** 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.

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.

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.

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dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

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

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.

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.

Gödel's completeness theorem of 1930 implicitly contains a definition of a universal computer, because the logical rules acting on some axioms of arithmetic will eventually prove as a theorem the result of any computation.

# REDIRECT Gödel's completeness theorem

One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it.

In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem, a weak form of the axiom of choice.

Isabelle has been used to formalize numerous theorems from mathematics and computer science, like Gödel's completeness theorem, Gödel's theorem about the consistency of the axiom of choice, the prime number theorem, correctness of security protocols, and properties of programming language semantics.

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.

* Gödel's completeness theorem, 1930

Some examples are the Hahn – Banach theorem, König's lemma, Brouwer fixed point theorem, Gödel's completeness theorem and Jordan curve theorem.

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.

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.

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

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.

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.

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

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 ".

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 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.

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 theorems are two fundamental theorems of mathematical logic which state inherent limitations of all but the most trivial axiomatic systems for mathematics.

The sentences whose existence is secured by the diagonal lemma can then, in turn, be used to prove fundamental limitative results such as Gödel's incompleteness theorems and Tarski's indefinability theorem.