Ideas of George Cantor, by Theme
[German, 1845  1918, Born in St Petersburg. Studied in Berlin. Taught at the University of Halle from 1872.]
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4. Formal Logic / F. Set Theory ST / 1. Set Theory
15901

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

15946

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

9616

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

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
13444

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

18098

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

4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
15505

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

4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10701

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

10865

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17831

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

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
13016

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

4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
14199

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

5. Theory of Logic / K. Features of Logics / 8. Enumerability
10082

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

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
13483

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

5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
8710

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

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
15910

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
15905

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
9983

Cantor took the ordinal numbers to be primary [Tait]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
17798

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
9971

Cantor introduced the distinction between cardinals and ordinals [Tait]

9892

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

15911

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
14136

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
15906

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

11015

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
15903

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

18251

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
15902

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

15908

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
13464

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
10112

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

15896

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
8733

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

17889

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

13447

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

10883

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

13528

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

9555

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
15893

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

18174

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

6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
18173

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

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
9992

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

6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
10232

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

6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
18176

Pure mathematics is pure set theory

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
8631

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

18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
8715

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

18. Thought / E. Abstraction / 2. Abstracta by Selection
13454

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

9145

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

27. Natural Reality / C. Space / 3. Points in Space
10863

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

28. God / A. Divine Nature / 2. Divine Nature
13465

Only God is absolutely infinite [Hart,WD]
