The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

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.

His ' matrix divisor ' ( vector bundle avant la lettre ) Riemann – Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The Grothendieck – Riemann – Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.

Argonne National Laboratory was a leader in automated theorem proving from the 1960s to the 2000s

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 it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

The Cook – Levin theorem states that the Boolean satisfiability problem is NP-complete, and in fact, this was the first decision problem proved to be NP-complete.

SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).

It was Pierre-Simon Laplace ( 1749 – 1827 ) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence.

In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.

An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.

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.

The culmination of their investigations, the Arzelà – Ascoli theorem, was a generalization of the Bolzano – Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions.

The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by NP-intermediate | Ladner's theorem.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

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.

* 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.

The term Bayesian refers to Thomas Bayes ( 1702 – 1761 ), who proved a special case of what is now called Bayes ' theorem in a paper titled " An Essay towards solving a Problem in the Doctrine of Chances ".

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.

Gauss proved the method under the assumption of normally distributed errors ( see Gauss – Markov theorem ; see also Gaussian ).

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

He also proved a result concerning Fermat numbers that is called Goldbach's theorem.

Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4.

J. Chernick proved a theorem in 1939 which can be used to construct a subset of Carmichael numbers.

In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, but not proved until 1976 ( by Kenneth Appel and Wolfgang Haken, using substantial computer assistance ).

Gödel's second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself.

Euclid's axioms seemed so intuitively obvious that any theorem proved from them was deemed true in an absolute, often metaphysical, sense.

The Pythagorean theorem is proved.

Cantor supposed that Thales proved his theorem by means of Euclid book I, prop 32 after the manner of Euclid book III, prop 31.

It improves Dirichlet's theorem on prime numbers in arithmetic progressions, by showing that by averaging over the modulus over a range, the mean error is much less than can be proved in a given case.

It was the first major theorem to be proved using a computer.

In 1890, in addition to exposing the flaw in Kempe's proof, Heawood proved the five color theorem and generalized the four color conjecture to surfaces of arbitrary genus — see below.

The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem.

He rediscovered the Descartes ' theorem in 1936 and published it as a poem.

J. Barkley Rosser ( 1936 ) strengthened the incompleteness theorem by finding a variation of the proof ( Rosser's trick ) that only requires the theory to be consistent, rather than ω-consistent.

This is known as the Eckart – Young theorem, as it was proved by those two authors in 1936 ( although it was later found to have been known to earlier authors ; see ).

König's lemma or König's infinity lemma is a theorem in graph theory due to Dénes Kőnig ( 1936 ).

In 1936 Maltsev proved the compactness theorem.

1936: Alfred Tarski proved his truth undefinability theorem.

Finally, Anatoly Ivanovich Maltsev ( Анато ́ лий Ива ́ нович Ма ́ льцев, 1936 ) proved the Löwenheim – Skolem theorem in its full generality.

* Proof of the undecidability of first order predicate logic ( Church's theorem 1936 )

Away from the thesis Wold and Cramér did some joint work, their best known result being the Cramér – Wold theorem ( 1936 ).

The kissing circles in this problem are sometimes known as Soddy circles, perhaps because Soddy chose to publish his version of the theorem in the form of a poem titled The Kiss Precise, which was printed in Nature ( June 20, 1936 ).

Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in mathematical logic, the foundations of mathematics, and in formal semantics.

Gödel also discovered the undefinability theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1936 publication of Tarski's work ( Murawski 1998 ).

According to the footnote of the undefinability theorem ( Satz I ) of the 1936 paper, the theorem and the sketch of the proof were added to the paper only after the paper was sent to print.

We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem Tarski actually proved in 1936.

The theorem just stated is a corollary of Post's theorem about the arithmetical hierarchy, proved some years after Tarski ( 1936 ).

The theorem was first proved by Stone ( 1936 ), and thus named in his honor.

In 1936 he proved the Freudenthal spectral theorem and in 1937 the Freudenthal suspension theorem.

In 1936, he proved Rosser's trick, a stronger version of Gödel's first incompleteness theorem which shows that the requirement for ω-consistency may be weakened to consistency.

* In 1936, he published a long paper that included Stone's representation theorem for Boolean algebras, an important result in mathematical logic and universal algebra.

" The fact that such self-reference can be expressed within arithmetic was not known until Gödel's paper appeared ; independent work of Alfred Tarski on his indefinability theorem was conducted around the same time but not published until 1936.