Revision #57 → #1057 · back to history
addedCardinality355927815661
addedComparing sets by one-to-one correspondence6722980ed3e9
addedSet of natural numbers (infinite set)481ed6bb9f81
addedFunction6959476fdc97
addedInjective function4a039844c9aa
addedSurjective function7be68dbfd7fd
addedBijective function61ad6beea54a
addedFunction multiplying by twodaeceaf5cb99
addedEquinumerous / same cardinality65359b03da25
addedEven numbers equinumerous with naturalsa52093b32ba9
addedWhole-greater-than-part fails for infinite setsf4efcfddceeb
addedEquinumerosity is an equivalence relationd672a44148c1
addedCardinal inequality (≤)bb9929693d03
addedStrictly smaller cardinality (<)9240f9ddc145
addedFour-element and five-element sets3a85d535f0ea
addedCardinal inequality is reflexive and transitive97650a936e30
addedSchröder–Bernstein theorem6a73bcb89fa1
addedOrder type of a well-ordered setb28c991e05ff
addedWell-ordering theoremc1910f3b841b
addedTotality/comparability of cardinals (equivalent to AC)9c599c6d3a3a
addedCountable / countably infinite set26f152a2231c
addedEven natural numbers are countable7114db194b4c
addedSquare numbers are countable935647505308
addedRational numbersd435387a02b9
addedRational numbers are countableea3b56984adf
addedAlgebraic numberad4873e0098e
addedTranscendental number1565a8e501cb
addedSet of algebraic numbers is countable98b935eb9907
addedAlmost all real numbers are transcendentalfdbedbb9732e
addedHilbert's Grand Hotel3ae6d2973d56
addedUncountable set2f79d2687bb7
addedReal numbers are uncountable (diagonal argument)05550524d726
addedPower set of the naturals is uncountabled4123adbd428
addedℝ and P(ℕ) equinumerous (cardinality of the continuum)0b16e5534534
addedCantor's theorem384c4c7ca7ba
addedInfinite hierarchy of infinities24ea1b319c38
addedCardinal number08fff7e43b8c
addedAxiom of cardinality / Hume's principlec8177eea6191
addedFrege–Russell cardinal numbers5ef7e0748e63
addedFinite cardinality (cardinality n)92a45a43d598
addedSet with cardinality 45c6a61d911f6
addedPeano axioms1e0a5097ca92
addedFinite cardinality corresponds uniquely with naturals53b011c09354
addedPigeonhole principle6c2c92572866
addedAddition principleecf1dbd52e62
addedMultiplication principle03be6df21d0a
addedInclusion–exclusion principlebd976aa1c8c6
addedFinite set99e41a705641
addedDedekind-finite set8e5d3b541e68
addedAleph numbers42665efa53e3
addedAleph-nought (ℵ₀)e360fa6dbff1
addedOrdinal numbersaad12bd4f0c4
addedOmega (smallest infinite ordinal)d341628658fc
addedFirst uncountable ordinal (ω₁)fa477bf25ea4
addedSuccessor cardinal051c619d9065
addedHartogs' theorem81b63055f66a
addedNo cardinality strictly between consecutive alephs4ab05edb365d
addedEvery infinite set has cardinality of some alepha0e13ffe9dc6
addedCardinal additiond96b5dd6d910
addedCardinal multiplicationfc55d215a645
addedAssociativity of cardinal arithmetic51b648e7941d
addedCommutativity of cardinal arithmetic6a599922f0b5
addedDistributivity of cardinal arithmeticd08c65e27ec5
addedKőnig's theorem9f35cd153536
addedκ + κ = κ for infinite cardinals8e89ca50d50e
addedCardinal exponentiation2aed68d1d464
addedZero to the power of zero is onea9aa38488b78
added2^κ equals cardinality of the power set9890dd80c15e
addedCardinality of continuum is some aleph (which is unprovable)a413d3260011
addedNo set of all cardinal numbers (proper class)4aeb511645bb
addedContinuum (real numbers)793e87fb1d19
addedCardinality of the continuum (𝔠)ea9c97ba98d2
addedAll intervals have the same cardinality as ℝ96758d013178
addedTangent function bijection to ℝ42f30441bba8
addedCantor set3a5199450921
addedBijection between a line and a squarec2ea0369ffe9
addedSpace-filling curves (Peano, Hilbert)6f59c325d3e8
addedℝⁿ has the same cardinality as ℝacad391edf29
addedCountable cartesian product equinumerous to ℝb9364f6d4d41
addedℝ strictly larger than ℕ1bbec1b1d054
addedContinuum Hypothesis36645bee8cc8
addedIndependence of CH from ZFC89bad5652223
addedGeneralized Continuum Hypothesisa3f895b4d916
addedBeth numbers42569e8d79ea
addedAxiom of Infinityc96c922863f6
addedAxiom of power set648b7eadb34a
addedAxiom of Choice6868ff23683b
addedWithout AC: surjection without injectionb2382ee93c74
addedWithout AC: incomparable sets (trichotomy fails)91b1dae8d0a4
addedTrichotomy and surjection⇒injection equivalent to AC3ceaef9f44d8
addedRepresentative cardinality function impossible without AC73d7ea9fc99a
addedFrege–Russell–Scott definition (Scott's trick)201d50c474b1
addedMultiplicative axiom equivalent to ACb51f9299d98c
addedZF+GCH derives ACac1b6446ee86
addedProper classf7aa73d6ae28
addedExamples of proper classes160f13c213b5
addedAxiom of limitation of size62f66074c529
addedAll proper classes have the same size90436d44d6b0
addedLarge cardinal axiomsc39c6828e9fd
addedInaccessible cardinaldd6978c7e296
addedVon Neumann hierarchyf2fc8d54f77e
addedInaccessible cardinal implies consistency of ZFCa50a13331883
addedMeasurable cardinal inconsistent with constructibilitye8efaf19efc0
addedReinhardt cardinal inconsistent with AC7485a0c3be9f
addedAxiom of Determinacya68a02f90137
addedAD implies regularity properties of reals75dfd6ecf848
addedAD inconsistent with AC93b424b88cdc
addedBanach–Tarski paradox8353b568c5b2
addedPartition of ℝ with more parts than realsd051df7090ef
addedLöwenheim–Skolem theorem8964d71e5bff
addedSkolem's paradox7a8bc92ba2d1
addedConsistent first-order set theory has a countable model8ab35fc66cab
addedLöwenheim–Skolem fails in higher-order logic606185f56ba0
addedEuclidean principle (whole greater than part)3776222cf726
addedAristotle's wheel paradoxc9e0e1ce2205
addedGalileo's paradoxbf86f368b78b
addedHume's principle09539e2c1977
addedBolzano's pairing of intervalsa955f1ee0dc3