Ideas from 'works' by George Cantor [1880], by Theme Structure
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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

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

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

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

There are infinite sets that are not enumerable

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

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

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
15905

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

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

Cantor introduced the distinction between cardinals and ordinals

9892

Cantor showed that ordinals are more basic than cardinals

6. Mathematics / A. Nature of Mathematics / 3. 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. Numbers / g. Real numbers
11015

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

15906

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

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

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
15902

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

15908

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
13464

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
10112

The naturals won't map onto the reals, so there are different sizes of infinity

6. Mathematics / B. Foundations for Mathematics / 5. 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

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

Cantor says (vaguely) that we abstract numbers from equal sized sets

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

Only God is absolutely infinite
