** Well-ordering theorem: Every set can be well-ordered.

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 notion of the Kolmogorov complexity can be used to state and prove impossibility results akin to Gödel's incompleteness theorem and Turing's halting problem.

which can be viewed as a version of the Pythagorean theorem.

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.

* 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.

Antiderivatives are important because they can be used to compute definite integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

The fundamental theorem of arithmetic states that any positive integer n can be represented uniquely as a product of powers of primes: where p < sub > 1 </ sub > < p < sub > 2 </ sub > < ... < p < sub > k </ sub > are primes and the a < sub > j </ sub > are positive integers.

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 Ham sandwich theorem: For any compact sets A < sub > 1 </ sub >,..., A < sub > n </ sub > in ℝ < sup > n </ sup > we can always find a hyperplane dividing each of them into two subsets of equal measure.

* 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.

The binomial theorem can be applied to the powers of any binomial.

For a binomial involving subtraction, the theorem can be applied as long as the opposite of the second term is used.

For positive values of a and b, the binomial theorem with n = 2 is the geometrically evident fact that a square of side can be cut into a square of side a, a square of side b, and two rectangles with sides a and b. With n = 3, the theorem states that a cube of side can be cut into a cube of side a, a cube of side b, three a × a × b rectangular boxes, and three a × b × b rectangular boxes.

A class of algorithms called SAT solvers can efficiently solve a large enough subset of SAT instances to be useful in various practical areas such as circuit design and automatic theorem proving, by solving SAT instances made by transforming problems that arise in those areas.

Schaefer's dichotomy theorem states that, for any restriction to Boolean operators that can be used to form these subformulae, the corresponding satisfiability problem is in P or NP-complete.

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.

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.

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.

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.

Practical applications are made impossible due to the no-cloning theorem, and the fact that quantum field theories preserve causality, so that quantum correlations cannot be used to transfer information.

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.

The " heuristic " approach of the Logic Theory Machine tried to emulate human mathematicians, and could not guarantee that a proof could be found for every valid theorem even in principle.

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

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

Compactness in this more general situation plays an extremely important role in mathematical analysis, because many classical and important theorems of 19th century analysis, such as the extreme value theorem, are easily generalized to this situation.

It is also an example of a more generalized version of the central limit theorem that is characteristic of all stable distributions, of which the Cauchy distribution is a special case.

The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalized quaternion, which are handled with the Brauer – Suzuki theorem: in particular there are no simple groups of 2-rank 1.

The original form of the theorem, contained in a third-century AD book The Mathematical Classic of Sun Zi ( 孫子算經 ) by Chinese mathematician Sun Tzu and later generalized with a complete solution called Da yan shu ( 大衍术 ) in a 1247 book by Qin Jiushao, the Shushu Jiuzhang ( 數書九章 Mathematical Treatise in Nine Sections ) is a statement about simultaneous congruences ( see modular arithmetic ).

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.

in light of the Jordan curve theorem and the generalized Stokes ' theorem, F < sub > γ </ sub >( z ) is independent of the particular choice of path γ, and thus F ( z ) is a well-defined function on U having F ( z < sub > 0 </ sub >)

For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals.

Marshall H. Stone considerably generalized the theorem and simplified the proof.

In differential geometry, Stokes ' theorem ( also called the generalized Stokes ' theorem ) is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus.

This approach was generalized by Karl Weierstrass to the Lindemann – Weierstrass theorem.

The virial theorem has been generalized in various ways, most notably to a tensor form.

The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux and Eugene Parker.

* Higher homotopy, generalized van Kampen's theorem

* Dylan G. L. Allegretti, Simplicial Sets and van Kampen's Theorem ( Discusses generalized versions of van Kampen's theorem applied to topological spaces and simplicial sets ).

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.

The lemma is generalized by ( and usually used in the proof of ) the Tietze extension theorem.

The theorem was originally proved by John Nash with the condition n ≥ m + 2 instead of n ≥ m + 1 and generalized by Nicolaas Kuiper, by a relatively easy trick.

A generalized form of the diagonal argument was used by Cantor to prove Cantor's theorem: for every set S the power set of S, i. e., the set of all subsets of S ( here written as P ( S )), is larger than S itself.

Fermat's little theorem is generalized by Euler's theorem: for any modulus n and any integer a coprime to n, we have

This can be further generalized to Carmichael's theorem.