green numbers give full details | back to texts | expand these ideas

9955 | Contextual definitions replace a complete sentence containing the expression |

10031 | Impredicative definitions quantify over the thing being defined |

10098 | The 'power set' of A is all the subsets of A |

10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} |

10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B |

10103 | Grouping by property is common in mathematics, usually using equivalence |

10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words |

10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set |

10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y |

10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y |

17900 | The Axiom of Reducibility made impredicative definitions possible |

10109 | ZFC can prove that there is no set corresponding to the concept 'set' |

10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical |

10111 | Asserting Excluded Middle is a hallmark of realism about the natural world |

10129 | A 'model' is a meaning-assignment which makes all the axioms true |

10105 | Differences between isomorphic structures seem unimportant |

10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness |

10126 | A 'consistent' theory cannot contain both a sentence and its negation |

10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency |

10127 | A 'complete' theory contains either any sentence or its negation |

10106 | Rational numbers give answers to division problems with integers |

10102 | The integers are answers to subtraction problems involving natural numbers |

10107 | Real numbers provide answers to square root problems |

9946 | Logicists say mathematics is applicable because it is totally general |

10125 | The classical mathematician believes the real numbers form an actual set |

17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones |

10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic |

17902 | A successor is the union of a set with its singleton |

10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle |

10130 | Set theory can prove the Peano Postulates |

10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it |

10131 | If mathematics is not about particulars, observing particulars must be irrelevant |

10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. |

10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. |

10095 | Type theory has only finitely many items at each level, which is a problem for mathematics |

17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals |

10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions |

10134 | Much infinite mathematics can still be justified finitely |

10123 | The intuitionists are the idealists of mathematics |

10124 | Gödel's First Theorem suggests there are truths which are independent of proof |

10110 | Corresponding to every concept there is a class (some of them sets) |