Page "Axiom of choice" ¶ 50
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## Some Related Sentences

** and Tarski's
** Tarski's definition of truth

** and theorem
** Well-ordering theorem: Every set can be well-ordered.
** 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.
** Gödel's completeness theorem for first-order logic: every consistent set of first-order sentences has a completion.
** 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
** For every non-empty set S there is a binary operation defined on S that makes it a group.
** For a list of persons with the given name Benjamin see
** The Apprentice Armorer's Illustrated Handbook For Making Mail
** For example, consider the map ( which is the " realification " of the complex square function ) where U
** For research into past life and evolution
** For museums and education
** For some verbs in some languages, the difference between perfective and imperfective conveys an additional meaning difference ; in such cases, the two aspects are typically translated using separate verbs in English.
** For ohm – SI unit of electrical resistance ; formerly also used upside down () to represent mho, the old name for the inverse of an ohm ( now siemens with symbol S ) used for electrical conductance.
** For Australian cars same ignition maps resulting same horsepower rating as in other markets.
** For instance, when a magazine claims that An MIT professor predicts that the robotic era is coming in 2050, the use of big-name " MIT " ( a world-renowned American university for the advanced research in math, science, and technology ) establishes the " strong " credibility.
** Episode 31 " How To Die For Nothing " ( Writer )
** Toxic Sludge Is Good For You: Lies, Damn Lies and the Public Relations Industry ( 1995 )
** Korean War: For the second time, United Nations troops recapture Seoul during Operation Ripper.
** For the last time, New Year's Day is legally on March 25 in England and Wales.
** For example: One specific person may be identified by all of the following identifiers: Jane Smith ; Jane Elizabeth Meredith Smith ; Jane E. M. Smith ; Jane E. Smith ; Janie Smith ; Janie ; Little Janie ( as opposed to her mother or sister or cousin, Big Janie ); Aunt Jane ; Auntie Janie ; Mom ; Grandmom ; Nana ; Kelly's mother ; Billy's grandmother ; Ms. Smith ; Dr. Smith ; Jane E. Smith, PhD ; and Fuzzy ( her jocular nickname at work ).
** For Your Eyes Only ( 1981 )
** " For happy the lily " ( reprise ) ( Ensemble ) ( See Versions ).
** Associativity of addition: For any integers, a, b and c, ( a + b ) + c = a + ( b + c ).
** Existence of additive identity: For any integer a, a + 0
** Existence of additive inverse: For any integer a, there exists an integer denoted by − a such that a + (− a )
** Commutativity of addition: For any two integers a and b, a + b = b + a.
** Existence of multiplicative identity: For any integer a, a · 1 =
** What I Lived For by Joyce Carol Oates
** The Archers 1968-1986: Looking For Love ( 1999 ) ISBN 0-563-55125-9

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