Click on the Idea Number for the full details | back to texts | expand these ideas

18194 | 'Forcing' can produce new models of ZFC from old models |

18195 | A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy |

18191 | Axiom of Infinity: completed infinite collections can be treated mathematically |

18193 | The Axiom of Foundation says every set exists at a level in the set hierarchy |

18169 | Axiom of Reducibility: propositional functions are extensionally predicative |

18168 | 'Propositional functions' are propositions with a variable as subject or predicate |

18171 | Cantor and Dedekind brought completed infinities into mathematics |

18190 | Completed infinities resulted from giving foundations to calculus |

18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities |

18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size |

18175 | For any cardinal there is always a larger one (so there is no set of all sets) |

18172 | Infinity has degrees, and large cardinals are the heart of set theory |

18196 | An 'inaccessible' cardinal cannot be reached by union sets or power sets |

18187 | Theorems about limits could only be proved once the real numbers were understood |

18182 | The extension of concepts is not important to me |

18177 | In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets |

18164 | Frege solves the Caesar problem by explicitly defining each number |

18178 | For Zermelo the successor of n is {n} (rather than n U {n}) |

18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) |

18180 | Von Neumann numbers are preferred, because they continue into the transfinite |

18163 | Mathematics rests on the logic of proofs, and on the set theoretic axioms |

18183 | Set theory brings mathematics into one arena, where interrelations become clearer |

18186 | Identifying geometric points with real numbers revealed the power of set theory |

18184 | Making set theory foundational to mathematics leads to very fruitful axioms |

18185 | Unified set theory gives a final court of appeal for mathematics |

18188 | The line of rationals has gaps, but set theory provided an ordered continuum |

18204 | Scientists posit as few entities as possible, but set theorist posit as many as possible |

18207 | Maybe applications of continuum mathematics are all idealisations |

18167 | We can get arithmetic directly from HP; Law V was used to get HP from the definition of number |

18205 | The theoretical indispensability of atoms did not at first convince scientists that they were real |

18206 | Science idealises the earth's surface, the oceans, continuities, and liquids |