Page "Axiom of choice" ¶ 104
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## Some Related Sentences

** and Gödel's
** What is Mathematics: Gödel's Theorem and Around by Karlis Podnieks.

** and completeness
** 3. 9 Knowledge of control objectives and techniques that ensure the completeness, accuracy, validity and authorization of transactions and data
** 5. 1 Evaluate the information security policies, standards and procedures for completeness and alignment with generally accepted practices.

** and theorem
** Well-ordering theorem: Every set can be well-ordered.
** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.
** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.
** Tychonoff's theorem stating that every product of compact topological spaces is compact.
** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.
** 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.
** Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem.
** The Nielsen – Schreier theorem, that every subgroup of a free group is free.
** The Hahn – Banach theorem in functional analysis, allowing the extension of linear functionals
** The theorem that every Hilbert space has an orthonormal basis.
** The Banach – Alaoglu theorem about compactness of sets of functionals.
** The Baire category theorem about complete metric spaces, and its consequences, such as the open mapping theorem and the closed graph theorem.
** The numbers and are not algebraic numbers ( see the Lindemann – Weierstrass theorem ); hence they are transcendental.
** Hilbert's basis theorem
** Bayes ' theorem
** More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ: U → R < sup > m </ sup >, where U is an open set in R < sup > n </ sup >, is almost everywhere differentiable.
** Lyapunov's central limit theorem
** Superposition theorem, in electronics
** " Kelvin's vorticity theorem for incompressible or barotropic flow ".
** Artin reciprocity law, a general theorem in number theory that provided a partial solution to Hilbert's ninth problem
** Various proofs of the four colour theorem.

** and for
** Atlas Autocode, a programming language developed for the Atlas Computer
** Altenberg bobsleigh, luge, and skeleton track, a venue for bobsleigh, luge and skeleton located in Altenberg, Saxony
** Altenberg, the German name for Vieille Montagne (" old mountain " in French ), the former zinc mine in Kelmis, Moresnet
**. am, Internet domain for Armenia
** Display aspect ratio-The aspect ratio for computer displays.
** Basil Fool for Christ ( Russian Orthodox Church )
** Raptio, large scale abduction of women, either for marriage or enslavement
** Lollia Paulina as she was a rival for Claudius ’ hand in marriage as proposed by the freedman Callistus.
** Sosibius Britannicus ’ tutor executed for plotting against Nero.
** World Day for Safety and Health at Work ( International )
** Ludwig Radermacher, Artium Scriptores, Vienna, 1951, pp. 200 – 202 ( rhetorical fragments only, adding Philodemus ' Rhetorica, which accounts for three of the nine fragments printed )
** ( Actual approximately − 11. 0 ) The diffuse reflector formula does for smaller phases.
** Minimum level required for POWER4 hardware and the last release that worked on the Micro Channel architecture
** The L stands for Linux affinity
** Support for mirroring with striping
** LVM ( Logical Volume Manager ) was incorporated into OSF / 1, and in 1995 for HP-UX, and the Linux LVM implementation is similar to the HP-UX LVM implementation.
** Theater for Social Change
** howitzers: capable of high angle fire, they are most often employed for indirect-fire.
** mortars: typically short-barreled, high-trajectory weapons designed primarily for an indirect-fire role.
** anti-tank artillery: weapons, usually mobile, designed for attacking tanks.
** anti-aircraft artillery: weapons, usually mobile, designed for attacking aircraft from the ground.
** Chromatic aberration, caused by differences in refractive index for different wavelengths of light
** The latest possible date for Easter Sunday, last in 1943.

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