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Ideas of George Cantor, by Text

[German, 1845 - 1918, Born in St Petersburg. Studied in Berlin. Taught at the University of Halle from 1872.]

1880 works
p.3 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Potter]
p.3 Cantor's theory concerns collections which can be counted, using the ordinals [Lavine]
p.7 CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Koellner]
p.7 Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Lavine]
p.11 Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Lavine]
p.14 There are infinite sets that are not enumerable [Smith,P]
p.16 Cantor's Theorem: for any set x, its power set P(x) has more members than x [Hart,WD]
p.17 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Maddy]
p.17 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Maddy]
p.19 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD]
p.19 Cantor: there is no size between naturals and reals, or between a set and its power set [Hart,WD]
p.22 Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Horsten]
p.27 Cantor says that maths originates only by abstraction from objects [Frege]
p.29 Only God is absolutely infinite [Hart,WD]
p.29 Cantor proposes that there won't be a potential infinity if there is no actual infinity [Hart,WD]
p.30 If a set is 'a many thought of as one', beginners should protest against singleton sets [Lewis]
p.38 A real is associated with an infinite set of infinite Cauchy sequences of rationals [Lavine]
p.40 Cantor introduced the distinction between cardinals and ordinals [Tait]
p.42 Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Lavine]
p.42 The Continuum Hypothesis says there are no sets between the natural numbers and reals [Shapiro]
p.43 Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Lavine]
p.48 It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Lavine]
p.49 Cantor took the ordinal numbers to be primary [Tait]
p.51 Cantor named the third realm between the finite and the Absolute the 'transfinite' [Lavine]
p.60 Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Hart,WD]
p.67 Continuum Hypothesis: there are no sets between N and P(N) [Wolf,RS]
p.79 Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Chihara]
p.92 Irrational numbers are the limits of Cauchy sequences of rational numbers [Lavine]
p.99 The naturals won't map onto the reals, so there are different sizes of infinity [George/Velleman]
p.106 Cantor proved that all sets have more subsets than they have members [Bostock]
p.113 The powerset of all the cardinal numbers is required to be greater than itself [Friend]
p.123 Cantor's sets were just collections, but Dedekind's were containers [Oliver/Smiley]
p.127 Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Shapiro]
p.145 Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Friend]
p.163 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Read]
p.183 Cantor proved that three dimensions have the same number of points as one dimension [Clegg]
p.185 The continuum is the powerset of the integers, which moves up a level [Clegg]
p.293 Cantor showed that ordinals are more basic than cardinals [Dummett]
p.304 A cardinal is an abstraction, from the nature of a set's elements, and from their order
p.414 Cantor presented the totality of natural numbers as finite, not infinite [Mayberry]
p.484 The Axiom of Union dates from 1899, and seems fairly obvious [Maddy]
I.1 p.21 Pure mathematics is pure set theory
1883 Grundlagen (Foundations of Theory of Manifolds)
p.52 Ordinals are generated by endless succession, followed by a limit ordinal [Lavine]
p.247 Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Lavine]
1885 Review of Frege's 'Grundlagen'
1932:440 p.60 The 'extension of a concept' in general may be quantitatively completely indeterminate
1897 The Theory of Transfinite Numbers
p.4 Cantor needed Power Set for the reals, but then couldn't count the new collections [Lavine]
p.85 p.22 A set is a collection into a whole of distinct objects of our intuition or thought
1899 Later Letters to Dedekind
p.366 Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Lake]
1915 Beitrage
1 p.599 We form the image of a cardinal number by a double abstraction, from the elements and from their order