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 | 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.205 | 10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice |
Ch.15 | p.205 | 10872 | Extensionality: Two sets are equal if and only if they have the same elements |
Ch.15 | p.205 | 10875 | Pairing: For any two sets there exists a set to which they both belong |
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 | 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 | 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) |