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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.]

p.3

10701

Cantor showed that supposed contradictions in infinity were just a lack of clarity [Potter]


p.7

15901

Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Lavine]


p.11

15902

Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Lavine]


p.14

10082

There are infinite sets that are not enumerable [Smith,P]


p.19

13454

Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD]


p.27

8631

Cantor says that maths originates only by abstraction from objects [Frege]


p.29

13465

Only God is absolutely infinite [Hart,WD]


p.29

13464

Cantor proposes that there won't be a potential infinity if there is no actual infinity [Hart,WD]


p.38

15903

A real is associated with an infinite set of infinite Cauchy sequences of rationals [Lavine]


p.40

9971

Cantor introduced the distinction between cardinals and ordinals [Tait]


p.42

15905

Cantor proved the points on a plane are in onetoone correspondence to the points on a line [Lavine]


p.43

15906

Cantor tried to prove points on a line matched naturals or reals  but nothing in between [Lavine]


p.48

15908

It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Lavine]


p.51

15910

Cantor named the third realm between the finite and the Absolute the 'transfinite' [Lavine]


p.60

13483

Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Hart,WD]


p.99

10112

The naturals won't map onto the reals, so there are different sizes of infinity [George/Velleman]


p.163

11015

Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Read]


p.293

9892

Cantor showed that ordinals are more basic than cardinals [Dummett]


p.304

14136

A cardinal is an abstraction, from the nature of a set's elements, and from their order


p.484

13016

The Axiom of Union dates from 1899, and seems fairly obvious [Maddy]

I.1

p.21

18176

Pure mathematics is pure set theory

1883

Grundlagen (Foundations of Theory of Manifolds)


p.52

15911

Ordinals are generated by endless succession, followed by a limit ordinal [Lavine]


p.247

15946

Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Lavine]

1885

Review of Frege's 'Grundlagen'

1932:440

p.60

9992

The 'extension of a concept' in general may be quantitatively completely indeterminate

1897

The Theory of Transfinite Numbers


p.4

15896

Cantor needed Power Set for the reals, but then couldn't count the new collections [Lavine]

p.85

p.22

9616

A set is a collection into a whole of distinct objects of our intuition or thought

1899

Later Letters to Dedekind


p.366

17831

Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Lake]

§1

p.599

9145

We form the image of a cardinal number by a double abstraction, from the elements and from their order
