attempt to construct a ToE is bound to fail.

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

Gödel's second incompleteness theorem ( 1931 ) shows that no formal system extending basic arithmetic can be used to prove its own consistency.

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

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.

Gödel's first incompleteness theorem shows that any consistent effective formal system that includes enough of the theory of the natural numbers is incomplete: there are true statements expressible in its language that are unprovable.

This is mostly of technical interest, since 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.

Hilbert lived for 12 years after Gödel's theorem, but he does not seem to have written any formal response to Gödel's work.

But doubtless the significance of Gödel's work to mathematics as a whole ( and not just to formal logic ) was amply and dramatically illustrated by its applicability to one of Hilbert's problems.

Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system ; then his incompleteness theorems showed that this is unattainable.

It is not in general possible for a logical system to have a formal negation operator such that there is a proof of " not " P exactly when there isn't a proof of P ; see Gödel's incompleteness theorems.