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proof and Gödel's
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 ".
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
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 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.

proof and completeness
A FAT usually includes a check of completeness, a verification against contractual requirements, a proof of functuality ( either by simulation or a conventional function test ) and a final inspection.
In August 1970, Gödel told Oskar Morgenstern that he was " satisfied " with the proof, but Morgenstern recorded in his diary entry for 29 August 1970, that Gödel would not publish because he was afraid that others might think " that he actually believes in God, whereas he is only engaged in a logical investigation ( that is, in showing that such a proof with classical assumptions ( completeness, etc.
The completeness theorem says that if a formula is logically valid then there is a finite deduction ( a formal proof ) of the formula.
This theorem by Henkin is the most directly obtained version of the completeness theorem in its simplest proof.
* The first proof of the completeness theorem.
The original completeness proof applies to all classical models, not some special proper subclass of intended ones.
In spite of its name, there is no purely algebraic proof of the theorem, since any proof must use the completeness of the reals ( or some other equivalent formulation of completeness ), which is not an algebraic concept.
New languages with new features are always being created, so proof of Turing completeness is always a challenge.
The proof is similar to the preceding statement ; the finite intersection property takes the role played by completeness.
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.
Trivial may also refer to any easy case of a proof, which for the sake of completeness cannot be ignored.
He was principally known for the " Henkin's completeness proof ": his version of the proof of the semantic completeness of standard systems of first-order logic.
) Henkin's 1949 proof is much easier to survey than Gödel's and has thus become the standard choice of completeness proof for presentation in introductory classes and texts.

proof and theorem
The reader will find it helpful to think of the special case when the primes are of degree 1, and even more particularly, to think of the proof of Theorem 10, a special case of this theorem.
The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.
It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.
In fact, Zermelo initially introduced the axiom of choice in order to formalize his proof of the well-ordering theorem.
Mordell's theorem had an ad hoc proof ; Weil began the separation of the infinite descent argument into two types of structural approach, by means of height functions for sizing rational points, and by means of Galois cohomology, which was not to be clearly named as that for two more decades.
He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).
The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.
* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.
Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent ithis first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.
A more general binomial theorem and the so-called " Pascal's triangle " were known in the 10th-century A. D. to Indian mathematician Halayudha and Persian mathematician Al-Karaji, in the 11th century to Persian poet and mathematician Omar Khayyam, and in the 13th century to Chinese mathematician Yang Hui, who all derived similar results .< ref > Al-Karaji also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using mathematical induction.
If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.
SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).
A copyright certificate for proof of the Fermat theorem, issued by the State Department of Intellectual Property of Ukraine
The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.
* ( Purely analytic proof of the theorem that between any two values which give results of opposite sign, there lies at least one real root of the equation ).
The first proof relies on a theorem about products of limits to show that the derivative exists.
In his 1799 doctorate in absentia, A new proof of the theorem that every integral rational algebraic function of one variable can be resolved into real factors of the first or second degree, Gauss proved the fundamental theorem of algebra which states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

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