### 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
 10859 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
 10857 Set theory made a closer study of infinity possible
 10864 Any set can always generate a larger set - its powerset, of subsets
 10865 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
 10870 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
 10872 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
 10875 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
 10876 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
 10878 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
 10877 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
 10879 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
 10871 Axiom of Existence: there exists at least one set
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
 10874 Specification: a condition applied to a set will always produce a new set
###### 6. Mathematics / A. Nature of Mathematics / 1. Mathematics
 10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable)
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / e. Ordinal numbers
 10860 An ordinal number is defined by the set that comes before it
 10861 Beyond infinity cardinals and ordinals can come apart
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
 10854 Transcendental numbers can't be fitted to finite equations
###### 6. Mathematics / A. Nature of Mathematics / 3. Numbers / k. Imaginary numbers
 10858 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
 10853 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
 10866 Cantor's account of infinities has the shaky foundation of irrational numbers
###### 6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
 10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals
 10868 The Continuum Hypothesis is not inconsistent with the axioms of set theory
 10869 The Continuum Hypothesis is independent of the axioms of set theory
###### 26. Natural Theory / B. Concepts of Nature / 3. Space / b. Points in space
 10863 Cantor proved that three dimensions have the same number of points as one dimension