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

15945 | Second-order set theory just adds a version of Replacement that quantifies over functions |

15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one |

15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size |

15937 | Those who reject infinite collections also want to reject the Axiom of Choice |

15936 | The Power Set is just the collection of functions from one collection to another |

15899 | Replacement was immediately accepted, despite having very few implications |

15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets |

15920 | Pure collections of things obey Choice, but collections defined by a rule may not |

15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules |

15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class |

15900 | The iterative conception of set wasn't suggested until 1947 |

15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate |

15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs |

15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement |

15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified |

15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain |

15934 | Mathematical proof by contradiction needs the law of excluded middle |

15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity |

15942 | Every rational number, unlike every natural number, is divisible by some other number |

15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true |

18250 | Cauchy gave a necessary condition for the convergence of a sequence |

15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios |

15912 | Counting results in well-ordering, and well-ordering makes counting possible |

15949 | The theory of infinity must rest on our inability to distinguish between very large sizes |

15947 | The infinite is extrapolation from the experience of indefinitely large size |

15940 | The intuitionist endorses only the potential infinite |

15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal |

15915 | Ordinals are basic to Cantor's transfinite, to count the sets |

15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal |

15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest |

15929 | Set theory will found all of mathematics - except for the notion of proof |

15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' |

15928 | Intuitionism rejects set-theory to found mathematics |