** 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.

This was rigorously proved and extended by Vladimir Arnold ( in 1963 for analytic Hamiltonian systems ) and Jürgen Moser ( in 1962 for smooth twist maps ), and the general result is known as the KAM theorem.

A mathematician feels that a theorem is better understood when it can be extended to apply in a broader setting than previously known.

In 1830, Évariste Galois, studying the permutations of the roots of a polynomial, extended Abel-Ruffini theorem by showing that, given a polynomial equation, one may decide if it is solvable by radicals, and, if it is, solve it.

) The double exponential complexity of the theory makes it infeasible to use the theorem provers on complicated formulas, but this behavior occurs only in the presence of nested quantifiers: Oppen and Nelson ( 1980 ) describe an automatic theorem prover which uses the simplex algorithm on an extended Presburger arithmetic without nested quantifiers.

In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem.

Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students ( for example in the Banach – Schauder theorem ) and other mathematicians ( in particular Bouwer and Poincaré and Birkhoff ).

However, the sampling theorem can be extended in a straightforward way to functions of arbitrarily many variables.

The original formulation of Cox's theorem is in, which is extended with additional results and more discussion in.

From the extended form of Bayes ' theorem,

The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Ibn al-Haytham ( Alhazen ).

The truth of de Moivre's theorem can be established by mathematical induction for natural numbers, and extended to all integers from there.

It may be computed using the extended Euclidean algorithm or using Fermat's little theorem as k < sup > q − 2 </ sup > mod q.

The use of the Bayes theorem has been extended in science and in other fields.

Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's last theorem, and Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor extended his techniques to prove the full modularity theorem in 2001.

* The relationship between the general and extended form of Brahmagupta's formula is similar to how the law of cosines extends the Pythagorean theorem.

Lucky numbers share some properties with primes, such as asymptotic behaviour according to the prime number theorem ; also, a version of Goldbach's conjecture has been extended to them.

This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory ( TDDFT ), which can be used to describe excited states.

This theorem is extended as a definition of noncommutative projective geometry by Michael Artin and J. J. Zhang, who add also some general ring-theoretic conditions ( e. g. Artin-Schelter regularity ).

The axioms I1, I2, and I3 were at first suspected to be inconsistent ( in ZFC ) as it was thought possible that Kunen's inconsistency theorem that Reinhardt cardinals are inconsistent with the axiom of choice could be extended to them, but this has not yet happened and they are now usually believed to be consistent.

It is enough to prove that h can be extended to an entire function, in which case the result follows by Liouville's theorem.