Ideas from 'Infinity: Quest to Think the Unthinkable' by Brian Clegg [2003], by Theme Structure

[found in 'Infinity' by Clegg,Brian [Robinson 2003,978-1-84119-650-3]].

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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
A set is 'well-ordered' if every subset has a first element
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Set theory made a closer study of infinity possible
Any set can always generate a larger set - its powerset, of subsets
The continuum is the powerset of the integers, which moves up a level
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Extensionality: Two sets are equal if and only if they have the same elements
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Pairing: For any two sets there exists a set to which they both belong
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
Unions: There is a set of all the elements which belong to at least one set in a collection
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinity: There exists a set of the empty set and the successor of each element
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
Powers: All the subsets of a given set form their own new powerset
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Choice: For every set a mechanism will choose one member of any non-empty subset
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
Axiom of Existence: there exists at least one set
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
Specification: a condition applied to a set will always produce a new set
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable)
6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
An ordinal number is defined by the set that comes before it
Beyond infinity cardinals and ordinals can come apart
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
Transcendental numbers can't be fitted to finite equations
6. Mathematics / A. Nature of Mathematics / 3. Numbers / k. Imaginary numbers
By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line
6. Mathematics / A. Nature of Mathematics / 3. Numbers / l. Zero
Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / a. The Infinite
Cantor's account of infinities has the shaky foundation of irrational numbers
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
The 'continuum hypothesis' says aleph-one is the cardinality of the reals
The Continuum Hypothesis is not inconsistent with the axioms of set theory
The Continuum Hypothesis is independent of the axioms of set theory
26. Natural Theory / B. Concepts of Nature / 3. Space / b. Points in space
Cantor proved that three dimensions have the same number of points as one dimension