green numbers give full details.     |    back to list of philosophers     |     expand these ideas

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.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.19 Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD]
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.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.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.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.99 The naturals won't map onto the reals, so there are different sizes of infinity [George/Velleman]
p.163 Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Read]
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.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