10073 | There cannot be a set theory which is complete |

10616 | Second-order arithmetic can prove new sentences of first-order |

10076 | The 'range' of a function is the set of elements in the output set created by the function |

10605 | Two functions are the same if they have the same extension |

10075 | A 'partial function' maps only some elements to another set |

10074 | A 'total function' maps every element to one element in another set |

10612 | An argument is a 'fixed point' for a function if it is mapped back to itself |

10615 | The Comprehension Schema says there is a property only had by things satisfying a condition |

10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system |

10602 | A 'natural deduction system' has no axioms but many rules |

10613 | No nice theory can define truth for its own language |

10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original |

10077 | A 'surjective' ('onto') function creates every element of the output set |

10079 | A 'bijective' function has one-to-one correspondence in both directions |

10070 | If everything that a theory proves is true, then it is 'sound' |

10086 | Soundness is true axioms and a truth-preserving proof system |

10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) |

10598 | A theory is 'negation complete' if it proves all sentences or their negation |

10597 | 'Complete' applies both to whole logics, and to theories within them |

10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved |

10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof |

10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating |

10087 | A theory is 'decidable' if all of its sentences could be mechanically proved |

10088 | Any consistent, axiomatized, negation-complete formal theory is decidable |

10081 | A set is 'enumerable' is all of its elements can result from a natural number function |

10083 | A set is 'effectively enumerable' if a computer could eventually list every member |

10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) |

10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable |

10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) |

10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system |

10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) |

10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals |

10619 | The truths of arithmetic are just true equations and their universally quantified versions |

10618 | All numbers are related to zero by the ancestral of the successor relation |

10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G |

10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers |

10850 | Baby Arithmetic is complete, but not very expressive |

10852 | Robinson Arithmetic (Q) is not negation complete |

10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic |

10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays |

10603 | The logic of arithmetic must quantify over properties of numbers to handle induction |

10848 | Multiplication only generates incompleteness if combined with addition and successor |

10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication |

10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation |