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.

In class theories such as Von Neumann – Bernays – Gödel set theory and Morse – Kelley set theory, there is a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

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.

Church subsequently modified his methods to include use of Herbrand – Gödel recursion and then proved ( 1936 ) that the Entscheidungsproblem is unsolvable: There is no generalized " effective calculation " ( method, algorithm ) that can determine whether or not a formula in either the recursive-or λ-calculus is " valid " ( more precisely: no method to show that a well formed formula has a " normal form ").

For example, if we can enumerate all such definable numbers by the Gödel numbers of their defining formulas then we can use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language.

He is best known for his book Gödel, Escher, Bach: an Eternal Golden Braid, first published in 1979.

Hofstadter's thesis about consciousness, first expressed in Gödel, Escher, Bach ( GEB ) but also present in several of his later books, is that it is an emergent consequence of seething lower-level activity in the brain.

Kurt Gödel is known to have read Cartesian Meditations.

A typical description of the problem is given in the book Gödel, Escher, Bach, by Douglas Hofstadter

Typical of these references is Gödel, Escher, Bach by Douglas Hofstadter, which accords the paradox a prominent place in a discussion of self-reference.

Exact solutions of great theoretical interest include the Gödel universe ( which opens up the intriguing possibility of time travel in curved spacetimes ), the Taub-NUT solution ( a model universe that is homogeneous, but anisotropic ), and Anti-de Sitter space ( which has recently come to prominence in the context of what is called the Maldacena conjecture ).

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel.

" A more elaborate version was given by Gottfried Leibniz ( 1646 CE to 1716 CE ); this is the version that Gödel studied and attempted to clarify with his ontological argument.

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.

Within philosophy familiar names include Daniel Dennett who writes from a computational systems perspective, John Searle known for his controversial Chinese room, Jerry Fodor who advocates functionalism, and Douglas Hofstadter, famous for writing Gödel, Escher, Bach, which questions the nature of words and thought.

Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna.

To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers.

This result, known as Tarski's undefinability theorem, was discovered independently by Gödel ( when he was working on the proof of the incompleteness theorem ) and by Alfred Tarski.

The name " quine " was coined by Douglas Hofstadter, in his popular science book Gödel, Escher, Bach: An Eternal Golden Braid, in the honor of philosopher Willard Van Orman Quine ( 1908 – 2000 ), who made an extensive study of indirect self-reference, and in particular for the following paradox-producing expression, known as Quine's paradox:

Kurt Gödel in 1932 showed that intuitionistic logic is not a finitely-many valued logic, and defined a system of Gödel logics intermediate between classical and intuitionistic logic ; such logics are known as intermediate logics.

In any event, Kurt Gödel in 1930 – 31 proved that while the logic of much of PM, now known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent.

This result, known as Tarski's undefinability theorem, was discovered independently by Gödel ( when he was working on the proof of the incompleteness theorem ) and by Alfred Tarski.

The stronger version of the incompleteness theorem that only assumes consistency, rather than ω-consistency, is now commonly known as Gödel's incompleteness theorem and as the Gödel – Rosser theorem.

In 1932, Kurt Gödel defined a system of Gödel logics intermediate between classical and intuitionistic logic ; such logics are known as intermediate logics.

Many researchers in axiomatic set theory have subscribed to what is known as set-theoretical Platonism, exemplified by mathematician Kurt Gödel.

The Institute is perhaps best known as the academic home of Albert Einstein, John von Neumann, Oskar Morgenstern and Kurt Gödel, after their immigration to the United States.

Given any statement, the number it is converted to is known as its Gödel number.

This possibility was first raised by Kurt Gödel in 1949, who discovered a solution to the equations of general relativity ( GR ) allowing CTCs known as the Gödel metric ; and since then other GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes.

Lucas is perhaps best known for his paper " Minds, Machines and Gödel ," arguing that an automaton cannot represent a human mathematician, essentially refuting computationalism.

This procedure is known variously as Gödel numbering, coding, and more generally, as arithmetization.

His results on the closure of non-deterministic space under complement, independently obtained in 1987 also by Neil Immerman ( the result known as the Immerman – Szelepcsényi theorem ), brought the Gödel Prize of ACM and EATCS to both of them in 1995.

Immerman is the winner, jointly with Róbert Szelepcsényi, of the 1995 Gödel Prize in theoretical computer science for proof of what is known as the Immerman – Szelepcsényi theorem, the result that nondeterministic space complexity classes are closed under complementation.

In string theory, Eric G. Gimon and Petr Hořava have argued that in a supersymmetric five-dimensional Gödel universe, quantum corrections to general relativity effectively cut off regions of spacetime with causality-violating closed timelike curves.

From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the identity of indiscernibles: two or more objects are identical ( are one and the same ) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness.

it is chiefly the rule of substitution which would have to be proved " ( Gödel 1944: 124 )

Physicists have long been aware that there are solutions to the theory of general relativity which contain closed timelike curves, or CTCs — see for example the Gödel metric.

For example, the conjunction of the Gödel sentence and any logically valid sentence will have this property.

Gödel finds ( b ) implausible, and thus seems to have believed the human mind was not equivalent to a finite machine, i. e., its power exceeded that of any finite machine.

On the other hand Gödel himself suggested the possibility of giving finitary consistency proofs using finitary methods that cannot be formalized in Peano arithmetic, so he seems to have had a more liberal view of what finitary methods might be allowed.

In 1950 Robinson proved that an essentially undecidable theory need not have an infinite number of axioms by coming up with a counterexample: Robinson arithmetic Q. Q is finitely axiomatizable because it lacks Peano arithmetic's axiom schema of induction ; nevertheless Q, like Peano arithmetic, is incomplete and undecidable in the sense of Gödel.

The paper is famous for the theorems it contains, which have many implications for consistency proofs in mathematics, and for the techniques that Gödel invented to prove these theorems.

Although Goethe's work was rejected by physicists, a number of philosophers and physicists have concerned themselves with it, including Thomas Johann Seebeck, Arthur Schopenhauer ( see: On Vision and Colors ), Hermann von Helmholtz, Rudolf Steiner, Ludwig Wittgenstein, Werner Heisenberg, Kurt Gödel, and Mitchell Feigenbaum.

* unstable: for example, the world lines of the dust particles in the Gödel solution have vanishing shear, expansion, and acceleration, but constant vorticity just balancing a constant Raychuadhuri scalar due to nonzero vacuum energy (" cosmological constant ").