### Ideas of George Cantor, by Theme

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

green numbers give full details    |    back to list of philosophers    |     expand these ideas
###### 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) Set
 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
 10865 The continuum is the powerset of the integers, which moves up a level [Clegg]
 10701 Cantor showed that supposed contradictions in infinity were just a lack of clarity [Potter]
###### 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 one-to-one 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 1-1 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 aleph-null 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 one-one 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-Time / 1. Space / c. 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]