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.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

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.

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

For a first order predicate calculus, with no (" proper ") axioms, Gödel's completeness theorem states that the theorems ( provable statements ) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.

* William McCune Argonne National Laboratory, author of Otter, the first high-performance theorem prover.

Developed original resolution and unification based first order theorem proving, co-editor of the " Handbook of Automated Reasoning ", recipient of the Herbrand Award 1996

According to (), the first historical mention of the statement of this theorem appears in ().

This position is a more refined version of the theorem first discovered by David Hume.

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 ")

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

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.

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem.

Andrey Kolmogorov later independently published this theorem in Problems Inform.

The proof of the theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

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

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.

Abel had sent most of his work to Berlin to be published in Crelles Journal, but he had saved what he regarded his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials.

The first complete proof of the theorem was provided by Abbati and published in 1803.

The virial theorem can be obtained directly from Lagrange's Identity as applied in classical gravitational dynamics, the original form of which was included in Lagrange's " Essay on the Problem of Three Bodies " published in 1772.

* Leonhard Euler produces the first published proof of Fermat's " little theorem ".

He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961 ; in 1962 he generalized the ideas in a 107 page paper that established the h-cobordism theorem.

1912 Plemelj published a very simple proof for the Fermat's last theorem for exponent n = 5, which was first given almost simultaneously by Dirichlet in 1828 and Legendre in 1830.

He wrote his first completeness theorem in modal logic at the age of 17, and had it published a year later.

The C < sup > 1 </ sup > theorem was published in 1954, the C < sup > k </ sup >- theorem in 1956.

A rigorous proof was published by Argand in 1806 ; it was here that, for the first time, the fundamental theorem of algebra was stated for polynomials with complex coefficients, rather than just real coefficients.

His solution contained a gap, which Cauchy considered minor, though this was not patched until the work of Norwegian mathematician Niels Henrik Abel, who published a proof in 1824, thus establishing the Abel – Ruffini theorem.

Moritz von Jacobi published the maximum power ( transfer ) theorem around 1840 ; it is also referred to as " Jacobi's law ".

However, the priority for this result ( now known as the Mohr – Mascheroni theorem ) belongs to the Dane Georg Mohr, who had previously published a proof in 1672.

It is named after Carl Friedrich Gauss who was aware of a version of the theorem but never published it, and Pierre Ossian Bonnet who published a special case in 1848.

The theorem was later published in 1779 in Étienne Bézout's Théorie générale des équations algébriques.

The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918.

In 1678, he published a now famous theorem on synthetic geometry in a triangle called Ceva ’ s Theorem.

He published this new theorem in De lineis rectis.

Ceva not only published his own theorem, but he also rediscovered and published Menelaus's theorem.