### 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
###### 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. Nature of 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. Nature of Numbers / g. Real numbers
 10854 Transcendental numbers can't be fitted to finite equations
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of 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. Nature of 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 / 5. The Infinite / a. The Infinite
 10866 Cantor's account of infinities has the shaky foundation of irrational numbers
###### 6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
 10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals
 10869 The Continuum Hypothesis is independent of the axioms of set theory