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.]

p.3

10701

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


p.3

15893

Cantor's theory concerns collections which can be counted, using the ordinals [Lavine]


p.7

15901

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


p.7

17889

CH: An infinite set of reals corresponds 11 either to the naturals or to the reals [Koellner]


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.16

13444

Cantor's Theorem: for any set x, its power set P(x) has more members than x [Hart,WD]


p.17

18174

Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Maddy]


p.17

18173

Cardinality strictly concerns oneone correspondence, to test infinite sameness of size [Maddy]


p.19

13454

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


p.19

13447

Cantor: there is no size between naturals and reals, or between a set and its power set [Hart,WD]


p.22

10883

Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Horsten]


p.27

8631

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


p.29

13464

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


p.29

13465

Only God is absolutely infinite [Hart,WD]


p.30

15505

If a set is 'a many thought of as one', beginners should protest against singleton sets [Lewis]


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.42

8733

The Continuum Hypothesis says there are no sets between the natural numbers and reals [Shapiro]


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.49

9983

Cantor took the ordinal numbers to be primary [Tait]


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.67

13528

Continuum Hypothesis: there are no sets between N and P(N) [Wolf,RS]


p.79

9555

Continuum Hypothesis: no cardinal greater than alephnull but less than cardinality of the continuum [Chihara]


p.92

18251

Irrational numbers are the limits of Cauchy sequences of rational numbers [Lavine]


p.99

10112

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


p.106

18098

Cantor proved that all sets have more subsets than they have members [Bostock]


p.113

8710

The powerset of all the cardinal numbers is required to be greater than itself [Friend]


p.123

14199

Cantor's sets were just collections, but Dedekind's were containers [Oliver/Smiley]


p.127

10232

Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Shapiro]


p.145

8715

Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Friend]


p.163

11015

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


p.183

10863

Cantor proved that three dimensions have the same number of points as one dimension [Clegg]


p.185

10865

The continuum is the powerset of the integers, which moves up a level [Clegg]


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.414

17798

Cantor presented the totality of natural numbers as finite, not infinite [Mayberry]


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
