66 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] |
22270 | Frege changed philosophy by extending logic's ability to check the grounds of thinking [Potter on Frege] |
13515 | To study abstract problems, some knowledge of set theory is essential [Hart,WD] |
8939 | We should not describe human laws of thought, but how to correctly track truth [Frege, by Fisher] |
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] |
4971 | I don't use 'subject' and 'predicate' in my way of representing a judgement [Frege] |
17745 | For Frege, 'All A's are B's' means that the concept A implies the concept B [Frege, by Walicki] |
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] |
7728 | Frege has a judgement stroke (vertical, asserting or judging) and a content stroke (horizontal, expressing) [Frege, by Weiner] |
16881 | The laws of logic are boundless, so we want the few whose power contains the others [Frege] |
7622 | In 1879 Frege developed second order logic [Frege, by Putnam] |
7729 | Frege replaced Aristotle's subject/predicate form with function/argument form [Frege, by Weiner] |
9950 | A quantifier is a second-level predicate (which explains how it contributes to truth-conditions) [Frege, by George/Velleman] |
9991 | For Frege the variable ranges over all objects [Frege, by Tait] |
10536 | Frege's domain for variables is all objects, but modern interpretations first fix the domain [Dummett on Frege] |
7730 | Frege introduced quantifiers for generality [Frege, by Weiner] |
7742 | Frege reduced most quantifiers to 'everything' combined with 'not' [Frege, by McCullogh] |
13506 | The universal quantifier can't really mean 'all', because there is no universal set [Hart,WD] |
13824 | Proof theory began with Frege's definition of derivability [Frege, by Prawitz] |
13609 | Frege produced axioms for logic, though that does not now seem the natural basis for logic [Frege, by Kaplan] |
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] |
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] |
17855 | It may be possible to define induction in terms of the ancestral relation [Frege, by Wright,C] |
10607 | Frege's logic has a hierarchy of object, property, property-of-property etc. [Frege, by Smith,P] |
13471 | Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD] |
11008 | Existence is not a first-order property, but the instantiation of a property [Frege, by Read] |
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] |
8130 | Qualities of experience are just representational aspects of experience ('Representationalism') [Harman, by Burge] |
13475 | The Fregean concept of GREEN is a function assigning true to green things, and false to the rest [Hart,WD] |
22280 | Frege's account was top-down and decompositional, not bottom-up and compositional [Frege, by Potter] |
7741 | The predicate 'exists' is actually a natural language expression for a quantifier [Frege, by Weiner] |