10859 | A set is 'well-ordered' if every subset has a first element |

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 |

10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice |

10872 | Extensionality: Two sets are equal if and only if they have the same elements |

10875 | Pairing: For any two sets there exists a set to which they both belong |

10876 | Unions: There is a set of all the elements which belong to at least one set in a collection |

10878 | Infinity: There exists a set of the empty set and the successor of each element |

10877 | Powers: All the subsets of a given set form their own new powerset |

10879 | Choice: For every set a mechanism will choose one member of any non-empty subset |

10871 | Axiom of Existence: there exists at least one set |

10874 | Specification: a condition applied to a set will always produce a new set |

10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) |

10860 | An ordinal number is defined by the set that comes before it |

10861 | Beyond infinity cardinals and ordinals can come apart |

10854 | Transcendental numbers can't be fitted to finite equations |

10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line |

10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless |

10866 | Cantor's account of infinities has the shaky foundation of irrational numbers |

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 |

10863 | Cantor proved that three dimensions have the same number of points as one dimension |