10775 | The axiom of choice now seems acceptable and obvious (if it is meaningful) |

10766 | Logic is either for demonstration, or for characterizing structures |

10767 | Elementary logic is complete, but cannot capture mathematics |

10769 | Second-order logic isn't provable, but will express set-theory and classic problems |

10762 | In sentential logic there is a simple proof that all truth functions can be reduced to 'not' and 'and' |

10776 | The main quantifiers extend 'and' and 'or' to infinite domains |

10774 | There are at least five unorthodox quantifiers that could be used |

10773 | The Löwenheim-Skolem property is a limitation (e.g. can't say there are uncountably many reals) |

10777 | Skolem mistakenly inferred that Cantor's conceptions were illusory |

10765 | Soundness would seem to be an essential requirement of a proof procedure |

10763 | Completeness and compactness together give axiomatizability |

10770 | If completeness fails there is no algorithm to list the valid formulas |

10772 | Compactness blocks infinite expansion, and admits non-standard models |

10771 | Compactness is important for major theories which have infinitely many axioms |

10764 | A complete logic has an effective enumeration of the valid formulas |

10768 | Effective enumeration might be proved but not specified, so it won't guarantee knowledge |