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.

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.

* 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 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 ).

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 used the blanket for late morning naps when hosts of the night had gone off to jobs and proved reluctant to leave him in their small rooms with their few possessions.

It is, however, a disarming disguise, or perhaps a shield, for not only has Mercer proved himself to be one of the few great lyricists over the years, but also one who can function remarkably under pressure.

The Peter family proved wonderful and helpful friends in the following days, Mrs. Peter, little Esther, and Raoul, who generously lent me his sleeping bag for my `` Watch on the Rhine ''.

for if this can be proved we shall surely be the gainers -- I mean, if there is a use in poetry as well as a delight ''.

Because of these involvements in the matter at stake, Boniface lacked the impartiality that is supposed to be an essential qualification for the position of arbiter, and in retrospect that would seem to be sufficient reason why the English embassies to the Curia proved so fruitless.

In fact, the world heard nothing but sabre-rattling, the same exercises which proved futile for the predecessors of the current President.

introduction of the `` dialogue '' principle proved strikingly effective at the thirty-fourth annual meeting of the Catholic Association for International Peace in Washington the last weekend in October.

Without exception Hino's brothers turned to either one or both of their father's occupations, but Hino showed a talent for neither and instead spent most of his time on the beach where he repaired nets and proved immensely popular as a storyteller.

Devoted to the improvement in business climate and increase in industrial construction in Rhode Island, it has proved a valuable mailing piece for this Division.

Here, this happy, roving son of good fortune proved that he could accept the disciplines of a new social-economic order fighting for its very existence and ideals in a truculent world.

Besides being most convenient, the line `` soon proved a good investment for the owners ''.

And she felt amply rewarded for her suffering when the evidence of Lee's quack shenanigans, gathered by the tape recorder under her friend's clothing, proved adequate in court for convicting Franklin D. Lee.

Alas, so it almost proved for us, too.

In this tangle of conflicting claims, the patent-sharing scheme adopted by the A.L.A.M. at its founding proved to be the best device for avoiding or mitigating the burdens of incessant litigation.

More than 300 teenagers last Sunday proved there is and as many more are expected to prove it again for Jim Kern and his wife Lynn from 4 to 8 p.m. Sunday at First Presbyterian Church.

We are left helpless to cope with it because we do not dare speak of it as anything real for fear that to do so would imply a commitment to that which has already been discredited and proved false.

It is the belief of Old Babylonian scholars such as Carruccio that Old Babylonians " may have used the abacus for the operations of addition and subtraction ; however, this primitive device proved difficult to use for more complex calculations ".

The stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of the Earth ’ s axis.

The 1976 definition of the astronomical unit was incomplete, in particular because it does not specify the frame of reference in which time is to be measured, but proved practical for the calculation of ephemerides: a fuller definition that is consistent with general relativity was proposed, and " vigorous debate " ensued until in August 2012 the International Astronomical Union adopted the current definition of 1 astronomical unit = 149597870700 meters.

Nevertheless, the increasing strength of the Arians proved a formidable task for Ambrose.

The investment proved a great success and a source of profit for Woodruff and Carnegie.

The English longbow proved its worth for the first time in Continental warfare at the Battle of Crécy.

The Weil conjecture on Tamagawa numbers proved resistant for many years.