13520 | A 'tautology' must include connectives |

13524 | Deduction Theorem: T∪{P}|-Q, then T|-(P→Q), which justifies Conditional Proof |

13522 | Universal Generalization: If we prove P(x) with no special assumptions, we can conclude ∀xP(x) |

13521 | Universal Specification: ∀xP(x) implies P(t). True for all? Then true for an instance |

13523 | Existential Generalization (or 'proof by example'): if we can say P(t), then we can say something is P |

13529 | Empty Set: ∃x∀y ¬(y∈x). The unique empty set exists |

13526 | Comprehension Axiom: if a collection is clearly specified, it is a set |

13534 | In first-order logic syntactic and semantic consequence (|- and |=) nicely coincide |

13535 | First-order logic is weakly complete (valid sentences are provable); we can't prove every sentence or its negation |

13531 | Model theory reveals the structures of mathematics |

13532 | Model theory 'structures' have a 'universe', some 'relations', some 'functions', and some 'constants' |

13519 | Model theory uses sets to show that mathematical deduction fits mathematical truth |

13533 | First-order model theory rests on completeness, compactness, and the Löwenheim-Skolem-Tarski theorem |

13537 | An 'isomorphism' is a bijection that preserves all structural components |

13539 | The LST Theorem is a serious limitation of first-order logic |

13538 | If a theory is complete, only a more powerful language can strengthen it |

13525 | Most deductive logic (unlike ordinary reasoning) is 'monotonic' - we don't retract after new givens |

13530 | An ordinal is an equivalence class of well-orderings, or a transitive set whose members are transitive |

13528 | Continuum Hypothesis: there are no sets between N and P(N) |

13527 | Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC |

13518 | Modern mathematics has unified all of its objects within set theory |