** 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.

* Every pair of congruence relations for an unknown integer x, of the form x ≡ k ( mod a ) and x ≡ l ( mod b ), has a solution, as stated by the Chinese remainder theorem ; in fact the solutions are described by a single congruence relation modulo ab.

In 1931 his incompleteness theorem showed that Hilbert's grand plan was impossible as stated.

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 ).

In cybernetics, the Good Regulator or Conant-Ashby theorem is stated " Every Good Regulator of a system must be a model of that system ".

The KAM theorem, as it was originally stated, could not be directly as a whole applied to the motions of the solar system.

The KAM theorem is usually stated in terms of trajectories in phase space of an integrable Hamiltonian system.

The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

Since there is a one-to-one correspondence between Borel regular measures in the interval and functions of bounded variation ( that assigns to each function of bounded variation the corresponding Lebesgue-Stieltjes measure, and the integral with respect to the Lebesgue-Stieltjes measure agrees with the Riemann-Stieltjes integral for continuous functions ), the above stated theorem generalizes the original statement of F. Riesz.

This classical statement, as well as the classical Divergence theorem and Green's Theorem, are simply special cases of the general formulation stated above.

Gödel's theorem, informally stated, asserts that any formal theory expressive enough for elementary arithmetical facts to be expressed and strong enough for them to be proved is either inconsistent ( both a statement and its denial can be derived from its axioms ) or incomplete, in the sense that there is a true statement about natural numbers that can't be derived in the formal theory.

On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics.

The equipartition theorem of classical mechanics, the basis of all classical thermodynamic theories, stated that an object's energy is partitioned equally among the object's vibrational modes.

* 700 BC – 600 BC: Baudhayana Sulbasutra, an orally transmitted Vedic Sanskrit text on altar construction, contains the earliest extant verbal statement of the Pythagorean theorem, which was likely known to ( but not stated by ) Old Babylonians ( 1800 BC to 1600 BC ).

Formally, the theorem can be stated as follows:

The theorem is named after Stefan Banach ( 1892 – 1945 ), and was first stated by him in 1922.

Sometimes this theorem is stated as follows: every non-zero single-variable polynomial with complex coefficients has exactly as many complex roots as its degree, if each root is counted up to its multiplicity.

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.

Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1712.

The theorem is named after Pierre de Fermat, who stated it in 1640.

Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy as the following: p divides a < sup > p − 1 </ sup > − 1 whenever p is prime and a is coprime to p.

Formally, the theorem is stated as follows: There exist unique integers q and r such that a = qd + r and 0 ≤ r < | d |, where | d | denotes the absolute value of d.

This is mostly of technical interest, since all true formal theories of arithmetic ( theories whose axioms are all true statements about natural numbers ) are ω-consistent, and thus Gödel's theorem as originally stated applies to them.

Then Cauchy's theorem can be stated as the integral of a function holomorphic in an open set taken around any cycle in the open set is zero.

The theorem is named after Andrey Nikolayevich Tychonoff, who proved it first in 1930 for powers of the closed unit interval and in 1935 stated the full theorem along with the remark that its proof was the same as for the special case.