Revision #37 → #1025 · back to history
addedAxiom of choice (informal)5602deb11b35
addedSmallest element of finite subsets of naturals78b1fae2c086
addedRussell's shoes and socks analogydc329736cd0d
addedChoice function0b6aedf4688e
addedAxiom of choice (formal axiom box)f273aebc3ddd
addedEquivalent form via Cartesian productd1c206ed2fd9
addedGeneralization to indexed families942b139777d0
addedPartition/transversal variant7049a0bfdff7
addedPowerset choice-function varianta7e709bc57ae
addedFinite collections need no choice (induction)6206ab5a045e
addedCanonical choice via explicit rulee35685ff3016
addedWell-orderable union gives choice functioncbebda80d8cd
addedSubsets of reals require choice9b02ecb1bed1
addedOpen interval has no least elementedf1a31906f7
addedWell-ordering of reals equivalent to choice function69efad273aea
addedConstruction of a non-measurable setb2c69fb40b64
addedBanach–Tarski paradox07567672b057
addedDiaconescu's theorem9f9dd2eec892
addedAC may fail with urelements (permutation models)09f10bb3e579
addedGödel: ZFC consistent if ZF consistentd39965f738b5
addedCohen: AC not a theorem of ZF (forcing)e19a3913c03c
addedAC independent of ZFc62930fc9672
addedPeano-arithmetic statements: ZF iff ZFC7289330cc3a4
addedZF + GCH implies ACdba749be7628
addedConstructibility and GCH imply AC5848a9ff4683
addedAxiom of global choice48e54edba0bd
addedGlobal choice from limitation of size855b142f8d3a
addedTarski's axiom stronger than AC3f33098b649c
addedTrichotomy of cardinalitiese86aabd8f71d
addedTarski's theorem about choice9700d40f5e7b
addedEvery surjection has a right inverse040142117195
addedDisjoint collection has transversalfdfe1eb2dfd6
addedRelation contains a functiona57e77e7822f
addedCartesian product of nonempty family is nonemptyaa5a1a06ddd0
addedMaximal disjoint subcollection meeting all members61675ef26224
addedMaximal collection with finite intersection property26a865cc8a7c
addedWell-ordering theorem6a45bf369f37
addedPowerset of an ordinal is well-orderablec51d2e5959f0
addedHausdorff maximal principle7ea65cc4a07f
addedAntichain principleb987df3722e1
addedZorn's lemma4b891c9abf93
addedKuratowski's lemma16064ad84b4c
addedTukey's lemma38dba439e447
addedKönig's theorem0a0605c97dbe
addedEvery vector space has a basisc85afe705eba
addedKrull's theorem (maximal ideal)fd123ebdd951
addedGroup structure on every nonempty set8d9e1d0ffc8b
addedEvery free abelian group is projectivee97734e3c5ec
addedBaer's criterion (divisible implies injective)bfa5f89f5b11
addedEvery set is a projective object in Set1b640343c32b
addedDual unit ball has an extreme point2eb2666c2852
addedTychonoff's theoremce0cd5c3f446
addedClosure of product equals product of closures9375939dbe31
addedMaximal consistent superset of first-order sentencesc2cef7b69463
addedLöwenheim–Skolem theoremfa3ef0f1a9e6
addedEvery connected graph has a spanning tree147ece39d68f
addedEvery small category has a skeletonbfc8cf5e1f87
addedWeakly equivalent small categories are equivalent605229977300
addedFreyd adjoint functor theorem13a517806706
addedAxiom of dependent choice (DC)63977ba7ec19
addedAxiom of countable choice (CC)94ff09c10c3c
addedUltrafilter lemma (Tarski)ecf3ece77c46
addedWell-orderability by rank below ordinal αa0658e1e0f6c
addedWell-orderability by Hartogs numberd4eec3cf236e
addedAxiom of countable choice (product form)bb6f9a47e3aa
addedAxiom of dependent choice (sequence form)9b07fd79410d
addedAxiom of choice for finite sets69963f30b6bd
addedUltrafilter lemma (filter extension)54d6b033550f
addedCountable union of countable sets is countablea6b557b1facb
addedInfinite set has countable-infinite subset70449e8fbb54
addedEight definitions of finite set equivalentc0fa32ab50e6
addedBorel determinacyd7f10c080b68
addedInfinite cardinal squared equals itself57ec11646294
addedVitali theorem (non-measurable sets)62eb73f46dbf
addedLebesgue-measurable non-Borel setsb9029f397085
addedHausdorff paradox5bbc85896f29
addedBanach–Tarski paradox6c562a4480f5
addedEvery field has an algebraic closuread311d5a2d70
addedEvery field extension has a transcendence basisb3577e9690b1
addedInfinite-dimensional spaces have infinite independent subsetdf6d3e93a80f
addedStone's representation theorema7f7ea0a8903
addedNielsen–Schreier theorem1f76e5b8062a
addedAdditive groups of R and C isomorphic3ae1460b2d68
addedAccumulation point: topological = sequential457c31a40521
addedContinuity: topological = sequentialaf043a8b04b7
addedCompactness: topological = sequential4ca875ec55e4
addedHahn–Banach theorem2793155324bd
addedEvery Hilbert space has orthonormal basisd52b484fc9ee
addedBanach–Alaoglu theorem553490390c50
addedBaire category theorem3a9d0fdd46ab
addedDiscontinuous linear map on infinite-dim TVS65c358803aa9
addedUniform space compact iff complete and totally boundedf02400537cc3
addedStone–Čech compactification existsb5cf419a25d9
addedUrysohn's lemma127fc595ef84
addedTietze extension theorema9ae77a846bc
addedExistence of partitions of unityae48dd091a69
addedGödel's completeness theoremb3662b0f86f1
addedCompactness theorembe382a369623
addedPartition principleace1bf67e1a2
addedInjection or surjection between two setsbdec41b9aa02
addedSurjection between two nonempty setsdc0ae1af380c
addedConverse Schröder–Bernstein theorem9a5adcdece6a
addedWeak partition principleefd0a3eff51f
addedNo infinite decreasing sequence of cardinalsa1fe9a2dcc5b
addedHahn embedding theoremee8c4df06d81
addedBaire property of all reals stronger than ¬AC587b2c8276d4
addedSolovay: ZF+DC with all reals measurable496c91d53a86
addedAxiom of determinacy implications01981a1eaa37
addedAC disprovable in New Foundations09d275ba872f
addedNegation of weak partition principlec74d6b755bfc
addedSequentially continuous but not continuous functionf01ea3d8399b
addedInfinite set without countably infinite subset394493feadec
addedReals as countable union of countable sets0304d2f32e9f
addedField with no algebraic closuree07be143ba45
addedField with two non-isomorphic algebraic closurese8c349aea673
addedVector space with no basis2c30f64b766b
addedVector space with bases of different cardinalities45534797eede
addedFree complete Boolean algebra on countably many generatorsb489ec8416b8
addedSet that cannot be linearly ordered2608a7859a0a
addedModel of ZF¬C with all sets in Rⁿ measurable9cb31e851095
addedGCH fails in all models of ZF¬Cf56146e52d5b
addedAxiom of choice in type theory0ea9cebe7b69