Commit aa0e5857 authored by ioa's avatar ioa
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the database of math-snippets

parent e199b214
((0 "<strong> 0.</strong> <img src=\"math-snippets/snip-0.png\">. ") (394 "<strong> 394.</strong> <img src=\"math-snippets/snip-8.png\">: Every well ordered set of non-empty linearly orderable sets has a choice function. ") (250 "<strong> 250.</strong> <img src=\"math-snippets/snip-5.png\">: For every natural number <img src=\"math-snippets/snip-6.png\">, every well ordered family of <img src=\"math-snippets/snip-7.png\"> element sets has a choice function. ") (414 "<strong> 414.</strong> Every <img src=\"math-snippets/snip-44.png\">-frame is a <img src=\"math-snippets/snip-45.png\">-frame. ") (273 "<strong> 273.</strong> There is a subset of <img src=\"math-snippets/snip-9.png\"> which is not Borel.") (396 "<strong> 396.</strong> <img src=\"math-snippets/snip-169.png\">: For each linearly ordered family of non-empty well orderable sets <img src=\"math-snippets/snip-13.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\"> <img src=\"math-snippets/snip-16.png\"> is a non-empty, finite subset of <img src=\"math-snippets/snip-17.png\">. ") (104 "<strong> 104.</strong> There is a regular uncountable aleph.") (219 "<strong> 219.</strong> <img src=\"math-snippets/snip-11.png\">, relatively prime to <img src=\"math-snippets/snip-7.png\">): For all non-zero <img src=\"math-snippets/snip-12.png\">, if <img src=\"math-snippets/snip-13.png\"> is a set of non-empty well orderable sets, then there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a non-empty, finite subset of <img src=\"math-snippets/snip-17.png\">, and <img src=\"math-snippets/snip-18.png\"> is relatively prime to <img src=\"math-snippets/snip-7.png\">. ") (183 "<strong> 183<img src=\"math-snippets/snip-19.png\">.</strong> There are no <img src=\"math-snippets/snip-20.png\"> minimal sets. That is, there are no sets <img src=\"math-snippets/snip-13.png\"> such that <font class=\"enum\">(1)</font> <img src=\"math-snippets/snip-21.png\"> is incomparable with <img src=\"math-snippets/snip-20.png\"> \\itemitem{(2)} <img src=\"math-snippets/snip-22.png\"> for every <img src=\"math-snippets/snip-23.png\"> and \\itemitem{(3)} <img src=\"math-snippets/snip-24.png\"> or <img src=\"math-snippets/snip-25.png\">. ") (121 "<strong> 121.</strong> <img src=\"math-snippets/snip-26.png\">: Every linearly ordered set of non-empty finite sets has a choice function. ") (215 "<strong> 215.</strong> If <img src=\"math-snippets/snip-27.png\"> can be linearly ordered implies <img src=\"math-snippets/snip-28.png\"> is finite), then <img src=\"math-snippets/snip-13.png\"> is finite.") (241 "<strong> 241.</strong> Every algebraic closure of <img src=\"math-snippets/snip-29.png\"> has a real closed subfield. ") (221 "<strong> 221.</strong> For all infinite <img src=\"math-snippets/snip-13.png\">, there is a non-principal measure on <img src=\"math-snippets/snip-30.png\">. ") (226 "<strong> 226.</strong> Let <img src=\"math-snippets/snip-31.png\"> be a commutative ring with identity, <img src=\"math-snippets/snip-32.png\"> a proper subring containing 1 and <img src=\"math-snippets/snip-33.png\"> a prime ideal in <img src=\"math-snippets/snip-32.png\">. Then there is a subring <img src=\"math-snippets/snip-3.png\"> of <img src=\"math-snippets/snip-31.png\"> and a prime ideal <img src=\"math-snippets/snip-34.png\"> in <img src=\"math-snippets/snip-3.png\"> such that (a) <img src=\"math-snippets/snip-35.png\"> (b) <img src=\"math-snippets/snip-36.png\"> (c) <img src=\"math-snippets/snip-37.png\"> is multiplicatively closed and (d) if <img src=\"math-snippets/snip-38.png\">, then <img src=\"math-snippets/snip-39.png\"> is multiplicatively closed. ") (122 "<strong> 122.</strong> <img src=\"math-snippets/snip-40.png\">: Every well ordered set of non-empty finite sets has a choice function. ") (377 "<strong> 377.</strong> Restricted Ordering Principle: For every infinite set <img src=\"math-snippets/snip-13.png\"> there is an infinite subset <img src=\"math-snippets/snip-65.png\"> of <img src=\"math-snippets/snip-13.png\"> such that <img src=\"math-snippets/snip-65.png\"> can be linearly ordered. De la Cruz/Di") (376 "<strong> 376.</strong> Restricted Kinna Wagner Principle: For every infinite set <img src=\"math-snippets/snip-13.png\"> there is an infinite subset <img src=\"math-snippets/snip-65.png\"> of <img src=\"math-snippets/snip-13.png\"> and a function <img src=\"math-snippets/snip-14.png\"> such that for every <img src=\"math-snippets/snip-66.png\">, if <img src=\"math-snippets/snip-67.png\"> then <img src=\"math-snippets/snip-68.png\"> is a non-empty proper subset of <img src=\"math-snippets/snip-69.png\">. De la Cruz/Di") (300 "<strong> 300.</strong> Any continuous surjection between extremally disconnected compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain. ") (110 "<strong> 110.</strong> Every vector space over <img src=\"math-snippets/snip-29.png\"> has a basis.") (167 "<strong> 167.</strong> <img src=\"math-snippets/snip-49.png\">, Partial Kinna-Wagner Principle: For every denumerable family <img src=\"math-snippets/snip-50.png\"> such that for all <img src=\"math-snippets/snip-51.png\">, <img src=\"math-snippets/snip-52.png\">, there is an infinite subset <img src=\"math-snippets/snip-53.png\"> and a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-54.png\">, <img src=\"math-snippets/snip-55.png\">.") (260 "<strong> 260.</strong> <img src=\"math-snippets/snip-56.png\">: If <img src=\"math-snippets/snip-47.png\"> is a transitive and connected relation in which every partially ordered subset has an upper bound, then <img src=\"math-snippets/snip-47.png\"> has a maximal element.") (308 "<strong> 308<img src=\"math-snippets/snip-57.png\">.</strong> If <img src=\"math-snippets/snip-34.png\"> is a prime and if <img src=\"math-snippets/snip-58.png\"> is a set of finite groups, then the weak direct product <img src=\"math-snippets/snip-59.png\"> has a maximal <img src=\"math-snippets/snip-34.png\">-subgroup. ") (380 "<strong> 380.</strong> <img src=\"math-snippets/snip-64.png\">: For every infinite family of non-empty well orderable sets, there is an infinite subfamily <img src=\"math-snippets/snip-65.png\"> of <img src=\"math-snippets/snip-13.png\"> which has a choice function. De la Cruz/Di") (330 "<strong> 330.</strong> <img src=\"math-snippets/snip-84.png\">: For every well ordered set <img src=\"math-snippets/snip-13.png\"> of well orderable sets such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-85.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for every <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\">. ") (255 "<strong> 255.</strong> <img src=\"math-snippets/snip-62.png\">: Every directed relation <img src=\"math-snippets/snip-63.png\"> in which every ramified subset <img src=\"math-snippets/snip-3.png\"> has an upper bound, has a maximal element.") (401 "<strong> 401.</strong> <img src=\"math-snippets/snip-87.png\">, The Kinna-Wagner Selection Principle for a linearly ordered set of finite sets: For every linearly ordered set of finite sets <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (382 "<strong> 382.</strong> DUMN: The disjoint union of metrizable spaces is normal.") (123 "<strong> 123.</strong> <img src=\"math-snippets/snip-70.png\">: Uniform weak ultrafilter principle: For each family <img src=\"math-snippets/snip-50.png\"> of infinite sets <img src=\"math-snippets/snip-71.png\"> such that <img src=\"math-snippets/snip-72.png\">, <img src=\"math-snippets/snip-16.png\"> is a non-principal ultrafilter on <img src=\"math-snippets/snip-17.png\">.") (225 "<strong> 225.</strong> Every proper filter on <img src=\"math-snippets/snip-73.png\"> can be extended to an ultrafilter. ") (342 "<strong> 342<img src=\"math-snippets/snip-41.png\">.</strong> (For <img src=\"math-snippets/snip-42.png\">, <img src=\"math-snippets/snip-6.png\">.) <img src=\"math-snippets/snip-43.png\">: Every infinite family of <img src=\"math-snippets/snip-7.png\">-element sets has an infinite subfamily with a choice function. ") (47 "<strong> 47<img src=\"math-snippets/snip-41.png\">.</strong> If <img src=\"math-snippets/snip-74.png\">, <img src=\"math-snippets/snip-75.png\">: Every well ordered collection of <img src=\"math-snippets/snip-7.png\">-element sets has a choice function.") (428 "<strong> 428.</strong> <img src=\"math-snippets/snip-76.png\"> B<img src=\"math-snippets/snip-77.png\">: There is a field <img src=\"math-snippets/snip-50.png\"> such that every vector space over <img src=\"math-snippets/snip-50.png\"> has a basis.") (338 "<strong> 338.</strong> <img src=\"math-snippets/snip-86.png\">: The union of a denumerable number of denumerable sets is well orderable. ") (336 "<strong> 336<img src=\"math-snippets/snip-41.png\">.</strong> (For <img src=\"math-snippets/snip-42.png\">, <img src=\"math-snippets/snip-6.png\">.) For every infinite set <img src=\"math-snippets/snip-13.png\">, there is an infinite <img src=\"math-snippets/snip-79.png\"> such that the set of all <img src=\"math-snippets/snip-7.png\">-element subsets of <img src=\"math-snippets/snip-65.png\"> has a choice function. ") (265 "<strong> 265.</strong> <img src=\"math-snippets/snip-46.png\">: Every relation <img src=\"math-snippets/snip-47.png\"> contains a <img src=\"math-snippets/snip-48.png\">-maximal transitive subset. ") (398 "<strong> 398.</strong> <img src=\"math-snippets/snip-179.png\">, The Kinna-Wagner Selection Principle for a linearly ordered family of sets: For every linearly ordered set <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (76 "<strong> 76.</strong> <img src=\"math-snippets/snip-82.png\"> (<img src=\"math-snippets/snip-73.png\">-MC): For every family <img src=\"math-snippets/snip-13.png\"> of pairwise disjoint non-empty sets, there is a function <img src=\"math-snippets/snip-14.png\"> such that for each <img src=\"math-snippets/snip-15.png\">, f(x) is a non-empty countable subset of <img src=\"math-snippets/snip-17.png\">.") (306 "<strong> 306.</strong> The set of <font class=\"icopy\">Vitali equivalence classes</font> is linearly orderable. (<i> Vitali equivalence classes</i> are equivalence classes of the real numbers under the relation <img src=\"math-snippets/snip-83.png\">.).") (32 "<strong> 32.</strong> <img src=\"math-snippets/snip-78.png\">: Every denumerable set of non-empty countable sets has a choice function. ") (115 "<strong> 115.</strong> The product of weakly Loeb <img src=\"math-snippets/snip-61.png\"> spaces is weakly Loeb. ") (120 "<strong> 120<img src=\"math-snippets/snip-92.png\">.</strong> If <img src=\"math-snippets/snip-93.png\">, <img src=\"math-snippets/snip-94.png\">: Every linearly ordered set of non-empty sets each of whose cardinality is in <img src=\"math-snippets/snip-95.png\"> has a choice function. ") (117 "<strong> 117.</strong> If <img src=\"math-snippets/snip-81.png\"> is a measurable cardinal, then <img src=\"math-snippets/snip-81.png\"> is the <img src=\"math-snippets/snip-81.png\">th inaccessible cardinal.") (95 "<strong> 95<img src=\"math-snippets/snip-96.png\">.</strong> Existence of Complementary Subspaces over a Field <img src=\"math-snippets/snip-50.png\">: If <img src=\"math-snippets/snip-50.png\"> is a field, then every vector space <img src=\"math-snippets/snip-97.png\"> over <img src=\"math-snippets/snip-50.png\"> has the property that if <img src=\"math-snippets/snip-98.png\"> is a subspace of <img src=\"math-snippets/snip-97.png\">, then there is a subspace <img src=\"math-snippets/snip-99.png\"> such that <img src=\"math-snippets/snip-100.png\"> and <img src=\"math-snippets/snip-101.png\"> generates <img src=\"math-snippets/snip-97.png\">.") (307 "<strong> 307.</strong> If <img src=\"math-snippets/snip-111.png\"> is the cardinality of the set of Vitali equivalence classes, then <img src=\"math-snippets/snip-112.png\">, where <img src=\"math-snippets/snip-1.png\"> is Hartogs aleph function and the <i> Vitali equivalence classes</i> are equivalence classes of the real numbers under the relation <img src=\"math-snippets/snip-113.png\">.") (339 "<strong> 339.</strong> Martin's Axiom <img src=\"math-snippets/snip-106.png\">: Whenever <img src=\"math-snippets/snip-107.png\"> is a non-empty, ccc quasi-order (ccc means every anti-chain is countable) and <img src=\"math-snippets/snip-108.png\"> is a family of <img src=\"math-snippets/snip-109.png\"> dense subsets of <img src=\"math-snippets/snip-110.png\">, then there is a <img src=\"math-snippets/snip-108.png\"> generic filter <img src=\"math-snippets/snip-2.png\"> in <img src=\"math-snippets/snip-110.png\">.") (124 "<strong> 124.</strong> Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and a scalar operator. (A set is amorphous if it is not the union of two disjoint infinite sets.) ") (78 "<strong> 78.</strong> Urysohn's Lemma: If <img src=\"math-snippets/snip-3.png\"> and <img src=\"math-snippets/snip-32.png\"> are disjoint closed sets in a normal space <img src=\"math-snippets/snip-104.png\">, then there is a continuous <img src=\"math-snippets/snip-105.png\"> which is 1 everywhere in <img src=\"math-snippets/snip-3.png\"> and 0 everywhere in <img src=\"math-snippets/snip-32.png\">.") (312 "<strong> 312.</strong> A subgroup of an amenable group is amenable. (<img src=\"math-snippets/snip-2.png\"> is <i> amenable</i> if there is a finitely additive measure <img src=\"math-snippets/snip-115.png\"> on <img src=\"math-snippets/snip-116.png\"> such that <img src=\"math-snippets/snip-117.png\"> and <img src=\"math-snippets/snip-118.png\">, <img src=\"math-snippets/snip-119.png\">.) ") (106 "<strong> 106.</strong> Baire Category Theorem for Compact Hausdorff Spaces: Every compact Hausdorff space is Baire.") (323 "<strong> 323.</strong> <img src=\"math-snippets/snip-136.png\">, The Kinna-Wagner Selection Principle for a family of well orderable sets: For every set <img src=\"math-snippets/snip-88.png\"> of well orderable sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-137.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (267 "<strong> 267.</strong> There is no infinite, free complete Boolean algebra. ") (62 "<strong> 62.</strong> <img src=\"math-snippets/snip-114.png\">: Every set of non-empty finite sets has a choice function. ") (196 "<strong> 196<img src=\"math-snippets/snip-19.png\">.</strong> <img src=\"math-snippets/snip-20.png\"> and <img src=\"math-snippets/snip-80.png\"> are not both measurable.") (284 "<strong> 284.</strong> A system of linear equations over a field <img src=\"math-snippets/snip-50.png\"> has a solution in <img src=\"math-snippets/snip-50.png\"> if and only if every finite sub-system has a solution in <img src=\"math-snippets/snip-50.png\">.") (361 "<strong> 361.</strong> In <img src=\"math-snippets/snip-142.png\">, the union of a denumerable number of analytic sets is analytic. ") (264 "<strong> 264.</strong> <img src=\"math-snippets/snip-126.png\">: Every connected relation <img src=\"math-snippets/snip-47.png\"> contains a <img src=\"math-snippets/snip-48.png\">-maximal partially ordered set.") (137 "<strong> 137<img src=\"math-snippets/snip-127.png\">.</strong> Suppose <img src=\"math-snippets/snip-128.png\">. If <img src=\"math-snippets/snip-14.png\"> is a 1-1 map from <img src=\"math-snippets/snip-129.png\"> into <img src=\"math-snippets/snip-130.png\"> then there are partitions <img src=\"math-snippets/snip-131.png\"> and <img src=\"math-snippets/snip-132.png\"> of <img src=\"math-snippets/snip-13.png\"> and <img src=\"math-snippets/snip-65.png\"> such that <img src=\"math-snippets/snip-14.png\"> maps <img src=\"math-snippets/snip-133.png\"> onto <img src=\"math-snippets/snip-134.png\">. ") (334 "<strong> 334.</strong> <img src=\"math-snippets/snip-60.png\">: For every set <img src=\"math-snippets/snip-13.png\"> of sets such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-52.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for every <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-18.png\"> is even.") (37 "<strong> 37.</strong> Lebesgue measure is countably additive.") (51 "<strong> 51.</strong> Cofinality Principle: Every linear ordering has a cofinal sub well ordering. ") (173 "<strong> 173.</strong> MPL: Metric spaces are para-Lindel&ouml;f.") (48 "<strong> 48<img src=\"math-snippets/snip-92.png\">.</strong> If <img src=\"math-snippets/snip-95.png\"> is a finite subset of <img src=\"math-snippets/snip-138.png\">, <img src=\"math-snippets/snip-139.png\">: For every <img src=\"math-snippets/snip-140.png\"> <img src=\"math-snippets/snip-75.png\">.") (341 "<strong> 341.</strong> Every Lindel&ouml;f metric space is second countable.") (34 "<strong> 34.</strong> <img src=\"math-snippets/snip-141.png\"> is regular.") (282 "<strong> 282.</strong> <img src=\"math-snippets/snip-102.png\">.") (33 "<strong> 33<img src=\"math-snippets/snip-41.png\">.</strong> If <img src=\"math-snippets/snip-74.png\">, <img src=\"math-snippets/snip-103.png\">: Every linearly ordered set of <img src=\"math-snippets/snip-7.png\"> element sets has a choice function.") (245 "<strong> 245.</strong> There is a function <img src=\"math-snippets/snip-152.png\"> such that for every <img src=\"math-snippets/snip-153.png\">, <img src=\"math-snippets/snip-154.png\">, <img src=\"math-snippets/snip-155.png\"> is a function from <img src=\"math-snippets/snip-73.png\"> onto <img src=\"math-snippets/snip-153.png\">.") (353 "<strong> 353.</strong> A countable product of first countable spaces is first countable.") (141 "<strong> 141.</strong> [14 P(<img src=\"math-snippets/snip-7.png\">)] with <img src=\"math-snippets/snip-145.png\">: Let <img src=\"math-snippets/snip-146.png\"> be a collection of sets such that <img src=\"math-snippets/snip-147.png\"> and suppose <img src=\"math-snippets/snip-31.png\"> is a symmetric binary relation on <img src=\"math-snippets/snip-148.png\"> such that for all finite <img src=\"math-snippets/snip-149.png\"> there is an <img src=\"math-snippets/snip-31.png\"> consistent choice function for <img src=\"math-snippets/snip-150.png\">. Then there is an <img src=\"math-snippets/snip-31.png\"> consistent choice function for <img src=\"math-snippets/snip-146.png\">.") (271 "<strong> 271<img src=\"math-snippets/snip-41.png\">.</strong> If <img src=\"math-snippets/snip-74.png\">, <img src=\"math-snippets/snip-151.png\">: The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs in at most <img src=\"math-snippets/snip-7.png\"> formulas.") (90 "<strong> 90.</strong> <img src=\"math-snippets/snip-144.png\">: Every linearly ordered set can be well ordered.") (58 "<strong> 58.</strong> There is an ordinal <img src=\"math-snippets/snip-153.png\"> such that <img src=\"math-snippets/snip-156.png\">. (<img src=\"math-snippets/snip-157.png\"> is Hartogs' aleph, the least <img src=\"math-snippets/snip-158.png\"> not <img src=\"math-snippets/snip-159.png\">.)") (82 "<strong> 82.</strong> <img src=\"math-snippets/snip-160.png\"> (see <font class=\"author\">Howard/Yorke</font> <font class=\"year\">1989</font>): If <img src=\"math-snippets/snip-13.png\"> is infinite then <img src=\"math-snippets/snip-30.png\"> is Dedekind infinite. (<img src=\"math-snippets/snip-13.png\"> is finite <img src=\"math-snippets/snip-161.png\"> is Dedekind finite.)") (128 "<strong> 128.</strong> Aczel's Realization Principle: On every infinite set there is a Hausdorff topology with an infinite set of non-isolated points. ") (419 "<strong> 419.</strong> UT(<img src=\"math-snippets/snip-201.png\">,cuf,cuf): The union of a denumerable set of cuf sets is cuf. (A set is <i> cuf</i> if it is a countable union of finite sets.)") (152 "<strong> 152.</strong> <img src=\"math-snippets/snip-162.png\">: Every non-well-orderable set is the union of a pairwise disjoint, well orderable family of denumerable sets. ") (189 "<strong> 189.</strong> <img src=\"math-snippets/snip-163.png\">: For every Abelian group <img src=\"math-snippets/snip-3.png\"> there is an injective Abelian group <img src=\"math-snippets/snip-2.png\"> and a one to one homomorphism from <img src=\"math-snippets/snip-3.png\"> into <img src=\"math-snippets/snip-2.png\">.") (324 "<strong> 324.</strong> <img src=\"math-snippets/snip-164.png\">, The Kinna-Wagner Selection Principle for a well ordered family of well orderable sets: For every well ordered set <img src=\"math-snippets/snip-88.png\"> of well orderable sets, there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-137.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (277 "<strong> 277.</strong> <img src=\"math-snippets/snip-165.png\">: Every non-well-orderable cardinal is decomposable.") (389 "<strong> 389.</strong> <img src=\"math-snippets/snip-180.png\">: Every denumerable family of two element subsets of <img src=\"math-snippets/snip-181.png\"> has a choice function. ") (187 "<strong> 187.</strong> Every pair of cardinal numbers has a greatest lower bound (in the usual cardinal ordering.) ") (30 "<strong> 30.</strong> Ordering Principle: Every set can be linearly ordered. ") (222 "<strong> 222.</strong> There is a non-principal measure on <img src=\"math-snippets/snip-143.png\">. ") (44 "<strong> 44.</strong> <img src=\"math-snippets/snip-120.png\">: Given a relation <img src=\"math-snippets/snip-31.png\"> such that for every subset <img src=\"math-snippets/snip-65.png\"> of a set <img src=\"math-snippets/snip-13.png\"> with <img src=\"math-snippets/snip-121.png\"> there is an <img src=\"math-snippets/snip-122.png\"> with <img src=\"math-snippets/snip-123.png\">, then there is a function <img src=\"math-snippets/snip-124.png\"> such that <img src=\"math-snippets/snip-125.png\">. ") (242 "<strong> 242.</strong> There is, up to an isomorphism, at most one algebraic closure of <img src=\"math-snippets/snip-168.png\">. ") (294 "<strong> 294.</strong> Every linearly ordered <img src=\"math-snippets/snip-170.png\">-set is well orderable. ") (111 "<strong> 111.</strong> <img src=\"math-snippets/snip-171.png\">: The union of an infinite well ordered set of 2-element sets is an infinite well ordered set. ") (354 "<strong> 354.</strong> A countable product of separable <img src=\"math-snippets/snip-61.png\"> spaces is separable.") (346 "<strong> 346.</strong> If <img src=\"math-snippets/snip-97.png\"> is a vector space without a finite basis then <img src=\"math-snippets/snip-97.png\"> contains an infinite, well ordered, linearly independent subset.") (272 "<strong> 272.</strong> There is an <img src=\"math-snippets/snip-166.png\"> such that neither <img src=\"math-snippets/snip-13.png\"> nor <img src=\"math-snippets/snip-167.png\"> has a perfect subset.") (146 "<strong> 146.</strong> <img src=\"math-snippets/snip-174.png\">: For every <img src=\"math-snippets/snip-61.png\"> topological space <img src=\"math-snippets/snip-175.png\">, if <img src=\"math-snippets/snip-13.png\"> is a continuous finite to one image of an A1 space then <img src=\"math-snippets/snip-175.png\"> is an A1 space. (<img src=\"math-snippets/snip-175.png\"> is A1 means if <img src=\"math-snippets/snip-176.png\"> covers <img src=\"math-snippets/snip-13.png\"> then <img src=\"math-snippets/snip-177.png\"> such that <img src=\"math-snippets/snip-178.png\"> ") (275 "<strong> 275.</strong> The sequence of cardinals <img src=\"math-snippets/snip-135.png\"> has a unique minimal upper bound. ") (358 "<strong> 358.</strong> <img src=\"math-snippets/snip-182.png\">, The Kinna-Wagner Selection Principle for a denumerable family of finite sets: For every denumerable set <img src=\"math-snippets/snip-88.png\"> of finite sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-137.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (385 "<strong> 385.</strong> Countable Ultrafilter Theorem: Every proper filter with a countable base over a set <img src=\"math-snippets/snip-104.png\"> (in <img src=\"math-snippets/snip-183.png\">) can be extended to an ultrafilter.") (422 "<strong> 422<img src=\"math-snippets/snip-41.png\">.</strong> <img src=\"math-snippets/snip-208.png\">, <img src=\"math-snippets/snip-209.png\">: The union of a well ordered set of <img src=\"math-snippets/snip-7.png\"> element sets can be well ordered. ") (344 "<strong> 344.</strong> If <img src=\"math-snippets/snip-192.png\"> is a family of non-empty sets, then there is a family <img src=\"math-snippets/snip-193.png\"> such that <img src=\"math-snippets/snip-194.png\">, <img src=\"math-snippets/snip-195.png\"> is an ultrafilter on <img src=\"math-snippets/snip-196.png\">. ") (80 "<strong> 80.</strong> <img src=\"math-snippets/snip-10.png\">: Every denumerable set of pairs has a choice function. ") (89 "<strong> 89.</strong> Antichain Principle: Every partially ordered set has a maximal antichain.") (325 "<strong> 325.</strong> Ramsey's Theorem II: <img src=\"math-snippets/snip-189.png\">, if A is an infinite set and the family of all <img src=\"math-snippets/snip-111.png\"> element subsets of <img src=\"math-snippets/snip-3.png\"> is partitioned into <img src=\"math-snippets/snip-7.png\"> sets <img src=\"math-snippets/snip-190.png\">, then there is an infinite subset <img src=\"math-snippets/snip-35.png\"> such that all <img src=\"math-snippets/snip-111.png\"> element subsets of <img src=\"math-snippets/snip-32.png\"> belong to the same <img src=\"math-snippets/snip-191.png\">. (Also, see form 17.) ") (161 "<strong> 161.</strong> Definability of cardinal addition in terms of <img src=\"math-snippets/snip-197.png\">: There is a first order formula whose only non-logical symbol is <img src=\"math-snippets/snip-197.png\"> (for cardinals) that defines cardinal addition.") (56 "<strong> 56.</strong> <img src=\"math-snippets/snip-198.png\">. (<img src=\"math-snippets/snip-199.png\"> is Hartogs' aleph, the least <img src=\"math-snippets/snip-158.png\"> not <img src=\"math-snippets/snip-200.png\">.) ") (420 "<strong> 420.</strong> UT(<img src=\"math-snippets/snip-201.png\">,<img src=\"math-snippets/snip-201.png\">,cuf): The union of a denumerable set of denumerable sets is cuf.") (4 "<strong> 4.</strong> Every infinite set is the union of some disjoint family of denumerable subsets. (Denumerable means <img src=\"math-snippets/snip-202.png\">.)") (53 "<strong> 53.</strong> For all infinite cardinals <img src=\"math-snippets/snip-111.png\">, <img src=\"math-snippets/snip-203.png\">. ") (140 "<strong> 140.</strong> Let <img src=\"math-snippets/snip-204.png\"> be the set of all (undirected) infinite cycles of reals (Graphs whose vertices are real numbers, connected, no loops and each vertex adjacent to exactly two others). Then there is a function <img src=\"math-snippets/snip-14.png\"> on <img src=\"math-snippets/snip-204.png\"> such that for all <img src=\"math-snippets/snip-205.png\">, <img src=\"math-snippets/snip-206.png\"> is a direction along <img src=\"math-snippets/snip-207.png\">. ") (191 "<strong> 191.</strong> <img src=\"math-snippets/snip-210.png\">: There is a set <img src=\"math-snippets/snip-104.png\"> such that for every set <img src=\"math-snippets/snip-211.png\">, there is an ordinal <img src=\"math-snippets/snip-153.png\"> and a function from <img src=\"math-snippets/snip-212.png\"> onto <img src=\"math-snippets/snip-211.png\">. ") (254 "<strong> 254.</strong> <img src=\"math-snippets/snip-213.png\">: Every directed relation <img src=\"math-snippets/snip-63.png\"> in which ramified subsets have least upper bounds, has a maximal element.") (174 "<strong> 174<img src=\"math-snippets/snip-19.png\">.</strong> <img src=\"math-snippets/snip-214.png\">: The representation theorem for multi-algebras with <img src=\"math-snippets/snip-215.png\"> unary operations: Assume <img src=\"math-snippets/snip-216.png\"> is a multi-algebra with <img src=\"math-snippets/snip-215.png\"> unary operations (and no other operations). Then there is an algebra <img src=\"math-snippets/snip-217.png\"> with <img src=\"math-snippets/snip-215.png\"> unary operations and an equivalence relation <img src=\"math-snippets/snip-218.png\"> on <img src=\"math-snippets/snip-32.png\"> such that <img src=\"math-snippets/snip-219.png\"> and <img src=\"math-snippets/snip-216.png\"> are isomorphic multi-algebras.") (237 "<strong> 237.</strong> The order of any group is divisible by the order of any of its subgroups, (i.e., if <img src=\"math-snippets/snip-1.png\"> is a subgroup of <img src=\"math-snippets/snip-2.png\"> then there is a set <img src=\"math-snippets/snip-3.png\"> such that <img src=\"math-snippets/snip-4.png\">.) ") (25 "<strong> 25.</strong> <img src=\"math-snippets/snip-220.png\"> is regular for all ordinals <img src=\"math-snippets/snip-221.png\">. ") (156 "<strong> 156.</strong> Theorem of Gelfand and Kolmogoroff: Two compact <img src=\"math-snippets/snip-61.png\"> spaces are homeomorphic if their rings of real valued continuous functions are isomorphic. ") (303 "<strong> 303.</strong> If <img src=\"math-snippets/snip-32.png\"> is a Boolean algebra, <img src=\"math-snippets/snip-222.png\"> and <img src=\"math-snippets/snip-104.png\"> is closed under <img src=\"math-snippets/snip-223.png\">, then there is a <img src=\"math-snippets/snip-48.png\">-maximal proper ideal <img src=\"math-snippets/snip-224.png\"> of <img src=\"math-snippets/snip-32.png\"> such that <img src=\"math-snippets/snip-225.png\">. ") (288 "<strong> 288<img src=\"math-snippets/snip-41.png\">.</strong> If <img src=\"math-snippets/snip-74.png\">, <img src=\"math-snippets/snip-188.png\">: Every denumerable set of <img src=\"math-snippets/snip-7.png\">-element sets has a choice function.") (65 "<strong> 65.</strong> The Krein-Milman Theorem: Let <img src=\"math-snippets/snip-95.png\"> be a compact convex set in a locally convex topological vector space <img src=\"math-snippets/snip-13.png\">. Then <img src=\"math-snippets/snip-95.png\"> has an extreme point. (An extreme point is a point which is not an interior point of any line segment which lies in <img src=\"math-snippets/snip-95.png\">.)") (347 "<strong> 347.</strong> Idemmultiple Partition Principle: If <img src=\"math-snippets/snip-28.png\"> is idemmultiple (<img src=\"math-snippets/snip-246.png\">) and <img src=\"math-snippets/snip-247.png\">, then <img src=\"math-snippets/snip-248.png\">.") (42 "<strong> 42.</strong> L&ouml;wenheim-Skolem Theorem: If a countable family of first order sentences is satisfiable in a set <img src=\"math-snippets/snip-88.png\"> then it is satisfiable in a countable subset of <img src=\"math-snippets/snip-88.png\">. ") (101 "<strong> 101.</strong> Partition Principle: If <img src=\"math-snippets/snip-104.png\"> is a partition of <img src=\"math-snippets/snip-88.png\">, then <img src=\"math-snippets/snip-227.png\">. ") (22 "<strong> 22.</strong> <img src=\"math-snippets/snip-228.png\">: If every member of an infinite set of cardinality <img src=\"math-snippets/snip-229.png\"> has power <img src=\"math-snippets/snip-230.png\">, then the union has power <img src=\"math-snippets/snip-230.png\">.") (119 "<strong> 119.</strong> <img src=\"math-snippets/snip-231.png\">,uniformly orderable with order type of the integers): Suppose <img src=\"math-snippets/snip-232.png\"> is a set and there is a function <img src=\"math-snippets/snip-14.png\"> such that for each <img src=\"math-snippets/snip-233.png\"> is an ordering of <img src=\"math-snippets/snip-234.png\"> of type <img src=\"math-snippets/snip-235.png\"> (the usual ordering of the integers), then <img src=\"math-snippets/snip-236.png\"> has a choice function. ") (397 "<strong> 397.</strong> <img src=\"math-snippets/snip-237.png\">: For each well ordered family of non-empty linearly orderable sets <img src=\"math-snippets/snip-13.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\"> <img src=\"math-snippets/snip-16.png\"> is a non-empty, finite subset of <img src=\"math-snippets/snip-17.png\">. ") (404 "<strong> 404.</strong> Every infinite set can be partitioned into infinitely many sets, each of which has at least two elements.") (17 "<strong> 17.</strong> Ramsey's Theorem I: If <img src=\"math-snippets/snip-3.png\"> is an infinite set and the family of all 2 element subsets of <img src=\"math-snippets/snip-3.png\"> is partitioned into 2 sets <img src=\"math-snippets/snip-13.png\"> and <img src=\"math-snippets/snip-65.png\">, then there is an infinite subset <img src=\"math-snippets/snip-35.png\"> such that all 2 element subsets of <img src=\"math-snippets/snip-32.png\"> belong to <img src=\"math-snippets/snip-13.png\"> or all 2 element subsets of <img src=\"math-snippets/snip-32.png\"> belong to <img src=\"math-snippets/snip-65.png\">. (Also, see form 325.)") (413 "<strong> 413.</strong> Every infinite set <img src=\"math-snippets/snip-104.png\"> is the union of a set, well-ordered by inclusion, of subsets which are non-equipollent to <img src=\"math-snippets/snip-104.png\">. ") (194 "<strong> 194.</strong> <img src=\"math-snippets/snip-249.png\"> or <img src=\"math-snippets/snip-250.png\">: If <img src=\"math-snippets/snip-251.png\">, <img src=\"math-snippets/snip-110.png\"> has domain <img src=\"math-snippets/snip-73.png\">, and <img src=\"math-snippets/snip-110.png\"> is in <img src=\"math-snippets/snip-252.png\">, then there is a sequence of elements <img src=\"math-snippets/snip-253.png\"> of <img src=\"math-snippets/snip-254.png\"> with <img src=\"math-snippets/snip-255.png\"> for all <img src=\"math-snippets/snip-256.png\">.") (29 "<strong> 29.</strong> If <img src=\"math-snippets/snip-257.png\"> and <img src=\"math-snippets/snip-258.png\"> and <img src=\"math-snippets/snip-259.png\"> are families of pairwise disjoint sets and <img src=\"math-snippets/snip-260.png\"> for all <img src=\"math-snippets/snip-261.png\">, then <img src=\"math-snippets/snip-262.png\">. ") (172 "<strong> 172.</strong> For every infinite set <img src=\"math-snippets/snip-104.png\">, if <img src=\"math-snippets/snip-104.png\"> is hereditarily countable (that is, every <img src=\"math-snippets/snip-263.png\"> is countable) then <img src=\"math-snippets/snip-264.png\">.") (310 "<strong> 310.</strong> The Measure Extension Theorem: Suppose that <img src=\"math-snippets/snip-283.png\"> is a subring (that is, <img src=\"math-snippets/snip-284.png\"> and <img src=\"math-snippets/snip-285.png\">) of a Boolean algebra <img src=\"math-snippets/snip-286.png\"> and <img src=\"math-snippets/snip-115.png\"> is a measure on <img src=\"math-snippets/snip-283.png\"> (that is, <img src=\"math-snippets/snip-287.png\">, <img src=\"math-snippets/snip-288.png\"> for <img src=\"math-snippets/snip-289.png\">, and <img src=\"math-snippets/snip-290.png\">.) then there is a measure on <img src=\"math-snippets/snip-286.png\"> that extends <img src=\"math-snippets/snip-115.png\">. ") (290 "<strong> 290.</strong> For all infinite <img src=\"math-snippets/snip-17.png\">, <img src=\"math-snippets/snip-266.png\">.") (109 "<strong> 109.</strong> Every field <img src=\"math-snippets/snip-50.png\"> and every vector space <img src=\"math-snippets/snip-97.png\"> over <img src=\"math-snippets/snip-50.png\"> has the property that each linearly independent set <img src=\"math-snippets/snip-267.png\"> can be extended to a basis.") (314 "<strong> 314.</strong> For every set <img src=\"math-snippets/snip-13.png\"> and every permutation <img src=\"math-snippets/snip-268.png\"> on <img src=\"math-snippets/snip-13.png\"> there are two reflections <img src=\"math-snippets/snip-269.png\"> and <img src=\"math-snippets/snip-270.png\"> on <img src=\"math-snippets/snip-13.png\"> such that <img src=\"math-snippets/snip-271.png\"> and for every <img src=\"math-snippets/snip-272.png\"> if <img src=\"math-snippets/snip-273.png\"> then <img src=\"math-snippets/snip-274.png\"> and <img src=\"math-snippets/snip-275.png\">. (A reflection is a permutation <img src=\"math-snippets/snip-276.png\"> such that <img src=\"math-snippets/snip-277.png\"> is the identity.)") (327 "<strong> 327.</strong> <img src=\"math-snippets/snip-384.png\">, The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set <img src=\"math-snippets/snip-88.png\"> of finite sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (281 "<strong> 281.</strong> There is a Hilbert space <img src=\"math-snippets/snip-1.png\"> and an unbounded linear operator on <img src=\"math-snippets/snip-1.png\">. ") (348 "<strong> 348.</strong> If <img src=\"math-snippets/snip-2.png\"> is a group and <img src=\"math-snippets/snip-13.png\"> and <img src=\"math-snippets/snip-65.png\"> both freely generate <img src=\"math-snippets/snip-2.png\"> then <img src=\"math-snippets/snip-291.png\">. ") (236 "<strong> 236.</strong> If <img src=\"math-snippets/snip-97.png\"> is a vector space with a basis and <img src=\"math-snippets/snip-104.png\"> is a linearly independent subset of <img src=\"math-snippets/snip-97.png\"> such that no proper extension of <img src=\"math-snippets/snip-104.png\"> is a basis for <img src=\"math-snippets/snip-97.png\">, then <img src=\"math-snippets/snip-104.png\"> is a basis for <img src=\"math-snippets/snip-97.png\">. ") (427 "<strong> 427.</strong> <img src=\"math-snippets/snip-76.png\"> AL20(<img src=\"math-snippets/snip-50.png\">): There is a field <img src=\"math-snippets/snip-50.png\"> such that every vector space <img src=\"math-snippets/snip-97.png\"> over <img src=\"math-snippets/snip-50.png\"> has the property that every independent subset of <img src=\"math-snippets/snip-97.png\"> can be extended to a basis. ") (41 "<strong> 41.</strong> <img src=\"math-snippets/snip-278.png\">: For every cardinal <img src=\"math-snippets/snip-111.png\">, <img src=\"math-snippets/snip-279.png\"> or <img src=\"math-snippets/snip-280.png\">. ") (262 "<strong> 262.</strong> <img src=\"math-snippets/snip-292.png\">: Every transitive relation <img src=\"math-snippets/snip-47.png\"> in which every ramified subset <img src=\"math-snippets/snip-3.png\"> has an upper bound, has a maximal element.") (203 "<strong> 203.</strong> <img src=\"math-snippets/snip-282.png\">(disjoint,<img src=\"math-snippets/snip-304.png\">: Every partition of <img src=\"math-snippets/snip-305.png\"> into non-empty subsets has a choice function.") (70 "<strong> 70.</strong> There is a non-trivial ultrafilter on <img src=\"math-snippets/snip-73.png\">.") (164 "<strong> 164.</strong> Every non-well-orderable set has an infinite subset with a Dedekind finite power set.") (52 "<strong> 52.</strong> Hahn-Banach Theorem: If <img src=\"math-snippets/snip-97.png\"> is a real vector space and <img src=\"math-snippets/snip-238.png\"> satisfies <img src=\"math-snippets/snip-239.png\"> and <img src=\"math-snippets/snip-240.png\"> and <img src=\"math-snippets/snip-104.png\"> is a subspace of <img src=\"math-snippets/snip-97.png\"> and <img src=\"math-snippets/snip-241.png\"> is linear and satisfies <img src=\"math-snippets/snip-242.png\"> then <img src=\"math-snippets/snip-14.png\"> can be extended to <img src=\"math-snippets/snip-243.png\"> such that <img src=\"math-snippets/snip-244.png\"> is linear and <img src=\"math-snippets/snip-245.png\">. ") (175 "<strong> 175.</strong> Transitivity Condition: For all sets <img src=\"math-snippets/snip-17.png\">, there is a set <img src=\"math-snippets/snip-294.png\"> abd a function <img src=\"math-snippets/snip-14.png\"> such that <img src=\"math-snippets/snip-294.png\"> is transitive and <img src=\"math-snippets/snip-14.png\"> is a one to one function from <img src=\"math-snippets/snip-17.png\"> onto <img src=\"math-snippets/snip-294.png\">. von") (73 "<strong> 73.</strong> <img src=\"math-snippets/snip-281.png\">, <img src=\"math-snippets/snip-43.png\">: For every <img src=\"math-snippets/snip-42.png\">, if <img src=\"math-snippets/snip-282.png\"> is an infinite family of <img src=\"math-snippets/snip-7.png\"> element sets, then <img src=\"math-snippets/snip-282.png\"> has an infinite subfamily with a choice function. ") (103 "<strong> 103.</strong> If <img src=\"math-snippets/snip-295.png\"> is a linear ordering and <img src=\"math-snippets/snip-296.png\"> then some initial segment of <img src=\"math-snippets/snip-110.png\"> is uncountable.") (258 "<strong> 258.</strong> <img src=\"math-snippets/snip-297.png\">: Every directed relation <img src=\"math-snippets/snip-47.png\"> in which linearly ordered subsets have upper bounds, has a maximal element.") (220 "<strong> 220<img src=\"math-snippets/snip-57.png\">.</strong> Suppose <img src=\"math-snippets/snip-298.png\"> and <img src=\"math-snippets/snip-34.png\"> is a prime. Any two elementary Abelian <img src=\"math-snippets/snip-34.png\">-groups (all non-trivial elements have order <img src=\"math-snippets/snip-34.png\">) of the same cardinality are isomorphic.") (365 "<strong> 365.</strong> For every uncountable set <img src=\"math-snippets/snip-3.png\">, if <img src=\"math-snippets/snip-3.png\"> has the same cardinality as each of its uncountable subsets then <img src=\"math-snippets/snip-299.png\">. ") (134 "<strong> 134.</strong> If <img src=\"math-snippets/snip-13.png\"> is an infinite <img src=\"math-snippets/snip-300.png\"> space and <img src=\"math-snippets/snip-301.png\"> is <img src=\"math-snippets/snip-302.png\">, then <img src=\"math-snippets/snip-65.png\"> is countable. (<img src=\"math-snippets/snip-302.png\"> is ``hereditarily <img src=\"math-snippets/snip-303.png\">''.)") (411 "<strong> 411.</strong> RCuc (Reflexive Compactness for uniformly convex Banach spaces): The closed unit ball of a uniformly convex Banach space is compact for the weak topology.") (26 "<strong> 26.</strong> <img src=\"math-snippets/snip-306.png\">: The union of denumerably many sets each of power <img src=\"math-snippets/snip-229.png\"> has power <img src=\"math-snippets/snip-230.png\">. ") (246 "<strong> 246.</strong> The monadic theory theory <img src=\"math-snippets/snip-307.png\"> of <img src=\"math-snippets/snip-308.png\"> is recursive. ") (86 "<strong> 86<img src=\"math-snippets/snip-19.png\">.</strong> <img src=\"math-snippets/snip-309.png\">: If <img src=\"math-snippets/snip-13.png\"> is a set of non-empty sets such that <img src=\"math-snippets/snip-310.png\">, then <img src=\"math-snippets/snip-13.png\"> has a choice function. ") (92 "<strong> 92.</strong> <img src=\"math-snippets/snip-311.png\">: Every well ordered family of non-empty subsets of <img src=\"math-snippets/snip-9.png\"> has a choice function.") (368 "<strong> 368.</strong> The set of all denumerable subsets of <img src=\"math-snippets/snip-142.png\"> has power <img src=\"math-snippets/snip-325.png\">. ") (224 "<strong> 224.</strong> There is a partition of the real line into <img src=\"math-snippets/snip-313.png\"> Borel sets <img src=\"math-snippets/snip-314.png\"> such that for some <img src=\"math-snippets/snip-315.png\">, <img src=\"math-snippets/snip-316.png\">, <img src=\"math-snippets/snip-317.png\">. (<img src=\"math-snippets/snip-318.png\"> for <img src=\"math-snippets/snip-319.png\"> is defined by induction, <img src=\"math-snippets/snip-320.png\"> is an open subset of <img src=\"math-snippets/snip-321.png\"> and for <img src=\"math-snippets/snip-322.png\">, <img src=\"math-snippets/snip-323.png\"> if <img src=\"math-snippets/snip-221.png\"> is even and <img src=\"math-snippets/snip-324.png\"> if <img src=\"math-snippets/snip-221.png\"> is odd.) ") (60 "<strong> 60.</strong> <img src=\"math-snippets/snip-173.png\">: Every set of non-empty, well orderable sets has a choice function.") (145 "<strong> 145.</strong> Compact <img src=\"math-snippets/snip-326.png\">-spaces are Dedekind finite. (A <i> <img src=\"math-snippets/snip-326.png\">-space</i> is a topological space in which the intersection of a countable collection of open sets is open.)") (43 "<strong> 43.</strong> <img src=\"math-snippets/snip-184.png\"> (DC), Principle of Dependent Choices: If <img src=\"math-snippets/snip-104.png\"> is a relation on a non-empty set <img src=\"math-snippets/snip-3.png\"> and <img src=\"math-snippets/snip-185.png\"> then there is a sequence <img src=\"math-snippets/snip-186.png\"> of elements of <img src=\"math-snippets/snip-3.png\"> such that <img src=\"math-snippets/snip-187.png\">. ") (20 "<strong> 20.</strong> If <img src=\"math-snippets/snip-327.png\"> and <img src=\"math-snippets/snip-328.png\"> are families of pairwise disjoint sets and <img src=\"math-snippets/snip-260.png\"> for all <img src=\"math-snippets/snip-261.png\">, then <img src=\"math-snippets/snip-329.png\">. ") (138 "<strong> 138<img src=\"math-snippets/snip-127.png\">.</strong> Suppose <img src=\"math-snippets/snip-256.png\">. If <img src=\"math-snippets/snip-14.png\"> is a partial map from <img src=\"math-snippets/snip-130.png\"> onto <img src=\"math-snippets/snip-129.png\"> (that is, the domain is a subset of <img src=\"math-snippets/snip-130.png\">), then there are partitions <img src=\"math-snippets/snip-131.png\"> and <img src=\"math-snippets/snip-132.png\"> of <img src=\"math-snippets/snip-13.png\"> and <img src=\"math-snippets/snip-65.png\"> such that <img src=\"math-snippets/snip-14.png\"> maps <img src=\"math-snippets/snip-330.png\"> onto <img src=\"math-snippets/snip-331.png\">. ") (278 "<strong> 278.</strong> In an integral domain <img src=\"math-snippets/snip-31.png\">, if every ideal is finitely generated then <img src=\"math-snippets/snip-31.png\"> has a maximal proper ideal.") (216 "<strong> 216.</strong> Every infinite tree has either an infinite chain or an infinite antichain. ") (7 "<strong> 7.</strong> There is no infinite decreasing sequence of cardinals.") (370 "<strong> 370.</strong> Weak Gelfand Extreme Point Theorem: If <img src=\"math-snippets/snip-3.png\"> is a non-trivial Gelfand algebra then the closed unit ball in the dual of <img src=\"math-snippets/snip-3.png\"> has an extreme point <img src=\"math-snippets/snip-346.png\">. ") (390 "<strong> 390.</strong> Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements.") (426 "<strong> 426.</strong> If <img src=\"math-snippets/snip-352.png\"> is a first countable topological space and <img src=\"math-snippets/snip-353.png\"> is a family such that for all <img src=\"math-snippets/snip-354.png\">, <img src=\"math-snippets/snip-355.png\"> is a local base at <img src=\"math-snippets/snip-17.png\">, then there is a family <img src=\"math-snippets/snip-356.png\"> such that for every <img src=\"math-snippets/snip-354.png\">, <img src=\"math-snippets/snip-357.png\"> is a countable local base at <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-358.png\">.") (75 "<strong> 75.</strong> If a set has at least two elements, then it can be partitioned into well orderable subsets, each of which has at least two elements.") (209 "<strong> 209.</strong> There is an ordinal <img src=\"math-snippets/snip-153.png\"> such that for all <img src=\"math-snippets/snip-13.png\">, if <img src=\"math-snippets/snip-13.png\"> is a denumerable union of denumerable sets then <img src=\"math-snippets/snip-332.png\"> cannot be partitioned into <img src=\"math-snippets/snip-20.png\"> non-empty sets.") (153 "<strong> 153.</strong> The closed unit ball of a Hilbert space is compact in the weak topology.") (429 "<strong> 429<img src=\"math-snippets/snip-57.png\">.</strong> (Where <img src=\"math-snippets/snip-34.png\"> is a prime) B: Every vector space over <img src=\"math-snippets/snip-335.png\"> has a basis. (<img src=\"math-snippets/snip-335.png\"> is the <img src=\"math-snippets/snip-34.png\"> element field.)") (198 "<strong> 198.</strong> For every set <img src=\"math-snippets/snip-104.png\">, if the only linearly orderable subsets of <img src=\"math-snippets/snip-104.png\"> are the finite subsets of <img src=\"math-snippets/snip-104.png\">, then either <img src=\"math-snippets/snip-104.png\"> is finite or <img src=\"math-snippets/snip-104.png\"> has an amorphous subset.") (359 "<strong> 359.</strong> If <img src=\"math-snippets/snip-258.png\"> and <img src=\"math-snippets/snip-259.png\"> are families of pairwise disjoint sets and <img src=\"math-snippets/snip-336.png\"> for all <img src=\"math-snippets/snip-261.png\">, then <img src=\"math-snippets/snip-337.png\">.") (163 "<strong> 163.</strong> Every non-well-orderable set has an infinite, Dedekind finite subset. ") (68 "<strong> 68.</strong> Nielsen-Schreier Theorem: Every subgroup of a free group is free. ") (279 "<strong> 279.</strong> The Closed Graph Theorem for operations between Fr&eacute;chet Spaces: Suppose <img src=\"math-snippets/snip-13.png\"> and <img src=\"math-snippets/snip-65.png\"> are Fr&eacute;chet spaces, <img src=\"math-snippets/snip-342.png\"> is linear and <img src=\"math-snippets/snip-343.png\"> is closed in <img src=\"math-snippets/snip-344.png\">. Then <img src=\"math-snippets/snip-345.png\"> is continuous.") (136 "<strong> 136<img src=\"math-snippets/snip-127.png\">.</strong> Surjective Cardinal Cancellation (depends on <img src=\"math-snippets/snip-128.png\">): For all cardinals <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-28.png\">, <img src=\"math-snippets/snip-347.png\"> implies <img src=\"math-snippets/snip-348.png\">.") (46 "<strong> 46<img src=\"math-snippets/snip-92.png\">.</strong> If <img src=\"math-snippets/snip-95.png\"> is a finite subset of <img src=\"math-snippets/snip-138.png\">, <img src=\"math-snippets/snip-349.png\">: For every <img src=\"math-snippets/snip-350.png\">, every set of <img src=\"math-snippets/snip-7.png\">-element sets has a choice function. ") (170 "<strong> 170.</strong> <img src=\"math-snippets/snip-226.png\">.") (147 "<strong> 147.</strong> <img src=\"math-snippets/snip-351.png\">: Every <img src=\"math-snippets/snip-61.png\"> topological space <img src=\"math-snippets/snip-175.png\"> can be covered by a well ordered family of discrete sets.") (10 "<strong> 10.</strong> <img src=\"math-snippets/snip-334.png\">: Every denumerable family of non-empty finite sets has a choice function. ") (421 "<strong> 421.</strong> <img src=\"math-snippets/snip-359.png\">: The union of a denumerable set of well orderable sets can be well ordered. ") (317 "<strong> 317.</strong> Weak Sikorski Theorem: If <img src=\"math-snippets/snip-32.png\"> is a complete, well orderable Boolean algebra and <img src=\"math-snippets/snip-14.png\"> is a homomorphism of the Boolean algebra <img src=\"math-snippets/snip-360.png\"> into <img src=\"math-snippets/snip-32.png\"> where <img src=\"math-snippets/snip-360.png\"> is a subalgebra of the Boolean algebra <img src=\"math-snippets/snip-3.png\">, then <img src=\"math-snippets/snip-14.png\"> can be extended to a homomorphism of <img src=\"math-snippets/snip-3.png\"> into <img src=\"math-snippets/snip-32.png\">.") (179 "<strong> 179<img src=\"math-snippets/snip-361.png\">.</strong> Suppose <img src=\"math-snippets/snip-362.png\"> is an ordinal. <img src=\"math-snippets/snip-363.png\">, <img src=\"math-snippets/snip-364.png\">).") (178 "<strong> 178<img src=\"math-snippets/snip-365.png\">.</strong> If <img src=\"math-snippets/snip-42.png\">, <img src=\"math-snippets/snip-6.png\"> and <img src=\"math-snippets/snip-366.png\">, <img src=\"math-snippets/snip-367.png\">, <img src=\"math-snippets/snip-368.png\">: If <img src=\"math-snippets/snip-13.png\"> is any set of <img src=\"math-snippets/snip-7.png\">-element sets then there is a function <img src=\"math-snippets/snip-14.png\"> with domain <img src=\"math-snippets/snip-13.png\"> such that for all <img src=\"math-snippets/snip-369.png\">, <img src=\"math-snippets/snip-370.png\"> and <img src=\"math-snippets/snip-371.png\">.") (369 "<strong> 369.</strong> If <img src=\"math-snippets/snip-142.png\"> is partitioned into two sets, at least one of them has cardinality <img src=\"math-snippets/snip-325.png\">.") (309 "<strong> 309.</strong> The Banach-Tarski Paradox: There are three finite partitions <img src=\"math-snippets/snip-372.png\">, <img src=\"math-snippets/snip-373.png\">, <img src=\"math-snippets/snip-374.png\"> and <img src=\"math-snippets/snip-375.png\"> of <img src=\"math-snippets/snip-376.png\"> such that <img src=\"math-snippets/snip-377.png\"> is congruent to <img src=\"math-snippets/snip-378.png\"> for <img src=\"math-snippets/snip-379.png\"> and <img src=\"math-snippets/snip-380.png\"> is congruent to <img src=\"math-snippets/snip-381.png\"> for <img src=\"math-snippets/snip-382.png\">. ") (235 "<strong> 235.</strong> If <img src=\"math-snippets/snip-97.png\"> is a vector space and <img src=\"math-snippets/snip-338.png\"> and <img src=\"math-snippets/snip-339.png\"> are bases for <img src=\"math-snippets/snip-97.png\"> then <img src=\"math-snippets/snip-340.png\"> and <img src=\"math-snippets/snip-341.png\"> are comparable.") (343 "<strong> 343.</strong> A product of non-empty, compact <img src=\"math-snippets/snip-61.png\"> topological spaces is non-empty. ") (98 "<strong> 98.</strong> The set of all finite subsets of a Dedekind finite set is Dedekind finite.") (116 "<strong> 116.</strong> Every compact <img src=\"math-snippets/snip-61.png\"> space is weakly Loeb. ") (35 "<strong> 35.</strong> The union of countably many meager subsets of <img src=\"math-snippets/snip-9.png\"> is meager. (Meager sets are the same as sets of the first category.)") (50 "<strong> 50.</strong> Sikorski's Extension Theorem: Every homomorphism of a subalgebra <img src=\"math-snippets/snip-32.png\"> of a Boolean algebra <img src=\"math-snippets/snip-3.png\"> into a complete Boolean algebra <img src=\"math-snippets/snip-385.png\"> can be extended to a homomorphism of <img src=\"math-snippets/snip-3.png\"> into <img src=\"math-snippets/snip-385.png\">.") (230 "<strong> 230.</strong> <img src=\"math-snippets/snip-386.png\">.") (387 "<strong> 387.</strong> DPO: Every infinite set has a non-trivial, dense partial order. (A partial ordering <img src=\"math-snippets/snip-552.png\"> on a set <img src=\"math-snippets/snip-13.png\"> is dense if <img src=\"math-snippets/snip-553.png\"> and is non-trivial if <img src=\"math-snippets/snip-554.png\">). ") (8 "<strong> 8.</strong> <img src=\"math-snippets/snip-389.png\">: Every denumerable family of non-empty sets has a choice function. ") (298 "<strong> 298.</strong> Every compact Hausdorff space has a Gleason cover.") (193 "<strong> 193.</strong> <img src=\"math-snippets/snip-383.png\">: Every Abelian group is a homomorphic image of a free projective Abelian group. ") (6 "<strong> 6.</strong> <img src=\"math-snippets/snip-265.png\">: The union of a denumerable family of denumerable subsets of <img src=\"math-snippets/snip-9.png\"> is denumerable. ") (393 "<strong> 393.</strong> <img src=\"math-snippets/snip-392.png\">: Every linearly ordered set of non-empty well orderable sets has a choice function. ") (229 "<strong> 229.</strong> If <img src=\"math-snippets/snip-393.png\"> is a partially ordered group, then <img src=\"math-snippets/snip-197.png\"> can be extended to a linear order on <img src=\"math-snippets/snip-2.png\"> if and only if for every finite set <img src=\"math-snippets/snip-394.png\">, with <img src=\"math-snippets/snip-395.png\"> the identity for <img src=\"math-snippets/snip-396.png\"> to <img src=\"math-snippets/snip-7.png\">, the signs <img src=\"math-snippets/snip-397.png\"> (<img src=\"math-snippets/snip-398.png\">) can be chosen so that <img src=\"math-snippets/snip-399.png\"> (where <img src=\"math-snippets/snip-400.png\"> is the normal sub-semi-group of <img src=\"math-snippets/snip-2.png\"> generated by <img src=\"math-snippets/snip-401.png\"> and <img src=\"math-snippets/snip-402.png\"> where <img src=\"math-snippets/snip-346.png\"> is the identity of <img src=\"math-snippets/snip-2.png\">.)") (407 "<strong> 407.</strong> Let <img src=\"math-snippets/snip-32.png\"> be a Boolean algebra, <img src=\"math-snippets/snip-403.png\"> a non-zero element of <img src=\"math-snippets/snip-32.png\"> and <img src=\"math-snippets/snip-404.png\"> a sequence of subsets of <img src=\"math-snippets/snip-32.png\"> such that for each <img src=\"math-snippets/snip-405.png\">, <img src=\"math-snippets/snip-406.png\"> has a supremum <img src=\"math-snippets/snip-407.png\">. Then there exists an ultrafilter <img src=\"math-snippets/snip-408.png\"> in <img src=\"math-snippets/snip-32.png\"> such that <img src=\"math-snippets/snip-409.png\"> and, for each <img src=\"math-snippets/snip-405.png\">, if <img src=\"math-snippets/snip-410.png\">, then <img src=\"math-snippets/snip-411.png\">.") (301 "<strong> 301.</strong> Any continuous surjection between Boolean spaces has an irreducible restriction to a closed subset of its domain.") (181 "<strong> 181.</strong> <img src=\"math-snippets/snip-387.png\">: Every set <img src=\"math-snippets/snip-13.png\"> of non-empty sets such that <img src=\"math-snippets/snip-388.png\"> has a choice function.") (313 "<strong> 313.</strong> <img src=\"math-snippets/snip-390.png\"> (the set of integers under addition) is amenable. (<img src=\"math-snippets/snip-2.png\"> is <i> amenable</i> if there is a finitely additive measure <img src=\"math-snippets/snip-115.png\"> on <img src=\"math-snippets/snip-116.png\"> such that <img src=\"math-snippets/snip-117.png\"> and <img src=\"math-snippets/snip-391.png\">, <img src=\"math-snippets/snip-119.png\">.) ") (329 "<strong> 329.</strong> <img src=\"math-snippets/snip-412.png\">: For every set <img src=\"math-snippets/snip-88.png\"> of well orderable sets such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-85.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for every <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\">. ") (243 "<strong> 243.</strong> Every principal ideal domain is a unique factorization domain. ") (355 "<strong> 355.</strong> <img src=\"math-snippets/snip-333.png\">, The Kinna-Wagner Selection Principle for a denumerable family of sets: For every denumerable set <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-137.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (315 "<strong> 315.</strong> <img src=\"math-snippets/snip-413.png\">, where <img src=\"math-snippets/snip-414.png\">") (418 "<strong> 418.</strong> DUM(<img src=\"math-snippets/snip-201.png\">): The countable disjoint union of metrizable spaces is metrizable.") (213 "<strong> 213.</strong> <img src=\"math-snippets/snip-425.png\">: If <img src=\"math-snippets/snip-426.png\"> then <img src=\"math-snippets/snip-13.png\"> has a choice function. ") (409 "<strong> 409.</strong> Suppose <img src=\"math-snippets/snip-415.png\"> is a locally finite graph (i.e. <img src=\"math-snippets/snip-2.png\"> is a non-empty set and <img src=\"math-snippets/snip-416.png\"> is a function from <img src=\"math-snippets/snip-2.png\"> to <img src=\"math-snippets/snip-116.png\"> such that for each <img src=\"math-snippets/snip-417.png\">, <img src=\"math-snippets/snip-418.png\"> and <img src=\"math-snippets/snip-419.png\"> are finite), <img src=\"math-snippets/snip-95.png\"> is a finite set of integers, and <img src=\"math-snippets/snip-345.png\"> is a function mapping subsets of <img src=\"math-snippets/snip-95.png\"> into subsets of <img src=\"math-snippets/snip-95.png\">. If for each finite subgraph <img src=\"math-snippets/snip-420.png\"> there is a function <img src=\"math-snippets/snip-421.png\"> such that for each <img src=\"math-snippets/snip-422.png\">, <img src=\"math-snippets/snip-423.png\">, then there is a function <img src=\"math-snippets/snip-276.png\"> such that for all <img src=\"math-snippets/snip-417.png\">, <img src=\"math-snippets/snip-424.png\">.") (305 "<strong> 305.</strong> There are <img src=\"math-snippets/snip-325.png\"> Vitali equivalence classes. (<i> Vitali equivalence classes</i> are equivalence classes of the real numbers under the relation <img src=\"math-snippets/snip-428.png\">.).") (36 "<strong> 36.</strong> Compact T<img src=\"math-snippets/snip-429.png\"> spaces are Loeb. (A space is <i> Loeb</i> if the set of non-empty closed sets has a choice function.)") (319 "<strong> 319.</strong> Measurable cardinals are inaccessible.") (162 "<strong> 162.</strong> Non-existence of infinite units: There is no infinite cardinal number <img src=\"math-snippets/snip-3.png\"> such that <img src=\"math-snippets/snip-430.png\"> and for all cardinals <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-28.png\">, <img src=\"math-snippets/snip-431.png\"> or <img src=\"math-snippets/snip-432.png\">.") (130 "<strong> 130.</strong> <img src=\"math-snippets/snip-433.png\"> is well orderable.") (259 "<strong> 259.</strong> <img src=\"math-snippets/snip-312.png\">: If <img src=\"math-snippets/snip-47.png\"> is a transitive and connected relation in which every well ordered subset has an upper bound, then <img src=\"math-snippets/snip-47.png\"> has a maximal element. ") (340 "<strong> 340.</strong> Every Lindel&ouml;f metric space is separable.") (351 "<strong> 351.</strong> A countable product of metrizable spaces is metrizable.") (159 "<strong> 159.</strong> The regular cardinals are cofinal in the class of ordinals. ") (16 "<strong> 16.</strong> <img src=\"math-snippets/snip-434.png\">: Every denumerable collection of non-empty sets each with power <img src=\"math-snippets/snip-435.png\"> has a choice function. ") (64 "<strong> 64.</strong> <img src=\"math-snippets/snip-293.png\"> (see <font class=\"author\">Howard/Yorke</font> <font class=\"year\">1989</font>): There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)") (276 "<strong> 276.</strong> <img src=\"math-snippets/snip-436.png\">: For every set <img src=\"math-snippets/snip-3.png\">, <img src=\"math-snippets/snip-437.png\"> is Dedekind finite if and only if <img src=\"math-snippets/snip-438.png\"> or <img src=\"math-snippets/snip-439.png\">.") (177 "<strong> 177.</strong> An infinite box product of regular <img src=\"math-snippets/snip-300.png\"> spaces, each of cardinality greater than 1, is neither first countable nor connected. ") (328 "<strong> 328.</strong> <img src=\"math-snippets/snip-441.png\">: For every well ordered set <img src=\"math-snippets/snip-13.png\"> such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-85.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that and for every <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\">. ") (144 "<strong> 144.</strong> Every set is almost well orderable.") (168 "<strong> 168.</strong> Dual Cantor-Bernstein Theorem: <img src=\"math-snippets/snip-442.png\"> and <img src=\"math-snippets/snip-443.png\"> implies <img src=\"math-snippets/snip-444.png\"> .") (286 "<strong> 286.</strong> Extended Krein-Milman Theorem: Let K be a quasicompact (sometimes called convex-compact), convex subset of a locally convex topological vector space, then K has an extreme point.") (287 "<strong> 287.</strong> The Hahn-Banach Theorem for Separable Normed Linear Spaces: Assume <img src=\"math-snippets/snip-97.png\"> is a separable normed linear space and <img src=\"math-snippets/snip-445.png\"> satisfies <img src=\"math-snippets/snip-239.png\"> and <img src=\"math-snippets/snip-446.png\"> and assume <img src=\"math-snippets/snip-14.png\"> is a linear function from a subspace <img src=\"math-snippets/snip-104.png\"> of <img src=\"math-snippets/snip-97.png\"> into <img src=\"math-snippets/snip-142.png\"> which satisfies <img src=\"math-snippets/snip-447.png\">, then <img src=\"math-snippets/snip-14.png\"> can be extended to <img src=\"math-snippets/snip-448.png\"> so that <img src=\"math-snippets/snip-449.png\"> is linear and <img src=\"math-snippets/snip-450.png\">. ") (263 "<strong> 263.</strong> <img src=\"math-snippets/snip-451.png\">: Every every relation <img src=\"math-snippets/snip-47.png\"> which is antisymmetric and connected contains a <img src=\"math-snippets/snip-48.png\">-maximal partially ordered subset. ") (118 "<strong> 118.</strong> Every linearly orderable topological space is normal. ") (212 "<strong> 212.</strong> <img src=\"math-snippets/snip-452.png\">: If <img src=\"math-snippets/snip-31.png\"> is a relation on <img src=\"math-snippets/snip-9.png\"> such that for all <img src=\"math-snippets/snip-453.png\">, there is a <img src=\"math-snippets/snip-454.png\"> such that <img src=\"math-snippets/snip-455.png\">, then there is a function <img src=\"math-snippets/snip-456.png\"> such that for all <img src=\"math-snippets/snip-453.png\">, <img src=\"math-snippets/snip-457.png\">. ") (251 "<strong> 251.</strong> The additive groups <img src=\"math-snippets/snip-458.png\"> and <img src=\"math-snippets/snip-459.png\"> are isomorphic. ") (285 "<strong> 285.</strong> Let <img src=\"math-snippets/snip-218.png\"> be a set and <img src=\"math-snippets/snip-466.png\">, then <img src=\"math-snippets/snip-14.png\"> has a fixed point if and only if <img src=\"math-snippets/snip-218.png\"> is not the union of three mutually disjoint sets <img src=\"math-snippets/snip-467.png\">, <img src=\"math-snippets/snip-468.png\"> and <img src=\"math-snippets/snip-469.png\"> such that <img src=\"math-snippets/snip-470.png\"> for <img src=\"math-snippets/snip-471.png\">. ") (85 "<strong> 85.</strong> <img src=\"math-snippets/snip-462.png\">: Every family of denumerable sets has a choice function. ") (299 "<strong> 299.</strong> Any extremally disconnected compact Hausdorff space is projective in the category of Boolean topological spaces.") (256 "<strong> 256.</strong> <img src=\"math-snippets/snip-427.png\">: Every partially ordered set <img src=\"math-snippets/snip-47.png\"> in which every forest <img src=\"math-snippets/snip-3.png\"> has an upper bound, has a maximal element.") (363 "<strong> 363.</strong> There are exactly <img src=\"math-snippets/snip-325.png\"> Borel sets in <img src=\"math-snippets/snip-142.png\">. ") (74 "<strong> 74.</strong> For every <img src=\"math-snippets/snip-460.png\"> the following are equivalent: <font class=\"enum\">(1)</font> <img src=\"math-snippets/snip-3.png\"> is closed and bounded. \\itemitem{(2)} Every sequence <img src=\"math-snippets/snip-461.png\"> has a convergent subsequence with limit in A. ") (143 "<strong> 143.</strong> <img src=\"math-snippets/snip-473.png\">: If <img src=\"math-snippets/snip-47.png\"> is a connected relation (<img src=\"math-snippets/snip-474.png\"> or <img src=\"math-snippets/snip-475.png\">) then <img src=\"math-snippets/snip-13.png\"> contains a <img src=\"math-snippets/snip-48.png\">-maximal transitive subset. ") (160 "<strong> 160.</strong> No Dedekind finite set can be mapped onto an aleph. ") (201 "<strong> 201.</strong> Linking Axiom for Boolean Algebras: Every Boolean algebra has a maximal linked system. (<img src=\"math-snippets/snip-555.png\"> is <i> linked</i> if <img src=\"math-snippets/snip-556.png\"> for all <img src=\"math-snippets/snip-211.png\"> and <img src=\"math-snippets/snip-557.png\">.) ") (234 "<strong> 234.</strong> There is a non-Ramsey set: There is a set <img src=\"math-snippets/snip-3.png\"> of infinite subsets of <img src=\"math-snippets/snip-73.png\"> such that for every infinite subset <img src=\"math-snippets/snip-480.png\"> of <img src=\"math-snippets/snip-73.png\">, <img src=\"math-snippets/snip-480.png\"> has a subset which is in <img src=\"math-snippets/snip-3.png\"> and a subset which is not in <img src=\"math-snippets/snip-3.png\">. ") (364 "<strong> 364.</strong> In <img src=\"math-snippets/snip-142.png\">, there is a measurable set that is not Borel. ") (204 "<strong> 204.</strong> For every infinite <img src=\"math-snippets/snip-13.png\">, there is a function from <img src=\"math-snippets/snip-13.png\"> onto <img src=\"math-snippets/snip-481.png\">. ") (63 "<strong> 63.</strong> <img src=\"math-snippets/snip-472.png\">: Weak ultrafilter principle: Every infinite set has a non-trivial ultrafilter.") (55 "<strong> 55.</strong> For all infinite cardinals <img src=\"math-snippets/snip-111.png\"> and <img src=\"math-snippets/snip-7.png\">, if <img src=\"math-snippets/snip-463.png\"> then, <img src=\"math-snippets/snip-464.png\"> or <img src=\"math-snippets/snip-465.png\">.") (252 "<strong> 252.</strong> The additive groups of <img src=\"math-snippets/snip-482.png\"> and <img src=\"math-snippets/snip-142.png\"> are isomorphic. ") (12 "<strong> 12.</strong> A Form of Restricted Choice for Families of Finite Sets: For every infinite set <img src=\"math-snippets/snip-3.png\"> and every <img src=\"math-snippets/snip-42.png\">, there is an infinite subset <img src=\"math-snippets/snip-32.png\"> of <img src=\"math-snippets/snip-3.png\"> such the set of all <img src=\"math-snippets/snip-7.png\"> element subsets of <img src=\"math-snippets/snip-32.png\"> has a choice function. ") (112 "<strong> 112.</strong> <img src=\"math-snippets/snip-483.png\">: For every family <img src=\"math-snippets/snip-13.png\"> of non-empty sets each of which can be linearly ordered there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-477.png\">, <img src=\"math-snippets/snip-478.png\"> is a non-empty finite subset of <img src=\"math-snippets/snip-28.png\">.") (28 "<strong> 28<img src=\"math-snippets/snip-57.png\">.</strong> (Where <img src=\"math-snippets/snip-34.png\"> is a prime) AL20(<img src=\"math-snippets/snip-335.png\">): Every vector space <img src=\"math-snippets/snip-97.png\"> over <img src=\"math-snippets/snip-335.png\"> has the property that every linearly independent subset can be extended to a basis. (<img src=\"math-snippets/snip-335.png\"> is the <img src=\"math-snippets/snip-34.png\"> element field.)") (2 "<strong> 2.</strong> Existence of successor cardinals: For every cardinal <img src=\"math-snippets/snip-111.png\"> there is a cardinal <img src=\"math-snippets/snip-7.png\"> such that <img src=\"math-snippets/snip-484.png\"> and <img src=\"math-snippets/snip-485.png\">. ") (208 "<strong> 208.</strong> For all ordinals <img src=\"math-snippets/snip-153.png\">, <img src=\"math-snippets/snip-479.png\">. ") (126 "<strong> 126.</strong> <img src=\"math-snippets/snip-476.png\">, Countable axiom of multiple choice: For every denumerable set <img src=\"math-snippets/snip-13.png\"> of non-empty sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-477.png\">, <img src=\"math-snippets/snip-478.png\"> is a non-empty finite subset of <img src=\"math-snippets/snip-28.png\">.") (149 "<strong> 149.</strong> <img src=\"math-snippets/snip-505.png\">: Every <img src=\"math-snippets/snip-61.png\"> topological space is a continuous, finite to one image of an A1 space.") (371 "<strong> 371.</strong> There is an infinite, compact, Hausdorff, extremally disconnected topological space. ") (266 "<strong> 266.</strong> <img src=\"math-snippets/snip-492.png\">: Every antisymmetric relation contains <img src=\"math-snippets/snip-48.png\">-maximal partially ordered subset.") (192 "<strong> 192.</strong> <img src=\"math-snippets/snip-493.png\"> sets: For every set <img src=\"math-snippets/snip-3.png\"> there is a projective set <img src=\"math-snippets/snip-13.png\"> and a function from <img src=\"math-snippets/snip-13.png\"> onto <img src=\"math-snippets/snip-3.png\">. ") (54 "<strong> 54.</strong> For all infinite cardinals <img src=\"math-snippets/snip-111.png\">, <img src=\"math-snippets/snip-111.png\"> adj <img src=\"math-snippets/snip-494.png\"> implies <img src=\"math-snippets/snip-111.png\"> is an aleph. (<img src=\"math-snippets/snip-111.png\"> <i> adj</i> <img src=\"math-snippets/snip-7.png\"> iff <img src=\"math-snippets/snip-484.png\"> and <img src=\"math-snippets/snip-495.png\">.) ") (87 "<strong> 87<img src=\"math-snippets/snip-19.png\">.</strong> <img src=\"math-snippets/snip-486.png\">: Given a relation <img src=\"math-snippets/snip-31.png\"> such that for every subset <img src=\"math-snippets/snip-65.png\"> of a set <img src=\"math-snippets/snip-13.png\"> with <img src=\"math-snippets/snip-487.png\">, there is an <img src=\"math-snippets/snip-15.png\"> with <img src=\"math-snippets/snip-488.png\"> then there is a function <img src=\"math-snippets/snip-489.png\"> such that (<img src=\"math-snippets/snip-490.png\">) <img src=\"math-snippets/snip-491.png\">.") (232 "<strong> 232.</strong> Every metric space <img src=\"math-snippets/snip-518.png\"> has a <img src=\"math-snippets/snip-270.png\">-point finite base. ") (79 "<strong> 79.</strong> <img src=\"math-snippets/snip-9.png\"> can be well ordered. ") (408 "<strong> 408.</strong> If <img src=\"math-snippets/snip-527.png\"> is a family of functions such that for each <img src=\"math-snippets/snip-528.png\">, <img src=\"math-snippets/snip-529.png\">, where <img src=\"math-snippets/snip-218.png\"> and <img src=\"math-snippets/snip-170.png\"> are non-empty sets, and <img src=\"math-snippets/snip-530.png\"> is a filter base on <img src=\"math-snippets/snip-224.png\"> such that <font class=\"enum\">1.</font> For all <img src=\"math-snippets/snip-531.png\"> and all finite <img src=\"math-snippets/snip-532.png\"> there is an <img src=\"math-snippets/snip-528.png\"> such that <img src=\"math-snippets/snip-533.png\"> is defined on <img src=\"math-snippets/snip-50.png\">, and \\itemitem{2.} For all <img src=\"math-snippets/snip-534.png\"> and all finite <img src=\"math-snippets/snip-532.png\"> there exist at most finitely many functions on <img src=\"math-snippets/snip-50.png\"> which are restrictions of the functions <img src=\"math-snippets/snip-533.png\"> with <img src=\"math-snippets/snip-528.png\">, \\noindent then there is a function <img src=\"math-snippets/snip-14.png\"> with domain <img src=\"math-snippets/snip-218.png\"> such that for each finite <img src=\"math-snippets/snip-532.png\"> and each <img src=\"math-snippets/snip-531.png\"> there is an <img src=\"math-snippets/snip-528.png\"> such that <img src=\"math-snippets/snip-535.png\">.") (345 "<strong> 345.</strong> Rasiowa-Sikorski Axiom: If <img src=\"math-snippets/snip-508.png\"> is a Boolean algebra, <img src=\"math-snippets/snip-211.png\"> is a non-zero element of <img src=\"math-snippets/snip-32.png\">, and <img src=\"math-snippets/snip-509.png\"> is a denumerable set of subsets of <img src=\"math-snippets/snip-32.png\"> then there is a maximal filter <img src=\"math-snippets/snip-50.png\"> of <img src=\"math-snippets/snip-32.png\"> such that <img src=\"math-snippets/snip-510.png\"> and for each <img src=\"math-snippets/snip-42.png\">, if <img src=\"math-snippets/snip-511.png\"> and <img src=\"math-snippets/snip-512.png\"> exists then <img src=\"math-snippets/snip-513.png\">.") (11 "<strong> 11.</strong> A Form of Restricted Choice for Families of Finite Sets: For every infinite set <img src=\"math-snippets/snip-3.png\">, <img src=\"math-snippets/snip-3.png\"> has an infinite subset <img src=\"math-snippets/snip-32.png\"> such that for every <img src=\"math-snippets/snip-42.png\">, <img src=\"math-snippets/snip-496.png\">, the set of all <img src=\"math-snippets/snip-7.png\"> element subsets of <img src=\"math-snippets/snip-32.png\"> has a choice function. ") (357 "<strong> 357.</strong> <img src=\"math-snippets/snip-498.png\">, The Kinna-Wagner Selection Principle for a denumerable family of denumerable sets: For every denumerable set <img src=\"math-snippets/snip-88.png\"> of denumerable sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-137.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (211 "<strong> 211.</strong> <img src=\"math-snippets/snip-499.png\">: Dependent choice for relations on <img src=\"math-snippets/snip-142.png\">: If <img src=\"math-snippets/snip-500.png\"> satisfies <img src=\"math-snippets/snip-501.png\"> then there is a sequence <img src=\"math-snippets/snip-502.png\"> of real numbers such that <img src=\"math-snippets/snip-503.png\">.") (302 "<strong> 302.</strong> Any continuous surjection between compact Hausdorff spaces has an irreducible restriction to a closed subset of its domain. ") (165 "<strong> 165.</strong> <img src=\"math-snippets/snip-504.png\">: Every well ordered family of non-empty, well orderable sets has a choice function. ") (249 "<strong> 249.</strong> If <img src=\"math-snippets/snip-345.png\"> is an infinite tree in which every element has exactly 2 immediate successors then <img src=\"math-snippets/snip-345.png\"> has an infinite branch. ") (352 "<strong> 352.</strong> A countable product of second countable spaces is second countable.") (61 "<strong> 61.</strong> <img src=\"math-snippets/snip-506.png\">)<img src=\"math-snippets/snip-507.png\">: For each <img src=\"math-snippets/snip-42.png\">, <img src=\"math-snippets/snip-6.png\">, every set of <img src=\"math-snippets/snip-7.png\"> element sets has a choice function. ") (49 "<strong> 49.</strong> Order Extension Principle: Every partial ordering can be extended to a linear ordering. ") (227 "<strong> 227.</strong> For all groups <img src=\"math-snippets/snip-2.png\">, if every finite subgroup of <img src=\"math-snippets/snip-2.png\"> can be fully ordered then <img src=\"math-snippets/snip-2.png\"> can be fully ordered.") (244 "<strong> 244.</strong> Every principal ideal domain has a maximal ideal.") (240 "<strong> 240.</strong> If a group <img src=\"math-snippets/snip-2.png\"> satisfies ``every ascending chain of subgroups is finite,\" then every subgroup of <img src=\"math-snippets/snip-2.png\"> is finitely generated.") (399 "<strong> 399.</strong> <img src=\"math-snippets/snip-514.png\">, The Kinna-Wagner Selection Principle for a set of linearly orderable sets: For every set of linearly orderable sets <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (88 "<strong> 88.</strong> <img src=\"math-snippets/snip-515.png\">: Every family of pairs has a choice function. ") (133 "<strong> 133.</strong> Every set is either well orderable or has an infinite amorphous subset. ") (269 "<strong> 269.</strong> For every cardinal <img src=\"math-snippets/snip-111.png\">, there is a set <img src=\"math-snippets/snip-3.png\"> such that <img src=\"math-snippets/snip-516.png\"> and there is a choice function on the collection of 2-element subsets of <img src=\"math-snippets/snip-3.png\">. ") (257 "<strong> 257.</strong> <img src=\"math-snippets/snip-517.png\">: Every transitive relation <img src=\"math-snippets/snip-47.png\"> in which every partially ordered subset has an upper bound, has a maximal element. ") (66 "<strong> 66.</strong> Every vector space over a field has a basis.") (372 "<strong> 372.</strong> Generalized Hahn-Banach Theorem: Assume that <img src=\"math-snippets/snip-13.png\"> is a real vector space, <img src=\"math-snippets/snip-519.png\"> is a Dedekind complete ordered vector space and <img src=\"math-snippets/snip-520.png\"> is a subspace of <img src=\"math-snippets/snip-13.png\">. If <img src=\"math-snippets/snip-521.png\"> is linear and <img src=\"math-snippets/snip-522.png\"> is sublinear and if <img src=\"math-snippets/snip-523.png\"> on <img src=\"math-snippets/snip-520.png\"> then <img src=\"math-snippets/snip-524.png\"> can be extended to a linear map <img src=\"math-snippets/snip-525.png\"> such that <img src=\"math-snippets/snip-526.png\"> on <img src=\"math-snippets/snip-13.png\">.") (77 "<strong> 77.</strong> A linear ordering of a set <img src=\"math-snippets/snip-110.png\"> is a well ordering if and only if <img src=\"math-snippets/snip-110.png\"> has no infinite descending sequences.") (127 "<strong> 127.</strong> An amorphous power of a compact <img src=\"math-snippets/snip-61.png\"> space, which as a set is well orderable, is well orderable. ") (304 "<strong> 304.</strong> There does not exist a <img src=\"math-snippets/snip-61.png\"> topological space <img src=\"math-snippets/snip-13.png\"> such that every infinite subset of <img src=\"math-snippets/snip-13.png\"> contains an infinite compact subset.") (362 "<strong> 362.</strong> In <img src=\"math-snippets/snip-142.png\">, every Borel set is analytic. ") (93 "<strong> 93.</strong> There is a non-measurable subset of <img src=\"math-snippets/snip-9.png\">.") (69 "<strong> 69.</strong> Every field has an algebraic closure. ") (57 "<strong> 57.</strong> If <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-28.png\"> are Dedekind finite sets then either <img src=\"math-snippets/snip-536.png\"> or <img src=\"math-snippets/snip-537.png\">. ") (412 "<strong> 412.</strong> RCh (Reflexive Compactness for Hilbert spaces): The closed unit ball of a Hilbert space is compact for the weak topology.") (125 "<strong> 125.</strong> There does not exist an infinite, compact connected <img src=\"math-snippets/snip-34.png\"> space. (A <img src=\"math-snippets/snip-34.png\"> <i> space</i> is a <img src=\"math-snippets/snip-61.png\"> space in which the intersection of any well orderable family of open sets is open.)") (188 "<strong> 188.</strong> <img src=\"math-snippets/snip-538.png\">: For every Abelian group <img src=\"math-snippets/snip-3.png\"> there is a projective Abelian group <img src=\"math-snippets/snip-2.png\"> and a homomorphism from <img src=\"math-snippets/snip-2.png\"> onto <img src=\"math-snippets/snip-3.png\">.") (139 "<strong> 139.</strong> Using the discrete topology on 2, <img src=\"math-snippets/snip-539.png\"> is compact.") (83 "<strong> 83.</strong> <img src=\"math-snippets/snip-549.png\"> (see <font class=\"author\">Howard/Yorke</font> <font class=\"year\">1989</font>): <img src=\"math-snippets/snip-345.png\">-finite is equivalent to finite.") (195 "<strong> 195.</strong> Every general linear system has a linear global reaction. ") (274 "<strong> 274.</strong> There is a cardinal number <img src=\"math-snippets/snip-17.png\"> and an <img src=\"math-snippets/snip-42.png\"> such that <img src=\"math-snippets/snip-540.png\"> adj<img src=\"math-snippets/snip-541.png\">. (The expression ``<img src=\"math-snippets/snip-17.png\"> adj<img src=\"math-snippets/snip-542.png\"> means there are cardinals <img src=\"math-snippets/snip-543.png\"> such that <img src=\"math-snippets/snip-544.png\"> and <img src=\"math-snippets/snip-545.png\"> and for all <img src=\"math-snippets/snip-546.png\"> and if <img src=\"math-snippets/snip-547.png\">, then <img src=\"math-snippets/snip-548.png\"> (Compare with [0 A]).") (23 "<strong> 23.</strong> <img src=\"math-snippets/snip-550.png\">: For every ordinal <img src=\"math-snippets/snip-153.png\">, if <img src=\"math-snippets/snip-3.png\"> and every member of <img src=\"math-snippets/snip-3.png\"> has cardinality <img src=\"math-snippets/snip-20.png\">, then <img src=\"math-snippets/snip-551.png\">. ") (392 "<strong> 392.</strong> <img src=\"math-snippets/snip-577.png\">: Every linearly ordered set of linearly orderable sets has a choice function. ") (335 "<strong> 335<img src=\"math-snippets/snip-41.png\">.</strong> Every quotient group of an Abelian group each of whose elements has order <img src=\"math-snippets/snip-558.png\"> has a set of representatives.") (113 "<strong> 113.</strong> Tychonoff's Compactness Theorem for Countably Many Spaces: The product of a countable set of compact spaces is compact.") (206 "<strong> 206.</strong> The existence of a non-principal ultrafilter: There exists an infinite set <img src=\"math-snippets/snip-13.png\"> and a non-principal ultrafilter on <img src=\"math-snippets/snip-13.png\">.") (157 "<strong> 157.</strong> Theorem of Goodner: A compact <img src=\"math-snippets/snip-559.png\"> space is extremally disconnected (the closure of every open set is open) if and only if each non-empty subset of <img src=\"math-snippets/snip-560.png\"> (set of continuous real valued functions on <img src=\"math-snippets/snip-13.png\">) which is pointwise bounded has a supremum.") (184 "<strong> 184.</strong> Existence of a double uniformization: For all <img src=\"math-snippets/snip-13.png\"> and <img src=\"math-snippets/snip-65.png\">, for all <img src=\"math-snippets/snip-561.png\">, if there is an infinite cardinal <img src=\"math-snippets/snip-81.png\"> satisfying: <font class=\"enum\">(1)</font> <img src=\"math-snippets/snip-562.png\">, <img src=\"math-snippets/snip-563.png\"> and \\itemitem{(2)} <img src=\"math-snippets/snip-564.png\">, <img src=\"math-snippets/snip-565.png\">, then <img src=\"math-snippets/snip-566.png\"> such that for all <img src=\"math-snippets/snip-567.png\"> such that <img src=\"math-snippets/snip-568.png\"> and <img src=\"math-snippets/snip-569.png\"> such that <img src=\"math-snippets/snip-570.png\">. (<img src=\"math-snippets/snip-571.png\"> is called a <i> double uniformization</i> of <img src=\"math-snippets/snip-218.png\">.)") (311 "<strong> 311.</strong> Abelian groups are amenable. (<img src=\"math-snippets/snip-2.png\"> is <i> amenable</i> if there is a finitely additive measure <img src=\"math-snippets/snip-115.png\"> on <img src=\"math-snippets/snip-116.png\"> such that <img src=\"math-snippets/snip-572.png\"> and <img src=\"math-snippets/snip-391.png\">, <img src=\"math-snippets/snip-119.png\">.)") (337 "<strong> 337.</strong> <img src=\"math-snippets/snip-573.png\">: If <img src=\"math-snippets/snip-13.png\"> is a well ordered collection of non-empty sets and there is a function <img src=\"math-snippets/snip-14.png\"> defined on <img src=\"math-snippets/snip-13.png\"> such that for every <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a linear ordering of <img src=\"math-snippets/snip-17.png\">, then there is a choice function for <img src=\"math-snippets/snip-13.png\">.") (131 "<strong> 131.</strong> <img src=\"math-snippets/snip-497.png\">: For every denumerable family <img src=\"math-snippets/snip-13.png\"> of pairwise disjoint non-empty sets, there is a function <img src=\"math-snippets/snip-14.png\"> such that for each <img src=\"math-snippets/snip-15.png\">, f(x) is a non-empty countable subset of <img src=\"math-snippets/snip-17.png\">.") (405 "<strong> 405.</strong> Every infinite set can be partitioned into sets each of which is countable and has at least two elements. ") (1 "<strong> 1.</strong> <img src=\"math-snippets/snip-440.png\">: The Axiom of Choice: Every set of non-empty sets has a choice function. ") (321 "<strong> 321.</strong> There does not exist an ordinal <img src=\"math-snippets/snip-153.png\"> such that <img src=\"math-snippets/snip-20.png\"> is weakly compact and <img src=\"math-snippets/snip-80.png\"> is measurable.") (316 "<strong> 316.</strong> If a linearly ordered set <img src=\"math-snippets/snip-574.png\"> has the fixed point property then <img src=\"math-snippets/snip-574.png\"> is complete. (<img src=\"math-snippets/snip-574.png\"> has the <i> fixed point property</i> if every function <img src=\"math-snippets/snip-575.png\"> satisfying <img src=\"math-snippets/snip-576.png\"> has a fixed point, and (<img src=\"math-snippets/snip-574.png\"> is {\\it complete} if every subset of <img src=\"math-snippets/snip-3.png\"> has a least upper bound.)") (13 "<strong> 13.</strong> Every Dedekind finite subset of <img src=\"math-snippets/snip-9.png\"> is finite. ") (142 "<strong> 142.</strong> <img src=\"math-snippets/snip-578.png\">: There is a set of reals without the property of Baire. ") (19 "<strong> 19.</strong> A real function is analytically representable if and only if it is in <font class=\"icopy\">Baire's classification</font>.") (388 "<strong> 388.</strong> Every infinite branching poset (a partially ordered set in which each element has at least two lower bounds) has either an infinite chain or an infinite antichain.") (200 "<strong> 200.</strong> For all infinite <img src=\"math-snippets/snip-17.png\">, <img src=\"math-snippets/snip-580.png\">.") (322 "<strong> 322.</strong> <img src=\"math-snippets/snip-581.png\">, The Kinna-Wagner Selection Principle for a well ordered family of sets: For every well ordered set <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (424 "<strong> 424.</strong> Every Lindel&ouml;f metric space is super second countable. ") (15 "<strong> 15.</strong> <img src=\"math-snippets/snip-584.png\"> (KW), The Kinna-Wagner Selection Principle: For every set <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (378 "<strong> 378.</strong> Restricted Choice for Families of Well Ordered Sets: For every infinite set <img src=\"math-snippets/snip-13.png\"> there is an infinite subset <img src=\"math-snippets/snip-65.png\"> of <img src=\"math-snippets/snip-13.png\"> such that the family of non-empty well orderable subsets of <img src=\"math-snippets/snip-65.png\"> has a choice function. De la Cruz/Di") (81 "<strong> 81<img src=\"math-snippets/snip-41.png\">.</strong> (For <img src=\"math-snippets/snip-42.png\">) <img src=\"math-snippets/snip-586.png\">: For every set <img src=\"math-snippets/snip-104.png\"> there is an ordinal <img src=\"math-snippets/snip-153.png\"> and a one to one function <img src=\"math-snippets/snip-587.png\">. (<img src=\"math-snippets/snip-588.png\"> and <img src=\"math-snippets/snip-589.png\">. (<img src=\"math-snippets/snip-590.png\"> is equivalent to form 1 (AC) and <img src=\"math-snippets/snip-591.png\"> is equivalent to the selection principle (form 15)).") (292 "<strong> 292.</strong> <img src=\"math-snippets/snip-592.png\">: For each linearly ordered family of non-empty sets <img src=\"math-snippets/snip-13.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\"> <img src=\"math-snippets/snip-16.png\"> is non-empty, finite subset of <img src=\"math-snippets/snip-17.png\">.") (295 "<strong> 295.</strong> DO: Every infinite set has a dense linear ordering. ") (356 "<strong> 356.</strong> <img src=\"math-snippets/snip-593.png\">, The Kinna-Wagner Selection Principle for a family of denumerable sets: For every set <img src=\"math-snippets/snip-88.png\"> of denumerable sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-137.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (148 "<strong> 148.</strong> <img src=\"math-snippets/snip-579.png\">: For every <img src=\"math-snippets/snip-61.png\"> topological space <img src=\"math-snippets/snip-175.png\">, if <img src=\"math-snippets/snip-13.png\"> is well ordered, then <img src=\"math-snippets/snip-13.png\"> has a well ordered base.") (18 "<strong> 18.</strong> <img src=\"math-snippets/snip-585.png\">: The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.") (403 "<strong> 403.</strong> <img src=\"math-snippets/snip-597.png\">, The Kinna-Wagner Selection Principle for a linearly ordered set of well orderable sets: For every linearly ordered set of well orderable sets <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (261 "<strong> 261.</strong> <img src=\"math-snippets/snip-598.png\">: Every transitive relation <img src=\"math-snippets/snip-47.png\"> in which every subset which is a tree has an upper bound, has a maximal element.") (155 "<strong> 155.</strong> <img src=\"math-snippets/snip-599.png\">: There are no non-trivial L&auml;uchli continua. (A <i> L&auml;uchli continuum</i> is a strongly connected continuum. {\\it Continuum} <img src=\"math-snippets/snip-600.png\"> compact, connected, Hausdorff space; and {\\it strongly connected} <img src=\"math-snippets/snip-600.png\"> every continuous real valued function is constant.)") (72 "<strong> 72.</strong> Artin-Schreier Theorem: Every field in which <img src=\"math-snippets/snip-601.png\"> is not the sum of squares can be ordered. (The ordering, <img src=\"math-snippets/snip-197.png\">, must satisfy (a) <img src=\"math-snippets/snip-602.png\"> for all <img src=\"math-snippets/snip-603.png\"> and (b) <img src=\"math-snippets/snip-604.png\"> and <img src=\"math-snippets/snip-605.png\">c.)") (31 "<strong> 31.</strong> <img src=\"math-snippets/snip-606.png\">: The countable union theorem: The union of a denumerable set of denumerable sets is denumerable. ") (59 "<strong> 59<img src=\"math-snippets/snip-607.png\">.</strong> If <img src=\"math-snippets/snip-574.png\"> is a partial ordering that is not a well ordering, then there is no set <img src=\"math-snippets/snip-32.png\"> such that <img src=\"math-snippets/snip-608.png\"> (the usual injective cardinal ordering on <img src=\"math-snippets/snip-32.png\">) is isomorphic to <img src=\"math-snippets/snip-574.png\">. ") (205 "<strong> 205.</strong> For all cardinals <img src=\"math-snippets/snip-111.png\"> and <img src=\"math-snippets/snip-7.png\">, if <img src=\"math-snippets/snip-609.png\"> and <img src=\"math-snippets/snip-610.png\"> then there is a cardinal <img src=\"math-snippets/snip-611.png\"> such that <img src=\"math-snippets/snip-612.png\">.") (39 "<strong> 39.</strong> <img src=\"math-snippets/snip-582.png\">: Every set <img src=\"math-snippets/snip-3.png\"> of non-empty sets such that <img src=\"math-snippets/snip-583.png\"> has a choice function.") (367 "<strong> 367.</strong> There is a Hamel basis for <img src=\"math-snippets/snip-142.png\"> as a vector space over <img src=\"math-snippets/snip-29.png\">. ") (373 "<strong> 373<img src=\"math-snippets/snip-41.png\">.</strong> (For <img src=\"math-snippets/snip-42.png\">, <img src=\"math-snippets/snip-6.png\">.) <img src=\"math-snippets/snip-596.png\">: Every denumerable set of <img src=\"math-snippets/snip-7.png\">-element sets has an infinite subset with a choice function. ") (333 "<strong> 333.</strong> <img src=\"math-snippets/snip-613.png\">: For every set <img src=\"math-snippets/snip-13.png\"> of sets such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-85.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for every <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-18.png\"> is odd.") (185 "<strong> 185.</strong> Every linearly ordered Dedekind finite set is finite. ") (366 "<strong> 366.</strong> There is a discontinuous function <img src=\"math-snippets/snip-614.png\"> such that for all real <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-28.png\">, <img src=\"math-snippets/snip-615.png\">.") (24 "<strong> 24.</strong> <img src=\"math-snippets/snip-594.png\">: Every denumerable collection of non-empty sets each with power <img src=\"math-snippets/snip-595.png\"> has a choice function. ") (289 "<strong> 289.</strong> If <img src=\"math-snippets/snip-104.png\"> is a set of subsets of a countable set and <img src=\"math-snippets/snip-104.png\"> is closed under chain unions, then <img src=\"math-snippets/snip-104.png\"> has a <img src=\"math-snippets/snip-48.png\">-maximal element. ") (199 "<strong> 199<img src=\"math-snippets/snip-41.png\">.</strong> (For <img src=\"math-snippets/snip-74.png\">) If all <img src=\"math-snippets/snip-616.png\">, Dedekind finite subsets of <img src=\"math-snippets/snip-617.png\"> are finite, then all <img src=\"math-snippets/snip-618.png\"> Dedekind finite subsets of <img src=\"math-snippets/snip-619.png\"> are finite. ") (190 "<strong> 190.</strong> There is a non-trivial injective Abelian group.") (154 "<strong> 154.</strong> Tychonoff's Compactness Theorem for Countably Many <img src=\"math-snippets/snip-61.png\"> Spaces: The product of countably many <img src=\"math-snippets/snip-61.png\"> compact spaces is compact.") (27 "<strong> 27.</strong> <img src=\"math-snippets/snip-620.png\">: The union of denumerably many sets each of power <img src=\"math-snippets/snip-215.png\"> has power <img src=\"math-snippets/snip-20.png\">. ") (210 "<strong> 210.</strong> The commutator subgroup of a free group is free.") (386 "<strong> 386.</strong> Every B compact (pseudo)metric space is Baire.") (214 "<strong> 214.</strong> <img src=\"math-snippets/snip-621.png\">: For every family <img src=\"math-snippets/snip-3.png\"> of infinite sets, there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-622.png\">, <img src=\"math-snippets/snip-478.png\"> is a non-empty subset of <img src=\"math-snippets/snip-28.png\"> and <img src=\"math-snippets/snip-623.png\">.") (40 "<strong> 40.</strong> <img src=\"math-snippets/snip-624.png\">: Every well orderable set of non-empty sets has a choice function. ") (38 "<strong> 38.</strong> <img src=\"math-snippets/snip-9.png\"> is not the union of a countable family of countable sets. ") (415 "<strong> 415.</strong> Every <img src=\"math-snippets/snip-44.png\">-compactly generated complete lattice is algebraic. ") (400 "<strong> 400.</strong> <img src=\"math-snippets/snip-625.png\">, The Kinna-Wagner Selection Principle for a linearly ordered set of linearly orderable sets: For every linearly ordered set of linearly orderable sets <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (202 "<strong> 202.</strong> <img src=\"math-snippets/snip-626.png\">: Every linearly ordered family of non-empty sets has a choice function. ") (384 "<strong> 384.</strong> Closed Filter Extendability for <img src=\"math-snippets/snip-300.png\"> Spaces. Every closed filter in a <img src=\"math-snippets/snip-300.png\"> topological space can be extended to a maximal closed filter.") (223 "<strong> 223.</strong> There is an infinite set <img src=\"math-snippets/snip-13.png\"> and a non-principal measure on <img src=\"math-snippets/snip-30.png\">. ") (150 "<strong> 150.</strong> <img src=\"math-snippets/snip-627.png\">: Every infinite set of denumerable sets has an infinite subset with a choice function. ") (238 "<strong> 238.</strong> Every elementary Abelian group (that is, for some prime <img src=\"math-snippets/snip-34.png\"> every non identity element has order <img src=\"math-snippets/snip-34.png\">) is the direct sum of cyclic subgroups. ") (129 "<strong> 129.</strong> For every infinite set <img src=\"math-snippets/snip-3.png\">, <img src=\"math-snippets/snip-3.png\"> admits a partition into sets of order type <img src=\"math-snippets/snip-628.png\">. (For every infinite <img src=\"math-snippets/snip-3.png\">, there is a set <img src=\"math-snippets/snip-629.png\"> such that <img src=\"math-snippets/snip-630.png\"> is a partition of <img src=\"math-snippets/snip-3.png\"> and for each <img src=\"math-snippets/snip-631.png\">, <img src=\"math-snippets/snip-632.png\"> is an ordering of <img src=\"math-snippets/snip-633.png\"> of type <img src=\"math-snippets/snip-634.png\">.) ") (231 "<strong> 231.</strong> <img src=\"math-snippets/snip-635.png\">: The union of a well ordered collection of well orderable sets is well orderable.") (332 "<strong> 332.</strong> A product of non-empty compact sober topological spaces is non-empty. ") (102 "<strong> 102.</strong> For all Dedekind finite cardinals <img src=\"math-snippets/snip-34.png\"> and <img src=\"math-snippets/snip-33.png\">, if <img src=\"math-snippets/snip-636.png\"> then <img src=\"math-snippets/snip-637.png\">.") (21 "<strong> 21.</strong> If <img src=\"math-snippets/snip-104.png\"> is well ordered, <img src=\"math-snippets/snip-258.png\"> and <img src=\"math-snippets/snip-259.png\"> are families of pairwise disjoint sets, and <img src=\"math-snippets/snip-260.png\"> for all <img src=\"math-snippets/snip-261.png\">, then <img src=\"math-snippets/snip-638.png\">. ") (395 "<strong> 395.</strong> <img src=\"math-snippets/snip-639.png\">: For each linearly ordered family of non-empty linearly orderable sets <img src=\"math-snippets/snip-13.png\">, there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\"> <img src=\"math-snippets/snip-16.png\"> is a non-empty, finite subset of <img src=\"math-snippets/snip-17.png\">. ") (94 "<strong> 94.</strong> <img src=\"math-snippets/snip-640.png\">: Every denumerable family of non-empty sets of reals has a choice function.") (158 "<strong> 158.</strong> In every Hilbert space <img src=\"math-snippets/snip-1.png\">, if the closed unit ball is sequentially compact, then <img src=\"math-snippets/snip-1.png\"> has an orthonormal basis.") (318 "<strong> 318.</strong> <img src=\"math-snippets/snip-313.png\"> is not measurable.") (45 "<strong> 45<img src=\"math-snippets/snip-41.png\">.</strong> If <img src=\"math-snippets/snip-74.png\">, <img src=\"math-snippets/snip-641.png\">: Every set of <img src=\"math-snippets/snip-7.png\">-element sets has a choice function.") (114 "<strong> 114.</strong> Every A-bounded <img src=\"math-snippets/snip-61.png\"> topological space is weakly Loeb. (<i> A-bounded</i> means amorphous subsets are relatively compact. {\\it Weakly Loeb} means the set of non-empty closed subsets has a multiple choice function.)") (169 "<strong> 169.</strong> There is an uncountable subset of <img src=\"math-snippets/snip-9.png\"> without a perfect subset.") (132 "<strong> 132.</strong> <img src=\"math-snippets/snip-642.png\">: Every infinite family of finite sets has an infinite subfamily with a choice function. ") (197 "<strong> 197.</strong> <img src=\"math-snippets/snip-643.png\"> is the union of three sets <img src=\"math-snippets/snip-282.png\"> with the property that for all <img src=\"math-snippets/snip-644.png\"> there is a straight line <img src=\"math-snippets/snip-645.png\"> such that <img src=\"math-snippets/snip-646.png\">. ") (233 "<strong> 233.</strong> If a field has an algebraic closure it is unique up to isomorphism.") (326 "<strong> 326.</strong> 2-SAT: Restricted Compactness Theorem for Propositional Logic III: If <img src=\"math-snippets/snip-647.png\"> is a set of formulas in a propositional language such that every finite subset of <img src=\"math-snippets/snip-647.png\"> is satisfiable and if every formula in <img src=\"math-snippets/snip-647.png\"> is a disjunction of at most two literals, then <img src=\"math-snippets/snip-647.png\"> is satisfiable. (A <i> literal</i> is a propositional variable or its negation.)") (97 "<strong> 97.</strong> Cardinal Representatives: For every set <img src=\"math-snippets/snip-3.png\"> there is a function <img src=\"math-snippets/snip-603.png\"> with domain <img src=\"math-snippets/snip-437.png\"> such that for all <img src=\"math-snippets/snip-648.png\">, (i) <img src=\"math-snippets/snip-649.png\"> and (ii) <img src=\"math-snippets/snip-650.png\">. ") (107 "<strong> 107.</strong> M.~Hall's Theorem: Let <img src=\"math-snippets/snip-651.png\"> be a collection of finite subsets (of a set <img src=\"math-snippets/snip-13.png\">) then if for each finite <img src=\"math-snippets/snip-652.png\"> there is an injective choice function on <img src=\"math-snippets/snip-50.png\">(<img src=\"math-snippets/snip-653.png\">) then there is an injective choice function on <img src=\"math-snippets/snip-3.png\">. (That is, a 1-1 function <img src=\"math-snippets/snip-14.png\"> such that <img src=\"math-snippets/snip-654.png\">.) (According to a theorem of P.~Hall (<img src=\"math-snippets/snip-653.png\">) is equivalent to <img src=\"math-snippets/snip-655.png\">. P.~Hall's theorem does not require the axiom of choice.) ") (14 "<strong> 14.</strong> BPI: Every Boolean algebra has a prime ideal.") (228 "<strong> 228.</strong> Every torsion free Abelian group can be fully ordered.") (293 "<strong> 293.</strong> For all sets <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-28.png\">, if <img src=\"math-snippets/snip-17.png\"> can be linearly ordered and there is a mapping of <img src=\"math-snippets/snip-17.png\"> onto <img src=\"math-snippets/snip-28.png\">, then <img src=\"math-snippets/snip-28.png\"> can be linearly ordered. ") (180 "<strong> 180.</strong> Every Abelian group has a divisible hull. (If <img src=\"math-snippets/snip-3.png\"> and <img src=\"math-snippets/snip-32.png\"> are groups, <i> <img src=\"math-snippets/snip-32.png\"> is a divisible hull of <img src=\"math-snippets/snip-3.png\"></i> means <img src=\"math-snippets/snip-32.png\"> is a divisible group, <img src=\"math-snippets/snip-3.png\"> is a subgroup of <img src=\"math-snippets/snip-32.png\"> and for every non-zero <img src=\"math-snippets/snip-656.png\">, <img src=\"math-snippets/snip-657.png\"> such that <img src=\"math-snippets/snip-658.png\">.) ") (96 "<strong> 96.</strong> L&ouml;wig's Theorem. If <img src=\"math-snippets/snip-338.png\"> and <img src=\"math-snippets/snip-339.png\"> are both bases for the vector space <img src=\"math-snippets/snip-97.png\"> then <img src=\"math-snippets/snip-659.png\">.") (247 "<strong> 247.</strong> Every atomless Boolean algebra is Dedekind infinite. ") (67 "<strong> 67.</strong> <img src=\"math-snippets/snip-660.png\"> (MC), The Axiom of Multiple Choice: For every set <img src=\"math-snippets/snip-88.png\"> of non-empty sets there is a function <img src=\"math-snippets/snip-14.png\"> such that <img src=\"math-snippets/snip-661.png\"> and <img src=\"math-snippets/snip-16.png\"> is finite). ") (9 "<strong> 9.</strong> Finite <img src=\"math-snippets/snip-662.png\"> Dedekind finite: <img src=\"math-snippets/snip-663.png\"> (see <font class=\"author\">Jech</font> <font class=\"year\">1973b</font>): <img src=\"math-snippets/snip-664.png\"> (\\ac{Howard/Yorke} \\cite{1989}): Every Dedekind finite set is finite.") (207 "<strong> 207<img src=\"math-snippets/snip-19.png\">.</strong> <img src=\"math-snippets/snip-665.png\">: The union of <img src=\"math-snippets/snip-20.png\"> sets each of cardinality <img src=\"math-snippets/snip-20.png\"> has cardinality less than <img src=\"math-snippets/snip-666.png\">. ") (253 "<strong> 253.</strong> &#321;o&#347;' Theorem: If <img src=\"math-snippets/snip-667.png\"> is a relational system, <img src=\"math-snippets/snip-13.png\"> any set and <img src=\"math-snippets/snip-668.png\"> an ultrafilter in <img src=\"math-snippets/snip-332.png\">, then <img src=\"math-snippets/snip-88.png\"> and <img src=\"math-snippets/snip-669.png\"> are elementarily equivalent. ") (135 "<strong> 135.</strong> If <img src=\"math-snippets/snip-13.png\"> is a <img src=\"math-snippets/snip-61.png\"> space with at least two points and <img src=\"math-snippets/snip-301.png\"> is hereditarily metacompact then <img src=\"math-snippets/snip-65.png\"> is countable. (A space is <i> metacompact</i> if every open cover has an open point finite refinement. If <img src=\"math-snippets/snip-32.png\"> and <img src=\"math-snippets/snip-385.png\"> are covers of a space <img src=\"math-snippets/snip-13.png\">, then <img src=\"math-snippets/snip-385.png\"> is a {\\it refinement} of <img src=\"math-snippets/snip-32.png\"> if <img src=\"math-snippets/snip-670.png\">. <img src=\"math-snippets/snip-32.png\"> is {\\it point finite} if <img src=\"math-snippets/snip-671.png\"> there are only finitely many <img src=\"math-snippets/snip-672.png\"> such that <img src=\"math-snippets/snip-673.png\">.) ") (71 "<strong> 71<img src=\"math-snippets/snip-19.png\">.</strong> <img src=\"math-snippets/snip-674.png\">: <img src=\"math-snippets/snip-675.png\"> or <img src=\"math-snippets/snip-676.png\">.") (105 "<strong> 105.</strong> There is a partially ordered set <img src=\"math-snippets/snip-574.png\"> such that for no set <img src=\"math-snippets/snip-32.png\"> is <img src=\"math-snippets/snip-608.png\"> (the ordering on <img src=\"math-snippets/snip-32.png\"> is the usual injective cardinal ordering) isomorphic to <img src=\"math-snippets/snip-574.png\">.") (430 "<strong> 430<img src=\"math-snippets/snip-57.png\">.</strong> (Where <img src=\"math-snippets/snip-34.png\"> is a prime) AL21<img src=\"math-snippets/snip-677.png\">: Every vector space over <img src=\"math-snippets/snip-335.png\"> has the property that for every subspace <img src=\"math-snippets/snip-104.png\"> of <img src=\"math-snippets/snip-97.png\">, there is a subspace <img src=\"math-snippets/snip-678.png\"> of <img src=\"math-snippets/snip-97.png\"> such that <img src=\"math-snippets/snip-679.png\"> and <img src=\"math-snippets/snip-680.png\"> generates <img src=\"math-snippets/snip-97.png\"> in other words such that <img src=\"math-snippets/snip-681.png\">. ") (297 "<strong> 297.</strong> Extremally disconnected compact Hausdorff spaces are projective in the category of all compact Hausdorff spaces.") (217 "<strong> 217.</strong> Every infinite partially ordered set has either an infinite chain or an infinite antichain.") (375 "<strong> 375.</strong> Tietze-Urysohn Extension Theorem: If <img src=\"math-snippets/snip-175.png\"> is a normal topological space, <img src=\"math-snippets/snip-3.png\"> is closed in <img src=\"math-snippets/snip-13.png\">, and <img src=\"math-snippets/snip-682.png\"> is continuous, then there exists a continuous function <img src=\"math-snippets/snip-683.png\"> which extends <img src=\"math-snippets/snip-14.png\">.") (171 "<strong> 171.</strong> If <img src=\"math-snippets/snip-684.png\"> is a partial order such that <img src=\"math-snippets/snip-110.png\"> is the denumerable union of finite sets and all antichains in <img src=\"math-snippets/snip-110.png\"> are finite then for each denumerable family <img src=\"math-snippets/snip-108.png\"> of dense sets there is a <img src=\"math-snippets/snip-108.png\"> generic filter. ") (99 "<strong> 99.</strong> Rado's Selection Lemma: Let <img src=\"math-snippets/snip-685.png\"> be a family of finite subsets (of <img src=\"math-snippets/snip-13.png\">) and suppose for each finite <img src=\"math-snippets/snip-686.png\"> there is a function <img src=\"math-snippets/snip-687.png\"> such that <img src=\"math-snippets/snip-688.png\">. Then there is an <img src=\"math-snippets/snip-689.png\"> such that for every finite <img src=\"math-snippets/snip-686.png\"> there is a finite <img src=\"math-snippets/snip-345.png\"> such that <img src=\"math-snippets/snip-690.png\"> and such that <img src=\"math-snippets/snip-14.png\"> and <img src=\"math-snippets/snip-691.png\"> agree on S.") (176 "<strong> 176.</strong> Every infinite, locally finite group has an infinite Abelian subgroup. (<i> Locally finite</i> means every finite subset generates a finite subgroup.) ") (218 "<strong> 218.</strong> <img src=\"math-snippets/snip-692.png\">, relatively prime to <img src=\"math-snippets/snip-7.png\">): <img src=\"math-snippets/snip-693.png\">, if <img src=\"math-snippets/snip-13.png\"> is a set of non-empty sets, then there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a non-empty, finite subset of <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-18.png\"> is relatively prime to <img src=\"math-snippets/snip-7.png\">. ") (379 "<strong> 379.</strong> <img src=\"math-snippets/snip-694.png\">: For every infinite family <img src=\"math-snippets/snip-13.png\"> of sets each of which has at least two elements, there is an infinite subfamily <img src=\"math-snippets/snip-65.png\"> of <img src=\"math-snippets/snip-13.png\"> and a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-695.png\">, <img src=\"math-snippets/snip-478.png\"> is a non-empty proper subset of <img src=\"math-snippets/snip-28.png\">. De la Cruz/Di") (248 "<strong> 248.</strong> For any <img src=\"math-snippets/snip-81.png\">, <img src=\"math-snippets/snip-81.png\"> is the cardinal number of an infinite complete Boolean algebra if and only if <img src=\"math-snippets/snip-696.png\">. ") (410 "<strong> 410.</strong> RC (Reflexive Compactness): The closed unit ball of a reflexive normed space is compact for the weak topology.") (3 "<strong> 3.</strong> <img src=\"math-snippets/snip-697.png\">: For all infinite cardinals <img src=\"math-snippets/snip-111.png\">, <img src=\"math-snippets/snip-697.png\">.") (291 "<strong> 291.</strong> For all infinite <img src=\"math-snippets/snip-17.png\">, <img src=\"math-snippets/snip-698.png\">.") (402 "<strong> 402.</strong> <img src=\"math-snippets/snip-699.png\">, The Kinna-Wagner Selection Principle for a well ordered set of linearly orderable sets: For every well ordered set of linearly orderable sets <img src=\"math-snippets/snip-88.png\"> there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-89.png\">, if <img src=\"math-snippets/snip-90.png\"> then <img src=\"math-snippets/snip-91.png\">. ") (5 "<strong> 5.</strong> <img src=\"math-snippets/snip-700.png\">: Every denumerable set of non-empty denumerable subsets of <img src=\"math-snippets/snip-9.png\"> has a choice function.") (391 "<strong> 391.</strong> <img src=\"math-snippets/snip-172.png\">: Every set of non-empty linearly orderable sets has a choice function. ") (381 "<strong> 381.</strong> DUM: The disjoint union of metrizable spaces is metrizable.") (320 "<strong> 320.</strong> No successor cardinal, <img src=\"math-snippets/snip-80.png\">, is measurable.") (349 "<strong> 349.</strong> <img src=\"math-snippets/snip-701.png\">: For every set <img src=\"math-snippets/snip-13.png\"> of non-empty denumerable sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\">. ") (100 "<strong> 100.</strong> Weak Partition Principle: For all sets <img src=\"math-snippets/snip-17.png\"> and <img src=\"math-snippets/snip-28.png\">, if <img src=\"math-snippets/snip-704.png\">, then it is not the case that <img src=\"math-snippets/snip-705.png\">. ") (416 "<strong> 416.</strong> Every non-compact topological space <img src=\"math-snippets/snip-104.png\"> is the union of a set that is well-ordered by inclusion and consists of open proper subsets of <img src=\"math-snippets/snip-104.png\">. ") (270 "<strong> 270.</strong> <img src=\"math-snippets/snip-706.png\">: The compactness theorem for propositional logic restricted to sets of formulas in which each variable occurs only in a finite number of formulas.") (331 "<strong> 331.</strong> If <img src=\"math-snippets/snip-707.png\"> is a family of compact non-empty topological spaces then there is a family <img src=\"math-snippets/snip-708.png\"> such that <img src=\"math-snippets/snip-194.png\">, <img src=\"math-snippets/snip-709.png\"> is an irreducible closed subset of <img src=\"math-snippets/snip-710.png\">. ") (280 "<strong> 280.</strong> There is a complete separable metric space with a subset which does not have the Baire property.") (186 "<strong> 186.</strong> Every pair of cardinal numbers has a least upper bound (in the usual cardinal ordering.) ") (425 "<strong> 425.</strong> For every first countable topological space <img src=\"math-snippets/snip-711.png\"> there is a family <img src=\"math-snippets/snip-712.png\"> such that <img src=\"math-snippets/snip-713.png\">, <img src=\"math-snippets/snip-714.png\"> countable local base at <img src=\"math-snippets/snip-17.png\">. ") (350 "<strong> 350.</strong> <img src=\"math-snippets/snip-715.png\">: For every denumerable set <img src=\"math-snippets/snip-13.png\"> of non-empty denumerable sets there is a function <img src=\"math-snippets/snip-14.png\"> such that for all <img src=\"math-snippets/snip-15.png\">, <img src=\"math-snippets/snip-16.png\"> is a finite, non-empty subset of <img src=\"math-snippets/snip-17.png\">.") (296 "<strong> 296.</strong> Part-<img src=\"math-snippets/snip-716.png\">: Every infinite set is the disjoint union of infinitely many infinite sets. ") (417 "<strong> 417.</strong> On every non-trivial Banach space there is a non-trivial linear functional (bounded or unbounded). ") (406 "<strong> 406.</strong> The product of compact Hausdorf spaces is countably compact.") (268 "<strong> 268.</strong> If <img src=\"math-snippets/snip-717.png\"> is a lattice isomorphic to the lattice of subalgebras of some unary universal algebra (a unary universal algebra is one with only unary or nullary operations) and <img src=\"math-snippets/snip-153.png\"> is an automorphism of <img src=\"math-snippets/snip-717.png\"> of order 2 (that is, <img src=\"math-snippets/snip-718.png\"> is the identity) then there is a unary algebra <img src=\"math-snippets/snip-719.png\"> and an isomorphism <img src=\"math-snippets/snip-269.png\"> from <img src=\"math-snippets/snip-717.png\"> onto the lattice of subalgebras of <img src=\"math-snippets/snip-720.png\"> with <img src=\"math-snippets/snip-721.png\"> (<img src=\"math-snippets/snip-722.png\">) for all <img src=\"math-snippets/snip-723.png\">.") (84 "<strong> 84.</strong> <img src=\"math-snippets/snip-724.png\"> (see <font class=\"author\">Howard/Yorke</font> <font class=\"year\">1989</font>): <img src=\"math-snippets/snip-725.png\"> is <img src=\"math-snippets/snip-345.png\">-finite if and only if <img src=\"math-snippets/snip-726.png\"> is Dedekind finite). ") (108 "<strong> 108.</strong> There is an ordinal <img src=\"math-snippets/snip-153.png\"> such that <img src=\"math-snippets/snip-727.png\"> is not the union of a denumerable set of denumerable sets.") (151 "<strong> 151.</strong> <img src=\"math-snippets/snip-702.png\"> (<img src=\"math-snippets/snip-703.png\">): The union of a well ordered set of denumerable sets is well orderable.") (239 "<strong> 239.</strong> AL20(<img src=\"math-snippets/snip-729.png\">): Every vector <img src=\"math-snippets/snip-97.png\"> space over <img src=\"math-snippets/snip-729.png\"> has the property that every linearly independent subset of <img src=\"math-snippets/snip-97.png\"> can be extended to a basis.") (283 "<strong> 283.</strong> Cardinality of well ordered subsets: For all <img src=\"math-snippets/snip-42.png\"> and for all infinite <img src=\"math-snippets/snip-17.png\">, <img src=\"math-snippets/snip-730.png\"> where <img src=\"math-snippets/snip-731.png\"> is the set of all well orderable subsets of <img src=\"math-snippets/snip-17.png\">. ") (166 "<strong> 166.</strong> <img src=\"math-snippets/snip-732.png\">: Every infinite family of pairs has an infinite subfamily with a choice function.") (91 "<strong> 91.</strong> <img src=\"math-snippets/snip-733.png\">: The power set of a well ordered set can be well ordered. ") (182 "<strong> 182.</strong> There is an aleph whose cofinality is greater than <img src=\"math-snippets/snip-728.png\">. "))
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