64 ideas
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] |
4465 | Note that "is" can assert existence, or predication, or identity, or classification [PG] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
4686 | Fallacies are errors in reasoning, 'formal' if a clear rule is breached, and 'informal' if more general [PG] |
7415 | Question-begging assumes the proposition which is being challenged [PG] |
7414 | What is true of a set is also true of its members [PG] |
6696 | The Ad Hominem Fallacy criticises the speaker rather than the argument [PG] |
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] |
4687 | Minimal theories of truth avoid ontological commitment to such things as 'facts' or 'reality' [PG] |
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] |
13462 | With the Axiom of Choice every set can be well-ordered [Hart,WD] |
13461 | We can choose from finite and evident sets, but not from infinite opaque ones [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] |
13460 | 'Well-ordering' must have a least member, so it does the natural numbers but not the integers [Hart,WD] |
13458 | A partial ordering becomes 'total' if any two members of its field are comparable [Hart,WD] |
13457 | A 'partial ordering' is irreflexive and transitive; the sets are ordered, but not the subsets [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] |
13513 | Models are ways the world might be from a first-order point of view [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] |
13512 | Modern model theory begins with the proof of Los's Conjecture in 1962 [Hart,WD] |
13496 | First-order logic is 'compact': consequences of a set are consequences of a finite subset [Hart,WD] |
6516 | Monty Hall Dilemma: do you abandon your preference after Monty eliminates one of the rivals? [PG] |
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] |
13463 | There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD] |
13491 | The axiom of infinity with separation gives a least limit ordinal ω [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] |
18933 | Not-Being obviously doesn't exist, and the five modes of Being are all impossible [Gorgias, by Diog. Laertius] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
24054 | Everything has a probability, something will happen, and probabilities add up [PG] |
3875 | If reality is just what we perceive, we would have no need for a sixth sense [PG] |
13480 | Fregean self-evidence is an intrinsic property of basic truths, rules and definitions [Hart,WD] |
3876 | If my team is losing 3-1, I have synthetic a priori knowledge that they need two goals for a draw [PG] |
13476 | The failure of key assumptions in geometry, mereology and set theory throw doubt on the a priori [Hart,WD] |
7734 | Maybe a mollusc's brain events for pain ARE of the same type (broadly) as a human's [PG] |
7735 | Maybe a frog's brain events for fear are functionally like ours, but not phenomenally [PG] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
9866 | Gorgias says rhetoric is the best of arts, because it enslaves without using force [Gorgias, by Plato] |
5864 | Destroy seriousness with laughter, and laughter with seriousness [Gorgias] |
3877 | Utilitarianism seems to justify the discreet murder of unhappy people [PG] |
6126 | Life is Movement, Respiration, Sensation, Nutrition, Excretion, Reproduction, Growth (MRS NERG) [PG] |
3874 | How could God know there wasn't an unknown force controlling his 'free' will? [PG] |
3873 | An omniscient being couldn't know it was omniscient, as that requires information from beyond its scope of knowledge [PG] |