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17774 | Definitions make our intuitions mathematically useful |

17773 | Proof shows that it is true, but also why it must be true |

17795 | Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation |

17796 | There is a semi-categorical axiomatisation of set-theory |

17800 | The misnamed Axiom of Infinity says the natural numbers are finite in size |

17801 | The set hierarchy doesn't rely on the dubious notion of 'generating' them |

17803 | Limitation of size is part of the very conception of a set |

17786 | The mainstream of modern logic sees it as a branch of mathematics |

17788 | First-order logic only has its main theorems because it is so weak |

17791 | Only second-order logic can capture mathematical structure up to isomorphism |

17787 | Big logic has one fixed domain, but standard logic has a domain for each interpretation |

17790 | No Löwenheim-Skolem logic can axiomatise real analysis |

17779 | 'Classificatory' axioms aim at revealing similarity in morphology of structures |

17778 | Axiomatiation relies on isomorphic structures being essentially the same |

17780 | 'Eliminatory' axioms get rid of traditional ideal and abstract objects |

17789 | No logic which can axiomatise arithmetic can be compact or complete |

17782 | Greek quantities were concrete, and ratio and proportion were their science |

17781 | Real numbers were invented, as objects, to simplify and generalise 'quantity' |

17798 | Cantor presented the totality of natural numbers as finite, not infinite |

17784 | Real numbers can be eliminated, by axiom systems for complete ordered fields |

17797 | Cantor extended the finite (rather than 'taming the infinite') |

17799 | Cantor's infinite is an absolute, of all the sets or all the ordinal numbers |

17775 | If proof and definition are central, then mathematics needs and possesses foundations |

17776 | The ultimate principles and concepts of mathematics are presumed, or grasped directly |

17777 | Foundations need concepts, definition rules, premises, and proof rules |

17804 | Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms |

17792 | 1st-order PA is only interesting because of results which use 2nd-order PA |

17793 | It is only 2nd-order isomorphism which suggested first-order PA completeness |

17802 | We don't translate mathematics into set theory, because it comes embodied in that way |

17805 | Set theory is not just another axiomatised part of mathematics |

17794 | Set theory is not just first-order ZF, because that is inadequate for mathematics |

17785 | Real numbers as abstracted objects are now treated as complete ordered fields |