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
Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Lavine]
Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Lavine]
A set is a collection into a whole of distinct objects of our intuition or thought
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
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
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
The Axiom of Union dates from 1899, and seems fairly obvious [Maddy]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
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
Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Cantor named the third realm between the finite and the Absolute the 'transfinite' [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Numbers / b. Types of number
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. Numbers / e. Ordinal numbers
Cantor introduced the distinction between cardinals and ordinals [Tait]
Cantor showed that ordinals are more basic than cardinals [Dummett]
Ordinals are generated by endless succession, followed by a limit ordinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Numbers / f. Cardinal numbers
A cardinal is an abstraction, from the nature of a set's elements, and from their order
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Read]
Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Numbers / h. Reals from Cauchy
A real is associated with an infinite set of infinite Cauchy sequences of rationals [Lavine]
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Lavine]
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Lavine]
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite
Cantor proposes that there won't be a potential infinity if there is no actual infinity [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / f. Uncountable infinities
The naturals won't map onto the reals, so there are different sizes of infinity [George/Velleman]
Cantor needed Power Set for the reals, but then couldn't count the new collections [Lavine]
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
The 'extension of a concept' in general may be quantitatively completely indeterminate
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Pure mathematics is pure set theory
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Cantor says that maths originates only by abstraction from objects [Frege]
18. Thought / E. Abstraction / 2. Abstracta by Selection
Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD]
We form the image of a cardinal number by a double abstraction, from the elements and from their order
28. God / A. Divine Nature / 2. Divine Nature
Only God is absolutely infinite [Hart,WD]