Click on the Idea Number for the full details | back to texts | expand these ideas

13466 | We are all post-Kantians, because he set the current agenda for philosophy |

13477 | The problems are the monuments of philosophy |

13515 | To study abstract problems, some knowledge of set theory is essential |

13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' |

13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do |

13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth |

13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent |

13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' |

13456 | Set theory articulates the concept of order (through relations) |

13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe |

13443 | ∈ relates across layers, while ⊆ relates within layers |

13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x |

13442 | Without the empty set we could not form a∩b without checking that a and b meet |

13493 | In the modern view, foundation is the heart of the way to do set theory |

13495 | Foundation Axiom: an nonempty set has a member disjoint from it |

13462 | With the Axiom of Choice every set can be well-ordered |

13461 | We can choose from finite and evident sets, but not from infinite opaque ones |

13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate |

13516 | If we accept that V=L, it seems to settle all the open questions of set theory |

13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension |

13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains |

13458 | A partial ordering becomes 'total' if any two members of its field are comparable |

13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets |

13490 | Von Neumann defines α<β as α∈β |

13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers |

13481 | Maybe sets should be rethought in terms of the even more basic categories |

13506 | The universal quantifier can't really mean 'all', because there is no universal set |

13505 | Model theory studies how set theory can model sets of sentences |

13511 | Model theory is mostly confined to first-order theories |

13513 | Models are ways the world might be from a first-order point of view |

13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 |

13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset |

13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that |

13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals |

13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar |

13489 | Von Neumann treated cardinals as a special sort of ordinal |

13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers |

13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) |

13491 | The axiom of infinity with separation gives a least limit ordinal ω |

13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton |

13446 | 19th century arithmetization of analysis isolated the real numbers from geometry |

13445 | Descartes showed a one-one order-preserving match between points on a line and the real numbers |

13509 | We can establish truths about infinite numbers by means of induction |

13447 | Cantor: there is no size between naturals and reals, or between a set and its power set |

13517 | If set theory is consistent, we cannot refute or prove the Continuum Hypothesis |

13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several |

13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic |

13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' |

13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions |

13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori |

13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest |

13467 | Leibniz was the first modern to focus on sentence-sized units (where empiricists preferred word-size) |