71 ideas
| 21584 | A sense of timelessness is essential to wisdom [Russell] |
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 21572 | Philosophical disputes are mostly hopeless, because philosophers don't understand each other [Russell] |
| 21571 | Philosophical systems are interesting, but we now need a more objective scientific philosophy [Russell] |
| 21574 | Hegel's confusions over 'is' show how vast systems can be built on simple errors [Russell] |
| 21587 | Philosophers sometimes neglect truth and distort facts to attain a nice system [Russell] |
| 21582 | Physicists accept particles, points and instants, while pretending they don't do metaphysics [Russell] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 21573 | When problems are analysed properly, they are either logical, or not philosophical at all [Russell] |
| 13469 | Tarski showed how we could have a correspondence theory of truth, without using 'facts' [Hart,WD] |
| 13504 | Truth for sentences is satisfaction of formulae; for sentences, either all sequences satisfy it (true) or none do [Hart,WD] |
| 13503 | A first-order language has an infinity of T-sentences, which cannot add up to a definition of truth [Hart,WD] |
| 13500 | Conditional Proof: infer a conditional, if the consequent can be deduced from the antecedent [Hart,WD] |
| 13502 | ∃y... is read as 'There exists an individual, call it y, such that...', and not 'There exists a y such that...' [Hart,WD] |
| 13456 | Set theory articulates the concept of order (through relations) [Hart,WD] |
| 13497 | Nowadays ZFC and NBG are the set theories; types are dead, and NF is only useful for the whole universe [Hart,WD] |
| 13443 | ∈ relates across layers, while ⊆ relates within layers [Hart,WD] |
| 13442 | Without the empty set we could not form a∩b without checking that a and b meet [Hart,WD] |
| 13493 | In the modern view, foundation is the heart of the way to do set theory [Hart,WD] |
| 13495 | Foundation Axiom: an nonempty set has a member disjoint from it [Hart,WD] |
| 13461 | We can choose from finite and evident sets, but not from infinite opaque ones [Hart,WD] |
| 13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
| 13516 | If we accept that V=L, it seems to settle all the open questions of set theory [Hart,WD] |
| 13441 | Naïve set theory has trouble with comprehension, the claim that every predicate has an extension [Hart,WD] |
| 13494 | The iterative conception may not be necessary, and may have fixed points or infinitely descending chains [Hart,WD] |
| 13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [Hart,WD] |
| 13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
| 13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
| 13490 | Von Neumann defines α<β as α∈β [Hart,WD] |
| 13481 | Maybe sets should be rethought in terms of the even more basic categories [Hart,WD] |
| 21588 | Logic gives the method of research in philosophy [Russell] |
| 21586 | The logical connectives are not objects, but are formal, and need a context [Russell] |
| 13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
| 13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
| 13505 | Model theory studies how set theory can model sets of sentences [Hart,WD] |
| 13511 | Model theory is mostly confined to first-order theories [Hart,WD] |
| 13513 | Models are ways the world might be from a first-order point of view [Hart,WD] |
| 13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
| 21585 | The tortoise won't win, because infinite instants don't compose an infinitely long time [Russell] |
| 13484 | Berry's Paradox: we succeed in referring to a number, with a term which says we can't do that [Hart,WD] |
| 13482 | The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD] |
| 13507 | The machinery used to solve the Liar can be rejigged to produce a new Liar [Hart,WD] |
| 13459 | The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD] |
| 13491 | The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD] |
| 13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
| 13492 | Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD] |
| 13446 | 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD] |
| 13509 | We can establish truths about infinite numbers by means of induction [Hart,WD] |
| 13474 | Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 21684 | Atomic facts may be inferrable from others, but never from non-atomic facts [Russell] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 22316 | A positive and negative fact have the same constituents; their difference is primitive [Russell] |
| 21576 | With asymmetrical relations (before/after) the reduction to properties is impossible [Russell] |
| 21575 | When we attribute a common quality to a group, we can forget the quality and just talk of the group [Russell] |
| 16065 | Constitution is identity (being in the same place), or it isn't (having different possibilities) [Wasserman] |
| 16067 | Constitution is not identity, because it is an asymmetric dependence relation [Wasserman] |
| 16069 | There are three main objections to seeing constitution as different from identity [Wasserman] |
| 16068 | The weight of a wall is not the weight of its parts, since that would involve double-counting [Wasserman] |
| 16074 | Relative identity may reject transitivity, but that suggests that it isn't about 'identity' [Wasserman] |
| 21580 | Science condemns sense-data and accepts matter, but a logical construction must link them [Russell] |
| 13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
| 13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
| 21583 | When sense-data change, there must be indistinguishable sense-data in the process [Russell] |
| 21577 | Empirical truths are particular, so general truths need an a priori input of generality [Russell] |
| 21579 | Objects are treated as real when they connect with other experiences in a normal way [Russell] |
| 21578 | Global scepticism is irrefutable, but can't replace our other beliefs, and just makes us hesitate [Russell] |
| 6416 | Other minds seem to exist, because their testimony supports realism about the world [Russell, by Grayling] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 21581 | We never experience times, but only succession of events [Russell] |