90 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] |
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] |
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] |
8108 | Aesthetics presupposes a distinctive sort of experience, and a unified essence for art [Gardner] |
8112 | Art works originate in the artist's mind, and appreciation is re-creating this mental object [Gardner] |
8111 | Aesthetic objectivists must explain pleasure being essential, but not in the object [Gardner] |
8109 | Aesthetic judgements necessarily require first-hand experience, unlike moral judgements [Gardner] |
21167 | Gravity is unusual, in that it always attracts and never repels [New Sci.] |
21176 | In the Big Bang general relativity fails, because gravity is too powerful [New Sci.] |
21147 | Quantum electrodynamics incorporates special relativity and quantum mechanics [New Sci.] |
21155 | Photons have zero rest mass, so virtual photons have infinite range [New Sci.] |
21161 | In the standard model all the fundamental force fields merge at extremely high energies [New Sci.] |
21146 | Electrons move fast, so are subject to special relativity [New Sci.] |
21148 | The strong force is repulsive at short distances, strong at medium, and fades at long [New Sci.] |
21151 | Gluons, the particles carrying the strong force, interact because of their colour charge [New Sci.] |
21152 | The strong force binds quarks tight, and the nucleus more weakly [New Sci.] |
21143 | Quarks in threes can build hadrons with spin ½ or with spin 3/2 [New Sci.] |
21150 | Three different colours of quark (as in the proton) can cancel out to give no colour [New Sci.] |
21142 | Classifying hadrons revealed two symmetry patterns, produced by three basic elements [New Sci.] |
21145 | The four fundamental forces (gravity, electromagnetism, weak and strong) are the effects of particles [New Sci.] |
21153 | The weak force explains beta decay, and the change of type by quarks and leptons [New Sci.] |
21154 | Three particles enable the weak force: W+ and W- are charged, and Z° is not [New Sci.] |
21156 | The weak force particles are heavy, so the force has a short range [New Sci.] |
21164 | Why do the charges of the very different proton and electron perfectly match up? [New Sci.] |
21170 | The Standard Model cannot explain dark energy, survival of matter, gravity, or force strength [New Sci.] |
21140 | Spin is a built-in ration of angular momentum [New Sci.] |
21149 | Quarks have red, green or blue colour charge (akin to electric charge) [New Sci.] |
21158 | Fermions, with spin ½, are antisocial, and cannot share quantum states [New Sci.] |
21165 | Spin is akin to rotation, and is easily measured in a magnetic field [New Sci.] |
21157 | Particles are spread out, with wave-like properties, and higher energy shortens the wavelength [New Sci.] |
21163 | The mass of protons and neutrinos is mostly binding energy, not the quarks [New Sci.] |
21168 | Gravitional mass turns out to be the same as inertial mass [New Sci.] |
21138 | Neutrons are slightly heavier than protons, and decay into them by emitting an electron [New Sci.] |
21144 | Top, bottom, charm and strange quarks quickly decay into up and down [New Sci.] |
21141 | Neutrinos were proposed as the missing energy in neutron beta decay [New Sci.] |
21169 | Only neutrinos spin anticlockwise [New Sci.] |
21166 | Standard antineutrinos have opposite spin and opposite lepton number [New Sci.] |
21171 | The symmetry of unified electromagnetic and weak forces was broken by the Higgs field [New Sci.] |
21179 | Supersymmetric string theory can be expressed using loop quantum gravity [New Sci.] |
21178 | String theory is now part of 11-dimensional M-Theory, involving p-branes [New Sci.] |
21175 | String theory might be tested by colliding strings to make bigger 'stringballs' [New Sci.] |
21177 | String theory offers a quantum theory of gravity, by describing the graviton [New Sci.] |
21162 | Only supersymmetry offers to incorporate gravity into the scheme [New Sci.] |
21159 | Supersymmetry has extra heavy bosons and heavy fermions [New Sci.] |
21173 | Supersymmetry says particles and superpartners were unities, but then split [New Sci.] |
21172 | The evidence for supersymmetry keeps failing to appear [New Sci.] |
21160 | The Higgs field means even low energy space is not empty [New Sci.] |
21174 | Dark matter must have mass, to produce gravity, and no electric charge, to not reflect light [New Sci.] |