This reduction has been accomplished by the general methods of linear algebra, i.e., by the primary decomposition theorem.

The latter theorem has been generalized by Yamabe and Yujobo, and Cairns to show that in Af there are families of such cubes.

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

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.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

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.

* Automated theorem proving, the proving of mathematical theorems by a computer program

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.

He stopped by his local library where he found a book about the theorem .< ref >

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

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

Automated theorem proving ( also known as ATP or automated deduction ) is the proving of mathematical theorems by a computer program.

In 1920, Thoralf Skolem simplified a previous result by Leopold Löwenheim, leading to the Löwenheim – Skolem theorem and, in 1930, to the notion of a Herbrand universe and a Herbrand interpretation that allowed ( un ) satisfiability of first-order formulas ( and hence the validity of a theorem ) to be reduced to ( potentially infinitely many ) propositional satisfiability problems.

* 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.

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.

A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate ( i. e. ) the subalgebra generated by those elements is associative.

A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative.

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

Second-order logic, for example, does not have a completeness theorem for its standard semantics ( but does have the completeness property for Henkin semantics ), and the same is true of all higher-order logics.

Leon Henkin ( 1950 ) defined these semantics and proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics.

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.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

The most important among them are Zorn's lemma and the well-ordering theorem.

The most basic example of the binomial theorem is the formula for the square of x + y:

It was introduced in 1971 by Stephen Cook in his seminal paper " The complexity of theorem proving procedures " and is considered by many to be the most important open problem in the field.

* Groups of sectional 2-rank at most 4, classified by the Gorenstein – Harada theorem.

For example, in most systems of logic ( but not in intuitionistic logic ) Peirce's law ((( P → Q )→ P )→ P ) is a theorem.

The intuitive statement of the four color theorem, i. e. ' that given any separation of a plane into contiguous regions, called a map, the regions can be colored using at most four colors so that no two adjacent regions have the same color ', needs to be interpreted appropriately to be correct.

In graph-theoretic terminology, the four-color theorem states that the vertices of every planar graph can be colored with at most four colors so that no two adjacent vertices receive the same color, or for short, " every planar graph is four-colorable " (; ).

His concept " Dyson's transform " led to one of the most important lemmas of Olivier Ramaré's theorem that every even integer can be written as a sum of no more than six primes.

* The transforms are linear operators and, with proper normalization, are unitary as well ( a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality ).

The simplest version of this theorem that suffices in practice for most needs, and has connections with the Löwenheim-Skolem theorem, says:

This is the most basic form of the completeness theorem.

The most general formulation of the theorem needs some preparation.

All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry.

The most fundamental results of this theory are Shannon's source coding theorem, which establishes that, on average, the number of bits needed to represent the result of an uncertain event is given by its entropy ; and Shannon's noisy-channel coding theorem, which states that reliable communication is possible over noisy channels provided that the rate of communication is below a certain threshold, called the channel capacity.

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 use of any parametric model is viewed skeptically by most experts in sampling human populations: " most sampling statisticians, when they deal with confidence intervals at all, limit themselves to statements about based on very large samples, where the central limit theorem ensures that these will have distributions that are nearly normal.

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).