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