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### Ideas of Brian Clegg, by Text

#### [British, fl. 2003, Technical consultant and freelance author.]

 2003 Infinity: Quest to Think the Unthinkable
 Ch. 6 p.61 10853 Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless
 Ch. 6 p.69 10854 Transcendental numbers can't be fitted to finite equations
 Ch.12 p.163 10858 By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line
 Ch.13 p.157 10857 Set theory made a closer study of infinity possible
 Ch.13 p.168 10859 A set is 'well-ordered' if every subset has a first element
 Ch.13 p.169 10860 An ordinal number is defined by the set that comes before it
 Ch.13 p.169 10861 Beyond infinity cardinals and ordinals can come apart
 Ch.14 p.179 10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals
 Ch.14 p.183 10863 Cantor proved that three dimensions have the same number of points as one dimension
 Ch.14 p.184 10864 Any set can always generate a larger set - its powerset, of subsets
 Ch.14 p.185 10865 The continuum is the powerset of the integers, which moves up a level
 Ch.15 p.193 10866 Cantor's account of infinities has the shaky foundation of irrational numbers
 Ch.15 p.203 10868 The Continuum Hypothesis is not inconsistent with the axioms of set theory
 Ch.15 p.204 10869 The Continuum Hypothesis is independent of the axioms of set theory
 Ch.15 p.205 10872 Extensionality: Two sets are equal if and only if they have the same elements
 Ch.15 p.205 10870 ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice
 Ch.15 p.205 10875 Pairing: For any two sets there exists a set to which they both belong
 Ch.15 p.205 10871 Axiom of Existence: there exists at least one set
 Ch.15 p.205 10874 Specification: a condition applied to a set will always produce a new set
 Ch.15 p.206 10878 Infinity: There exists a set of the empty set and the successor of each element
 Ch.15 p.206 10877 Powers: All the subsets of a given set form their own new powerset
 Ch.15 p.206 10876 Unions: There is a set of all the elements which belong to at least one set in a collection
 Ch.15 p.206 10879 Choice: For every set a mechanism will choose one member of any non-empty subset
 Ch.17 p.218 10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable)