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.

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

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.

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.

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.

Since most physicists would consider the statement of the underlying rules to suffice as the definition of a " theory of everything ", most physicists argue that Gödel's Theorem does not mean that a ToE cannot exist.

Or if there is a measurable cardinal, then iterating the definable powerset operation rather than the full one yields Gödel's constructible universe, L, which does not satisfy the statement " there is a measurable cardinal " ( even though it contains the measurable cardinal as an ordinal ).

If one is to use Gödel's technique to prove the proposition that T cannot prove, one must first prove ( the mathematical statement representing ) the consistency of T, a daunting and perhaps impossible task.

In Gödel's first result he showed how to construct, for any sufficiently powerful and consistent finitely axiomatizable system — such as necessary to axiomatize the elementary theory of arithmetic — a statement that can be shown to be true, but that does not follow from the rules of the system.

the sentence employed to prove Gödel's first incompleteness theorem says " This statement is not provable.

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.

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.

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.

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

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 incompleteness theorems

# REDIRECT Gödel's incompleteness theorems

# REDIRECT Gödel's incompleteness theorems

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.

However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that " most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently.

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.

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

Gödel's incompleteness theorems cast unexpected light on these two related questions.

Gödel's first incompleteness theorem showed that Principia could not be both consistent and complete.

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

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.

The actual notion of computation was isolated soon after, starting with Gödel's incompleteness theorem.

Gödel's incompleteness theorem shows that no consistent, recursively enumerable theory ( that is, one whose theorems form a recursively enumerable set ) in which the concept of natural numbers can be expressed, can include all true statements about them.