17892 | For clear questions posed by reason, reason can also find clear answers |

10041 | Impredicative Definitions refer to the totality to which the object itself belongs |

8679 | We perceive the objects of set theory, just as we perceive with our senses |

9188 | Gödel proved that first-order logic is complete, and second-order logic incomplete |

10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science |

10042 | Reference to a totality need not refer to a conjunction of all its elements |

10620 | Originally truth was viewed with total suspicion, and only demonstrability was accepted |

17886 | The limitations of axiomatisation were revealed by the incompleteness theorems |

10071 | Second Incompleteness: nice theories can't prove their own consistency |

17883 | Gödel's Theorems did not refute the claim that all good mathematical questions have answers |

17888 | The undecidable sentence can be decided at a 'higher' level in the system |

10038 | A logical system needs a syntactical survey of all possible expressions |

18062 | Set-theory paradoxes are no worse than sense deception in physics |

10132 | There can be no single consistent theory from which all mathematical truths can be derived |

10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers |

17885 | Gödel eventually hoped for a generalised completeness theorem leaving nothing undecidable |

10614 | The real reason for Incompleteness in arithmetic is inability to define truth in a language |

10072 | First Incompleteness: arithmetic must always be incomplete |

9590 | Arithmetical truth cannot be fully and formally derived from axioms and inference rules |

10118 | First Incompleteness: a decent consistent system is syntactically incomplete |

10611 | There is a sentence which a theory can show is true iff it is unprovable |

10122 | Second Incompleteness: a decent consistent system can't prove its own consistency |

10867 | 'This system can't prove this statement' makes it unprovable either way |

10039 | Some arithmetical problems require assumptions which transcend arithmetic |

10043 | Mathematical objects are as essential as physical objects are for perception |

10271 | Basic mathematics is related to abstract elements of our empirical ideas |

10045 | Impredicative definitions are admitted into ordinary mathematics |