79 ideas
| 13466 | We are all post-Kantians, because he set the current agenda for philosophy [Hart,WD] |
| 2930 | The main aim of philosophy must be to determine the order of rank among values [Nietzsche] |
| 13477 | The problems are the monuments of philosophy [Hart,WD] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 20143 | Scientific knowledge is nothing without a prior philosophical 'faith' [Nietzsche] |
| 23722 | Objectivity is not disinterestedness (impossible), but the ability to switch perspectives [Nietzsche] |
| 4417 | Only that which has no history is definable [Nietzsche] |
| 23715 | Psychologists should be brave and proud, and prefer truth to desires, even when it is ugly [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] |
| 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] |
| 4421 | Philosophers have never asked why there is a will to truth in the first place [Nietzsche] |
| 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] |
| 23719 | Forgetfulness is a strong positive ability, not mental laziness [Nietzsche] |
| 4420 | There is only 'perspective' seeing and knowing, and so the best objectivity is multiple points of view [Nietzsche] |
| 4414 | Philosophers invented "free will" so that our virtues would be permanently interesting to the gods [Nietzsche] |
| 4419 | People who think in words are orators rather than thinkers, and think about facts instead of thinking facts [Nietzsche] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 4411 | It is a delusion to separate the man from the deed, like the flash from the lightning [Nietzsche] |
| 21103 | Moral questions can only be decided by common opinion [Hume] |
| 3793 | We must question the very value of moral values [Nietzsche] |
| 4408 | The concept of 'good' was created by aristocrats to describe their own actions [Nietzsche] |
| 23716 | A strong rounded person soon forgets enemies, misfortunes, and even misdeeds [Nietzsche] |
| 20129 | All animals strive for the ideal conditions to express their power, and hate any hindrances [Nietzsche] |
| 4409 | Only the decline of aristocratic morality led to concerns about "egoism" [Nietzsche] |
| 3259 | Nietzsche rejects impersonal morality, and asserts the idea of living well [Nietzsche, by Nagel] |
| 4416 | Basic justice is the negotiation of agreement among equals, and the imposition of agreement [Nietzsche] |
| 4418 | A masterful and violent person need have nothing to do with contracts [Nietzsche] |
| 4407 | Plato, Spinoza and Kant are very different, but united in their low estimation of pity [Nietzsche] |
| 4415 | Guilt and obligation originated in the relationship of buying and selling, credit and debt [Nietzsche] |
| 23718 | If we say birds of prey could become lambs, that makes them responsible for being birds of prey [Nietzsche] |
| 23717 | Modern nihilism is now feeling tired of mankind [Nietzsche] |
| 23721 | Old tribes always felt an obligation to the earlier generations, and the founders [Nietzsche] |
| 21099 | People must have agreed to authority, because they are naturally equal, prior to education [Hume] |
| 20142 | The state begins with brutal conquest of a disorganised people, not with a 'contract' [Nietzsche] |
| 21100 | The idea that society rests on consent or promises undermines obedience [Hume] |
| 20495 | We no more give 'tacit assent' to the state than a passenger carried on board a ship while asleep [Hume] |
| 21101 | The people would be amazed to learn that government arises from their consent [Hume] |
| 6703 | Poor people lack the knowledge or wealth to move to a different state [Hume] |
| 21102 | We all know that the history of property is founded on injustices [Hume] |
| 23720 | Punishment makes people harder, more alienated, and hostile [Nietzsche] |
| 4410 | The truly great haters in world history have always been priests [Nietzsche] |