** The Vitali theorem on the existence of non-measurable sets which states that there is a subset of the real numbers that is not Lebesgue measurable.

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.

has no zero in F. By contrast, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

For example, the prime number theorem states that the number of prime numbers less than or equal to N is asymptotically equal to N / ln N. Therefore the proportion of prime integers is roughly 1 / ln N, which tends to 0.

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.

* The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.

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.

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.

In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

The no-hair theorem states that, once it achieves a stable condition after formation, a black hole has only three independent physical properties: mass, charge, and angular momentum.

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.

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.

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.

Goldstone's theorem in quantum field theory states that in a system with broken continuous symmetry, there may exist excitations with arbitrarily low energy, called the Goldstone bosons.

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.

Moreover, the Banach fixed point theorem states that every contraction mapping on a nonempty complete metric space has a unique fixed point, and that for any x in M the iterated function sequence x, f ( x ), f ( f ( x )), f ( f ( f ( x ))), ... converges to the fixed point.

Fermat's little theorem states that all prime numbers have the above property.

This is a generalization of the Heine – Borel theorem, which states that any closed and bounded subspace S of R < sup > n </ sup > is compact and therefore complete.

The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.

The fundamental theorem of calculus states that antidifferentiation is the same as integration.

The Nyquist – Shannon sampling theorem states that a signal can be exactly reconstructed from its samples if the sampling frequency is greater than twice the highest frequency of the signal ; but requires an infinite number of samples.

If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:

The prime number theorem, conjectured on 31 May, gives a good understanding of how the prime numbers are distributed among the integers.

The theorem is stating two things: first, that 1200 can be represented as a product of primes, and second, no matter how this is done, there will always be four 2s, one 3, two 5s, and no other primes in the product.

For compact groups, the Peter – Weyl theorem explains how one may get harmonics by choosing one irreducible representation out of each equivalence class of representations.

) However, the fundamental theorem of arithmetic gives no insight into how to obtain an integer's prime factorization ; it only guarantees its existence.

The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.

With finite samples, approximation results measure how close a limiting distribution approaches the statistic's sample distribution: For example, with 10, 000 independent samples the normal distribution approximates ( to two digits of accuracy ) the distribution of the sample mean for many population distributions, by the Berry – Esseen theorem.

The structured program theorem does not address how to write and analyze a usefully structured program.

The Nyquist-Shannon sampling theorem provides an important guideline as to how much digital data is needed to accurately portray a given analog signal.

To see how the binomial theorem relates to the simple construction of Pascal's triangle, consider the problem of calculating the coefficients of the expansion of ( x + 1 )< sup > n + 1 </ sup > in terms of the corresponding coefficients of ( x + 1 )< sup > n </ sup > ( setting y

Bayes ' theorem was named after the Reverend Thomas Bayes ( 1702 – 61 ), who studied how to compute a distribution for the probability parameter of a binomial distribution ( in modern terminology ).

Taylor's theorem gives a precise bound on how good the approximation is.

The interest here is in how large n must be, in terms of the dimension m of M. The Whitney embedding theorem states that n = 2m is enough.

Shannon's theorem shows how to compute a channel capacity from a statistical description of a channel, and establishes that given a noisy channel with capacity C and information transmitted at a line rate R, then if

As an illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric: from this symmetry, Noether's theorem dictates that the angular momentum of the system be conserved, as a consequence of its laws of motion.

Gödel's incompleteness theorem places a severe limit on how weak a finitistic system can be while still proving the consistency of Peano arithmetic.

There are three articles centered on the Lisp programming language, where Hofstadter first details the language itself, and then shows how it relates to Gödel's incompleteness theorem.

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

For example, under the same conditions on f as in the classical theorem, given any natural number n ( no matter how large ), there exists ( that is, we can construct ) a real number c < sub > n </ sub > in the interval such that the absolute value of f ( c < sub > n </ sub >) is less than 1 / n.

The basic theorem states that, under a certain market price process ( the classical random walk ), in the absence of taxes, bankruptcy costs, agency costs, and asymmetric information, and in an efficient market, the value of a firm is unaffected by how that firm is financed.

Toponogov's theorem affords a characterization of sectional curvature in terms of how " fat " geodesic triangles appear when compared to their Euclidean counterparts.

* showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient.

In probability theory, the Girsanov theorem ( named after Igor Vladimirovich Girsanov ) describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure.

The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which describes the probability that an underlying instrument ( such as a share price or interest rate ) will take a particular value or values to the risk-neutral measure which is a very useful tool for pricing derivatives on the underlying.

But Tarski did not remember his proof, and it remains a mystery how he could do it without the compactness theorem.