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
The continuum is the powerset of the integers, which moves up a level
Set theory made a closer study of infinity possible
Any set can always generate a larger set - its powerset, of subsets
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
Beyond infinity cardinals and ordinals can come apart
An ordinal number is defined by the set that comes before it
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 is independent of the axioms of set theory
The 'continuum hypothesis' says aleph-one is the cardinality of the reals
The Continuum Hypothesis is not inconsistent with 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