Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

However, shortly after this positive result, Kurt Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems ( 1931 ), showing that in any sufficiently strong axiomatic system there are true statements which cannot be proved in the system.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

The debate began when Church proposed to Kurt Gödel that one should define the " effectively computable " functions as the λ-definable functions.

Stephen Kleene ( 1952 ) adds to the list the functions " reckonable in the system S < sub > 1 </ sub >" of Kurt Gödel 1936, and Emil Post's ( 1943, 1946 ) " canonical called normal systems ".

For example, the view that numbers are Platonic objects was revived by Kurt Gödel as a result of certain puzzles that he took to arise from the phenomenological accounts.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel — in a roundtable discussion during the Conference on Epistemology held jointly with the Society meetings — tentatively announced the first expression of his incompleteness theorem.

Logical Dilemmas: The Life and Work of Kurt Gödel.

Kurt Gödel is known to have read Cartesian Meditations.

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.

In letters to his mother, who was not a churchgoer and had raised Kurt and his brother as freethinkers, Gödel argued at length for a belief in an afterlife.

* Kurt Gödel ( 1995 ).

It was first proved by Kurt Gödel in 1929.

* Stanford Encyclopedia of Philosophy: " Kurt Gödel " -- by Juliette Kennedy.

* MacTutor biography: Kurt Gödel.

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.

* Kurt Gödel An example of a new type of cosmological solutions of Einstein ’ s field equations of gravitation Rev.

# REDIRECT Kurt Gödel

Kurt Friedrich Gödel (; ; April 28, 1906 – January 14, 1978 ) was an Austrian American logician, mathematician, and philosopher.

The ancestors of Kurt Gödel were often active in Brünn's cultural life.

The theorems were proven by Kurt Gödel in 1931, and are important in the philosophy of mathematics.

Some notable mathematicians include Archimedes of Syracuse, Leonhard Euler, Carl Gauss, Johann Bernoulli, Jacob Bernoulli, Aryabhata, Brahmagupta, Bhaskara II, Nilakantha Somayaji, Omar Khayyám, Muhammad ibn Mūsā al-Khwārizmī, Bernhard Riemann, Gottfried Leibniz, Andrey Kolmogorov, Euclid of Alexandria, Jules Henri Poincaré, Srinivasa Ramanujan, Alexander Grothendieck, David Hilbert, Alan Turing, von Neumann, Kurt Gödel, Joseph-Louis Lagrange, Georg Cantor, William Rowan Hamilton, Carl Jacobi, Évariste Galois, Nikolay Lobachevsky, Rene Descartes, Joseph Fourier, Pierre-Simon Laplace, Alonzo Church, Nikolay Bogolyubov and Pierre de Fermat.

Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency.

" He went on to compare Cohen to Kurt Gödel, saying: " Nothing more dramatic than their work has happened in the history of the subject.

The theorems, proven by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics.

The completeness of sentential calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann ( 1924 ), von Neumann ( 1927 ) and Herbrand ( 1931 ).

Due to limitations first exhibited by Kurt Gödel ( 1931 ), it may be impossible to predict the convergence time itself by a halting program, otherwise the halting problem could be solved.

Kurt Waldemar Tank ( February 24, 1898 – June 5, 1983 ) was a German aeronautical engineer and test pilot who led the design department at Focke-Wulf from 1931 to 1945.

Only in the last chapter is it revealed that the punched cards represent a program which proves two theorems which in reality would not be discovered until 1931 by Kurt Gödel.

The terms " diagonal lemma " or " fixed point " do not appear in Kurt Gödel's epochal 1931 article, or in Tarski ( 1936 ).

While at Princeton, he came very close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931.

After the coming of sound he made a trilogy of films that secured his reputation: Westfront 1918 ( 1930 ), The Threepenny Opera ( 1931 ) with Lotte Lenya ( based on the Bertolt Brecht and Kurt Weill musical ), and Kameradschaft ( 1931 ).

Hugo Winckler and Theodor Makridis Bey conducted the first excavations 1906, 1907, and 1911 – 13, which were resumed in 1931 under Kurt Bittel, followed by Peter Neve ( site director 1963, general director 1978 – 94 ).

It was first isolated in 1931, by Adolf Friedrich Johann Butenandt and Kurt Tscherning.

In 1931, Kurt Gödel published his famous incompleteness theorems, which he proved in part by showing how to represent syntax within first-order arithmetic.

In 1931, the mathematician and logician Kurt Gödel proved that any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

The German Albatross Books had pioneered the idea of a line of color-coded paperback editions in 1931 under Kurt Enoch ; Penguin Books in Britain had refined the idea in 1935 and had 1 million books in print by the following year.

Promoted to the rank of Colonel ( Oberst ) in 1925 and Lieutenant General ( Generalleutnant ) in 1931, Stülpnagel played a leading role in the Reichswehr in conjunction with Kurt von Schleicher and Freiherr von Bussche during the Weimar era.

* Love Life ( 1948 ), with Kurt Weill

* Calypso ( Bokononism ), songs from The Books of Bokonon from Kurt Vonnegut's novel Cat's Cradle ( 1963 )

Another version is the Collector's Edition, a two-disc set featuring a remastered transfer with a 5. 1 audio track, two commentaries ( one by John Carpenter and Kurt Russell, another by producer Debra Hill and Joe Alves ), a making-of featurette, the first issue of a comic book series titled John Carpenter's Snake Plissken Chronicles, and a ten-minute deleted opening sequence.

Himmler also gained authority over all of Germany's uniformed law enforcement agencies, which were amalgamated into the new Ordnungspolizei ( Orpo: Order Police ), which became an agency of the SS under Kurt Daluege.

Kurt Walker ( born August 9, 1959 ), professionally known by his stage name Kurtis Blow, is an American rapper and record producer.

The Mercury / Polygram single, co-produced by Kurt, included the collaborative efforts of Stephanie Mills, Whitney Houston, New Edition, El Debarge, James " JT " Taylor, The Fat Boys, Menudo ( Ricky Martin ), Teena Marie and Run DMC.

* Kurt Russell ( born 1951 ), an American actor

* In The Sirens of Titan by Kurt Vonnegut ( 1959 ), previously uninhabited Mars is populated by brainwashed transplants from Earth, leading to the invasion of Earth by the newly created Martian army.

The writer Theodor Fontane, painter Adolf von Menzel, and Dadaist Kurt Schwitters were all guests ; Karl Liebknecht, the Spartacus Communist movement leader read a lot here and even made some key political speeches from the pavement terrace, while author Erich Kästner wrote part of his 1929 bestseller for children, Emil und die Detektive ( Emil and the Detectives ), on the same terrace and made the café the setting for an important scene in the book.

According to Dusted Magazine, Sakamoto's use of squelching bounce sounds and mechanical beats was later incorporated in early electro and hip hop music productions such as “ Message II ( Survival )” ( 1982 ) by Melle Mel and Duke Bootee, “ Magic ’ s Wand ” ( 1982 ) by Whodini and Thomas Dolby, Twilight 22 ’ s “ Electric Kingdom ” ( 1983 ), and Kurt Mantronik's Mantronix: The Album ( 1985 ).

In The Importance of Being Earnest ( 1952 ), Dame Edith Evans reprised her celebrated interpretation of Lady Bracknell ; The Importance of Being Earnest ( 1992 ) by Kurt Baker used an all-black cast ; and Oliver Parker's The Importance of Being Earnest ( 2002 ) incorporated some material cut during the preparation of the original stage production.

This cycle was the first metabolic cycle discovered ( Hans Krebs and Kurt Henseleit, 1932 ), five years before the discovery of the TCA cycle.

Within the politically heated turmoil of the early Weimar Republic, the media frequently used the term Fememord to refer to right-wing political homicides, e. g. the murder of Jewish politicians such as Kurt Eisner ( 1919 ), Matthias Erzberger ( 1921 ), or Walther Rathenau ( 1922 ) by right-wing groups such as Organisation Consul.

Its members also belonged to other art movements and groups during the Weimar Republic era, such as architect Walter Gropius ( founder of Bauhaus ), and Kurt Weill and Bertolt Brecht ( agitprop theatre ).

( Despite recent finds indicating a Roman presence near the modern city of Minden, its location remains disputed ; other sites near Minden or Rinteln have been suggested by the historian Hans Delbrück ( 1848 – 1929 ) and the military writer Kurt Pastenaci ( 1894 – 1961 ), respectively.

His successor was Kurt Georg Kiesinger ( CDU ), who formed a grand coalition with the SPD.

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.

* Kurt Kuhlmey ( 1913 – 1993 ), Bundeswehr major general

* Kurt Plenzat ( 1914 – 1998 ), military officer

His next film, The Thing ( 1982 ), is notable for its high production values, including innovative special effects by Rob Bottin, special visual effects by matte artist Albert Whitlock, a score by Ennio Morricone and a cast including rising star Kurt Russell and respected character actors such as Wilford Brimley, Richard Dysart, Charles Hallahan, Keith David, and Richard Masur.

) Hart continued to write plays after parting with Kaufman, such as Christopher Blake ( 1946 ) and Light Up the Sky ( 1948 ), as well as the book for the musical Lady In The Dark ( 1941 ), with songs by Kurt Weill and Ira Gershwin.

Orff was a friend of Kurt Huber, one of the founders of the resistance movement Die Weiße Rose ( the White Rose ), who was condemned to death by the Volksgerichtshof and executed by the Nazis in 1943.