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.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

Experimentally, however, the total kinetic energy is found to be much greater: in particular, assuming the gravitational mass is due to only the visible matter of the galaxy, stars far from the center of galaxies have much higher velocities than predicted by the virial theorem.

The five color theorem, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century ; however, proving that four colors suffice turned out to be significantly harder.

Because the four color theorem is true, this is always possible ; however, because the person drawing the map is focused on the one large region, he fails to notice that the remaining regions can in fact be colored with three colors.

If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root ; this is the fundamental theorem of algebra.

In the letter, Germain claimed to have proved the theorem for n = p – 1, where p is a prime number of the form p = 8k + 7 ; however, her proof contained a weak assumption.

) In second order logic, however, the well-ordering theorem is strictly stronger than the axiom of choice: from the well-ordering theorem one may deduce the axiom of choice, but from the axiom of choice one cannot deduce the well-ordering 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 ).

As now formulated, the theorem of Wiener does not have any obvious association with Tauberian theorems, which deal with infinite series ; the translation from results formulated for integrals, or using the language of functional analysis and Banach algebras, is however a relatively routine process.

Condorcet's jury theorem proved that if the average member votes better than a roll of dice, then adding more members increases the number of majorities that can come to a correct vote ( however correctness is defined ).

None of these computations can show however why the Cayley – Hamilton theorem should be valid for matrices of all possible sizes n, so a uniform proof for all n is needed.

In an 1853 paper on the theory of equations and Galois theory he formulated the Kronecker – Weber theorem, however without offering a definitive proof ( the theorem was proved completely much later by David Hilbert ).

Professor Nambu had proposed a theory known as spontaneous symmetry breaking based on what was known to happen in superconductivity in condensed matter ; however, the theory predicted massless particles ( the Goldstone's theorem ), a clearly incorrect prediction.

Subsequent authors have shown that this version of the theorem is not generally true, however.

For a composite r ( that is, like the Rabin algorithm's ) there is no efficient method known for the finding of m. If, however is prime ( as are p and q in the Rabin algorithm ), the Chinese remainder theorem can be applied to solve for m.

The Handbook of Applied Cryptography by Menezes, Oorschot and Vanstone considers this equivalence probable, however, as long as the finding of the roots remains a two-part process ( 1. roots and and 2. application of the Chinese remainder theorem ).

By the Helmholtz decomposition theorem however, all vector fields can be describable in terms of a scalar potential and corresponding vector potential.

The Peano existence theorem however proves that even for ƒ merely continuous, solutions are guaranteed to exist locally in time ; the problem is that there is no guarantee of uniqueness.

It is, however, a theorem that if R is a local ring or if R is a graded ring and the r < sub > i </ sub > are all homogeneous, then a sequence is an R-sequence only if every permutation of it is an R-sequence.

The axiom of choice was formulated in 1904 by Ernst Zermelo in order to formalize his proof of the well-ordering theorem.

Transmission, Gregory Chaitin also presents this theorem in J. ACM – Chaitin's paper was submitted October 1966 and revised in December 1968, and cites both Solomonoff's and Kolmogorov's papers.

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.

His ' matrix divisor ' ( vector bundle avant la lettre ) Riemann – Roch theorem from 1938 was a very early anticipation of later ideas such as moduli spaces of bundles.

His first ( pre-IHÉS ) breakthrough in algebraic geometry was the Grothendieck – Hirzebruch – Riemann – Roch theorem, a far-reaching generalisation of the Hirzebruch – Riemann – Roch theorem proved algebraically ; in this context he also introduced K-theory.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The Grothendieck – Riemann – Roch theorem was announced by Grothendieck at the initial Mathematische Arbeitstagung in Bonn, in 1957.

Argonne National Laboratory was a leader in automated theorem proving from the 1960s to the 2000s

Following Desargues ' thinking, the sixteen-year-old Pascal produced, as a means of proof, a short treatise on what was called the " Mystic Hexagram ", Essai pour les coniques (" Essay on Conics ") and sent it — his first serious work of mathematics — to Père Mersenne in Paris ; it is known still today as Pascal's theorem.

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.

SAT was the first known NP-complete problem, as proved by Stephen Cook in 1971 ( see Cook's theorem for the proof ).

It was Pierre-Simon Laplace ( 1749 – 1827 ) who introduced a general version of the theorem and used it to approach problems in celestial mechanics, medical statistics, reliability, and jurisprudence.

In mathematics, a Gödel code was the basis for the proof of Gödel's incompleteness theorem.

An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous: here continuity is a local property of the function, and uniform continuity the corresponding global property.

The full significance of Bolzano's theorem, and its method of proof, would not emerge until almost 50 years later when it was rediscovered by Karl Weierstrass.

The culmination of their investigations, the Arzelà – Ascoli theorem, was a generalization of the Bolzano – Weierstrass theorem to families of continuous functions, the precise conclusion of which was that it was possible to extract a uniformly convergent sequence of functions from a suitable family of functions.

The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by NP-intermediate | Ladner's theorem.

The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

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 concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray Solomonoff, who published it in 1960, describing it in " A Preliminary Report on a General Theory of Inductive Inference " as part of his invention of algorithmic probability.

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.

* William McCune Argonne National Laboratory, author of Otter, the first high-performance theorem prover.

Developed original resolution and unification based first order theorem proving, co-editor of the " Handbook of Automated Reasoning ", recipient of the Herbrand Award 1996

According to (), the first historical mention of the statement of this theorem appears in ().

This position is a more refined version of the theorem first discovered by David Hume.

The first proof relies on a theorem about products of limits to show that the derivative exists.

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.

* 1799: Doctoral dissertation on the Fundamental theorem of algebra, with the title: Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse (" New proof of the theorem that every integral algebraic function of one variable can be resolved into real factors ( i. e., polynomials ) of the first or second degree ")

The first case was done by the Gorenstein – Walter theorem which showed that the only simple groups are isomorphic to L < sub > 2 </ sub >( q ) for q odd or A < sub > 7 </ sub >, the second and third cases were done by the Alperin – Brauer – Gorenstein theorem which implies that the only simple groups are isomorphic to L < sub > 3 </ sub >( q ) or U < sub > 3 </ sub >( q ) for q odd or M < sub > 11 </ sub >, and the last case was done by Lyons who showed that U < sub > 3 </ sub >( 4 ) is the only simple possibility.

The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Gödel — in a roundtable discussion during the Conference on Epistemology held jointly with the Society meetings — tentatively announced the first expression of his incompleteness theorem.

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem.