68 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] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
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] |
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] |
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] |
9117 | The smallest heap has four objects: three on the bottom, one on the top [Hart,WD, by Sorensen] |
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] |
13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
18430 | We accept properties because of type/tokens, reference, and quantification [Edwards] |
18432 | Quineans say that predication is primitive and inexplicable [Edwards] |
18437 | Resemblance nominalism requires a second entity to explain 'the rose is crimson' [Edwards] |
18434 | That a whole is prior to its parts ('priority monism') is a view gaining in support [Edwards] |
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] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
20624 | Work degrades into heat, but not vice versa [Close] |
20623 | First Law: energy can change form, but is conserved overall [Close] |
20625 | Third Law: total order and minimum entropy only occurs at absolute zero [Close] |
20628 | The electric and magnetic are tightly linked, and viewed according to your own motion [Close] |
20622 | All motions are relative and ambiguous, but acceleration is the same in all inertial frames [Close] |
20635 | The general relativity equations relate curvature in space-time to density of energy-momentum [Close] |
20627 | Electric fields have four basic laws (two by Gauss, one by Ampère, one by Faraday) [Close] |
20630 | Light isn't just emitted in quanta called photons - light is photons [Close] |
20637 | In general relativity the energy and momentum of photons subjects them to gravity [Close] |
20629 | Electro-magnetic waves travel at light speed - so light is electromagnetism! [Close] |
20632 | In QED, electro-magnetism exists in quantum states, emitting and absorbing electrons [Close] |
20642 | Photon exchange drives the electro-magnetic force [Close] |
20639 | Quantum fields contain continual rapid creation and disappearance [Close] |
20631 | Dirac showed how electrons conform to special relativity [Close] |
20641 | Electrons get their mass by interaction with the Higgs field [Close] |
20626 | Modern theories of matter are grounded in heat, work and energy [Close] |
20633 | The Higgs field is an electroweak plasma - but we don't know what stuff it consists of [Close] |
20640 | Space-time is indeterminate foam over short distances [Close] |