73 ideas
18290 | But what is the reasoning of the body, that it requires the wisdom you seek? [Nietzsche] |
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] |
18303 | Reject wisdom that lacks laughter [Nietzsche] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
18305 | To love truth, you must know how to lie [Nietzsche] |
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] |
16489 | Is it possible to state every possible truth about the whole course of nature without using 'not'? [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] |
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] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
16490 | Some facts about experience feel like logical necessities [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] |
16488 | It is hard to explain how a sentence like 'it is not raining' can be found true by observation [Russell] |
20757 | The powerful self behind your thoughts and feelings is your body [Nietzsche] |
18289 | Forget the word 'I'; 'I' is performed by the intelligence of your body [Nietzsche] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
16491 | If we define 'this is not blue' as disbelief in 'this is blue', we eliminate 'not' as an ingredient of facts [Russell] |
18299 | The will is constantly frustrated by the past [Nietzsche] |
18297 | We created meanings, to maintain ourselves [Nietzsche] |
18293 | The noble man wants new virtues; the good man preserves what is old [Nietzsche] |
18301 | We only really love children and work [Nietzsche] |
18307 | I want my work, not happiness! [Nietzsche] |
18291 | Virtues can destroy one another, through jealousy [Nietzsche] |
18287 | People now find both wealth and poverty too much of a burden [Nietzsche] |
18295 | If you want friends, you must be a fighter [Nietzsche] |
18286 | The greatest experience possible is contempt for your own happiness, reason and virtue [Nietzsche] |
18296 | An enduring people needs its own individual values [Nietzsche] |
18294 | The state coldly claims that it is the people, but that is a lie [Nietzsche] |
18304 | Saints want to live as they desire, or not to live at all [Nietzsche] |
18300 | Whenever we have seen suffering, we have wanted the revenge of punishment [Nietzsche] |
18302 | Man and woman are deeply strange to one another! [Nietzsche] |
4786 | Russell's 'at-at' theory says motion is to be at the intervening points at the intervening instants [Russell, by Psillos] |
18292 | I can only believe in a God who can dance [Nietzsche] |
18298 | Not being a god is insupportable, so there are no gods! [Nietzsche] |
18288 | Heaven was invented by the sick and the dying [Nietzsche] |
18306 | We don't want heaven; now that we are men, we want the kingdom of earth [Nietzsche] |