Gödel's completeness theorem establishes the completeness of a certain commonly used type of deductive system.

Gödel's completeness theorem is a fundamental theorem in mathematical logic that establishes a correspondence between semantic truth and syntactic provability in first-order logic.

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.

The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.

Lagrange's four-square theorem of 1770 states that every natural number is the sum of at most four squares ; since three squares are not enough, this theorem establishes g ( 2 ) = 4.

Moreover, if the relation '≥' in the above expression is actually an equality, then and hence ; the definition of z then establishes a relation of linear dependence between u and v. This establishes the theorem.

By induction, Hilbert's basis theorem establishes that, the ring of all polynomials in n variables with coefficients in, is a Noetherian ring.

* 1854 – Clausius establishes the importance of dQ / T ( Clausius's theorem ), but does not yet name the quantity.

The theorem establishes Shannon's channel capacity for such a communication link, a bound on the maximum amount of error-free digital data ( that is, information ) that can be transmitted with a specified bandwidth in the presence of the noise interference, assuming that the signal power is bounded, and that the Gaussian noise process is characterized by a known power or power spectral density.

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

Hilbert's Nullstellensatz ( German for " theorem of zeros ," or more literally, " zero-locus-theorem " – see Satz ) is a theorem which establishes a fundamental relationship between geometry and algebra.

One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it.

The recursion theorem establishes the existence of such a fixed point, assuming that F is computable, and is sometimes called ( Kleene's ) fixed point theorem for this reason.

Gödel's second incompleteness theorem establishes that logical systems of arithmetic can never contain a valid proof of their own consistency.

The Completeness theorem establishes an equivalence in first-order logic, between the formal provability of a formula, and its truth in all possible models.

Post's theorem establishes a close connection between the arithmetical hierarchy of sets of natural numbers and the Turing degrees.

* 1977: D. Sullivan establishes his theorem on the existence and uniqueness of Lipschitz and quasiconformal structures on topological manifolds of dimension different from 4.

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

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

* 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:

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 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.

This led in 1828 to an important theorem, the Theorema Egregium ( remarkable theorem ), establishing an important property of the notion of curvature.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

The study of mathematical proof is particularly important in logic, and has applications to automated theorem proving and formal verification of software.

Notably he was the first to use discharging for proving the theorem, which turned out to be important in the unavoidability portion of the subsequent Appel-Haken proof.

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.

Cantor recovered soon thereafter, and subsequently made further important contributions, including his famous diagonal argument and theorem.

An important consequence of the completeness theorem is that it is possible to recursively enumerate the semantic consequences of any effective first-order theory, by enumerating all the possible formal deductions from the axioms of the theory, and use this to produce an enumeration of their conclusions.

The theorem has several important consequences, some of which are also sometimes called " Hahn – Banach theorem ":

An important consequence of the KAM theorem is that for a large set of initial conditions the motion remains perpetually quasiperiodic.

One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem.

Some of the more important results in the study of monoids are the Krohn-Rhodes theorem and the star height problem.

An important step in the evolution of classical model theory occurred with the birth of stability theory ( through Morley's theorem on uncountably categorical theories and Shelah's classification program ), which developed a calculus of independence and rank based on syntactical conditions satisfied by theories.

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.

This bold attempt is entirely factitious and verbal, and it is only his employment of various terms not generally used in such a connection ( axiom, theorem, corollary, etc.

It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization ( which leads to the fundamental theorem of arithmetic on uniqueness of prime factorizations ), and the Euclidean algorithm for finding the greatest common divisor of two numbers.

This reveals the intimate connection between the Hahn – Banach theorem and convexity.

The theorem demonstrates a connection between integration and differentiation.

Because of the connection between the Riemann zeta function and π ( x ), the Riemann hypothesis has considerable importance in number theory: if established, it would yield a far better estimate of the error involved in the prime number theorem than is available today.

This use is met most often in connection with Thevenin's theorem in circuit theory.

Remembrance of the story of the bull's hide and the foundation of Carthage is preserved in mathematics in connection with the Isoperimetric problem which is sometimes called Dido's Problem ( and similarly the Isoperimetric theorem is sometimes called Dido's Theorem ).

The local existence and uniqueness theorem for geodesics states that geodesics on a smooth manifold with an affine connection exist, and are unique.

The Curry – Howard isomorphism implies a connection between logic and programming: every proof of a theorem of intuitionistic logic corresponds to a reduction of a typed lambda term, and conversely.

The CPT theorem appeared for the first time, implicitly, in the work of Julian Schwinger in 1951 to prove the connection between spin and statistics.

In a deeper way, this theorem relates the topology of the domain of integration to the structure of the differential forms themselves ; the precise connection is known as De Rham's theorem.

Specific integers known to be far larger than Graham's number have since appeared in many serious mathematical proofs ( e. g., in connection with Friedman's various finite forms of Kruskal's theorem ).

The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties.

The holonomy of the connection, around a closed loop is, as a consequence of Stokes ' theorem, determined by the magnetic flux through a surface bounded by the loop.

* Green-Kubo relations-there is a deep connection between the fluctuation theorem and the Green-Kubo relations for linear transport coefficients-like shear viscosity or thermal conductivity

If the homology groups are torsion-free, the Betti numbers are independent of F. The connection of p-torsion and the Betti number for characteristic p, for p a prime number, is given in detail by the universal coefficient theorem ( based on Tor functors, but in a simple case ).

The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality ( when those apply ), and the universal coefficient theorem of homology theory.

A proof with a large number of cases leaves an impression that the theorem is only true by coincidence, and not because of some underlying principle or connection.

* In mathematics, the modularity theorem ( formerly the Taniyama – Shimura conjecture ) establishes a connection between elliptic curves and modular forms.

Lie listed his results as 3 direct and 3 converse theorems, the infinitesimal variant of Cartan's theorem was essentially his 3rd converse theorem, hence Serre has called it in an influential book, the " third Lie theorem ", the name which is historically somewhat misleading, but more often used in the recent decade in the connection to many generalizations.