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
Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation
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
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
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
5. Theory of Logic / K. Features of Logics / 8. Enumerability
There are infinite sets that are not enumerable
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
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Cantor named the third realm between the finite and the Absolute the 'transfinite'
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
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
Cantor introduced the distinction between cardinals and ordinals
Cantor showed that ordinals are more basic than cardinals
Ordinals are generated by endless succession, followed by a limit ordinal
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 tried to prove points on a line matched naturals or reals - but nothing in between
Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1
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
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers
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
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
Cantor needed Power Set for the reals, but then couldn't count the new collections
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
18. Thought / D. Concepts / 6. Abstract Concepts / b. Abstracta from selection
Cantor says (vaguely) that we abstract numbers from equal sized sets
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