79 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] |
| 2056 | Philosophers are always switching direction to something more interesting [Plato] |
| 2083 | Either a syllable is its letters (making parts as knowable as whole) or it isn't (meaning it has no parts) [Plato] |
| 2086 | Understanding mainly involves knowing the elements, not their combinations [Plato] |
| 13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
| 2082 | A rational account is essentially a weaving together of things with names [Plato] |
| 2052 | Eristic discussion is aggressive, but dialectic aims to help one's companions in discussion [Plato] |
| 15854 | A primary element has only a name, and no logos, but complexes have an account, by weaving the names [Plato] |
| 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] |
| 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] |
| 10216 | We master arithmetic by knowing all the numbers in our soul [Plato] |
| 13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
| 2060 | There seem to be two sorts of change: alteration and motion [Plato] |
| 13488 | Mass words do not have plurals, or numerical adjectives, or use 'fewer' [Hart,WD] |
| 2084 | If a word has no parts and has a single identity, it turns out to be the same kind of thing as a letter [Plato] |
| 15844 | A sum is that from which nothing is lacking, which is a whole [Plato] |
| 15843 | The whole can't be the parts, because it would be all of the parts, which is the whole [Plato] |
| 2080 | Things are only knowable if a rational account (logos) is possible [Plato] |
| 16126 | Expertise is knowledge of the whole by means of the parts [Plato] |
| 2050 | It is impossible to believe something which is held to be false [Plato] |
| 2076 | How can a belief exist if its object doesn't exist? [Plato] |
| 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] |
| 2045 | Perception is infallible, suggesting that it is knowledge [Plato] |
| 2067 | Our senses could have been separate, but they converge on one mind [Plato] |
| 2068 | With what physical faculty do we perceive pairs of opposed abstract qualities? [Plato] |
| 2069 | Thought must grasp being itself before truth becomes possible [Plato] |
| 2078 | You might mistake eleven for twelve in your senses, but not in your mind [Plato] |
| 2089 | An inadequate rational account would still not justify knowledge [Plato] |
| 2085 | Parts and wholes are either equally knowable or equally unknowable [Plato] |
| 2091 | Without distinguishing marks, how do I know what my beliefs are about? [Plato] |
| 2087 | A rational account might be seeing an image of one's belief, like a reflection in a mirror [Plato] |
| 2090 | A rational account involves giving an image, or analysis, or giving a differentiating mark [Plato] |
| 2081 | Maybe primary elements can be named, but not receive a rational account [Plato] |
| 2088 | A rational account of a wagon would mean knowledge of its hundred parts [Plato] |
| 2047 | What evidence can be brought to show whether we are dreaming or not? [Plato] |
| 2053 | If you claim that all beliefs are true, that includes beliefs opposed to your own [Plato] |
| 2054 | Clearly some people are superior to others when it comes to medicine [Plato] |
| 2059 | How can a relativist form opinions about what will happen in the future? [Plato] |
| 13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
| 1655 | If goodness needs true opinion but not knowledge, you can skip the 'examined life' [Vlastos on Plato] |
| 2058 | God must be the epitome of goodness, and we can only approach a divine state by being as good as possible [Plato] |
| 2057 | There must always be some force of evil ranged against good [Plato] |