Although he spoke very little Czech himself, Gödel automatically became a Czechoslovakian citizen at age 12 when the Austro-Hungarian empire broke up at the end of World War I.

According to his classmate Klepetař, like most residents of the predominantly German Sudetenlaender, " Gödel considered himself always Austrian and an exile in Czechoslovakia " (" ein Österreicher im Exil in der Tschechoslowakei ").

One of the earliest attempts to use incompleteness to reason about human intelligence was by Gödel himself in his 1951 Gibbs lecture entitled " Some basic theorems on the foundations of mathematics and their philosophical implications ".

The first big success was by Gödel himself ( before he proved the incompleteness theorems ) who proved the completeness theorem for first-order logic, showing that any logical consequence of a series of axioms is provable.

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.

The problems were popularised by their occurrence in the 1979 book Gödel, Escher, Bach by Douglas Hofstadter, himself a composer of Bongard problems.

Gödel filled it out in pencil and wrote a cover letter, but he never returned it.

Kurt Gödel wrote the first paper on provability logic, which applies modal logic — the logic of necessity and possibility — to the theory of mathematical proof, but Gödel never developed the subject to any significant extent.

Rather in correspondence with Church ( ca 1934 – 5 ), Gödel proposed axiomatizing the notion of " effective calculability "; indeed, in a 1935 letter to Kleene, Church reported that:

Many years later in a letter to Davis ( ca 1965 ), Gödel would confess that " he was, at the time of these lectures, not at all convinced that his concept of recursion comprised all possible recursions ".

While Gödel never published anything bearing on his independent discovery of undefinability, he did describe it in a 1931 letter to John von Neumann.

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.

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

Shortly before his death, Cohen gave a lecture describing his solution to problem of the Continuum Hypothesis at the Gödel centennial conference, in Vienna in 2006.

Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory.

On the other hand, the status of the first and second problems is even more complicated: there is not any clear mathematical consensus as to whether the results of Gödel ( in the case of the second problem ), or Gödel and Cohen ( in the case of the first problem ) give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, a formalization which is quite reasonable but is not necessarily the only possible one.

Kurt Gödel and Paul Cohen.

By 1963 – 4 Gödel would disavow Herbrand – Gödel recursion and the λ-calculus in favor of the Turing machine as the definition of " algorithm " or " mechanical procedure " or " formal system ".

J. R. Lucas in Minds, Machines and Gödel ( 1963 ), and later in his book The Freedom of the Will ( 1970 ), lays out an anti-mechanist argument closely following the one described by Putnam, including reasons for why the human mind can be considered consistent.

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.

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.

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 demonstrated that any non-contradictory formal system, which was comprehensive enough to include at least arithmetic, cannot demonstrate its completeness by way of its own axioms.

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 ).

The following lemma, which Gödel adapted from Skolem's proof of the Löwenheim-Skolem theorem, lets us sharply reduce the complexity of the generic formula for which we need to prove the theorem:

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

Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic.

According to Gödel mathematical logic was " a science prior to all others, which contains the ideas and principles underlying all sciences.

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.

Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that ( since it would then prove its own consistency, which Gödel had shown was impossible ).

Strange loops take form in human consciousness as the complexity of active symbols in the brain inevitably lead to the same kind of self-reference which Gödel proved was inherent in any complex logical or arithmetical system in his Incompleteness Theorem.

Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent.

These set theories cannot prove their own Gödel sentences – provided that they are consistent, which is generally believed.

In Douglas Hofstadter's Gödel, Escher, Bach, there is a narrative between Achilles and the Tortoise ( characters borrowed from Lewis Carroll, who in turn borrowed them from Zeno ), and within this story they find a book entitled " Provocative Adventures of Achilles and the Tortoise Taking Place in Sundry Spots of the Globe ", which they begin to read, the Tortoise taking the part of the Tortoise, and Achilles taking the part of Achilles.

Kurt Gödel considered his incompleteness theorem as analogous to Richard's paradox which, in the original version runs as follows:

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 ).

In footnote 48a, Gödel stated that a planned second part of the paper would establish a link between consistency proofs and type theory, but Gödel did not publish a second part of the paper before his death.

Let be a Gödel numbering of the computable functions ; a map from the natural numbers to the class of unary ( partial ) computable functions.

Let T denote the set of L-sentences true in N, and T * the set of Gödel numbers of the sentences in T. The following theorem answers the question: Can T * be defined by a formula of first-order arithmetic?

Tarski's undefinability theorem ( general form ): Let ( L, N ) be any interpreted formal language which includes negation and has a Gödel numbering g ( x ) such that for every L-formula A ( x ) there is a formula B such that B ↔ A ( g ( B )) holds.

Let T * be the set of Gödel numbers of L-sentences true in N. Then there is no L-formula True ( n ) which defines T *.

Let be an enumeration of Gödel numbers of computable functions.