61 ideas
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
18405 | A 'teepee' argument has several mutually supporting planks to it [Cappelen/Dever] |
10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman] |
10101 | Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman] |
10103 | Grouping by property is common in mathematics, usually using equivalence [George/Velleman] |
10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman] |
10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman] |
10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman] |
10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman] |
17900 | The Axiom of Reducibility made impredicative definitions possible [George/Velleman] |
10109 | ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman] |
10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman] |
10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
10129 | A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman] |
10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman] |
10126 | A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman] |
10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman] |
10127 | A 'complete' theory contains either any sentence or its negation [George/Velleman] |
10106 | Rational numbers give answers to division problems with integers [George/Velleman] |
10102 | The integers are answers to subtraction problems involving natural numbers [George/Velleman] |
10107 | Real numbers provide answers to square root problems [George/Velleman] |
9946 | Logicists say mathematics is applicable because it is totally general [George/Velleman] |
10125 | The classical mathematician believes the real numbers form an actual set [George/Velleman] |
17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman] |
10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman] |
17902 | A successor is the union of a set with its singleton [George/Velleman] |
10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman] |
10130 | Set theory can prove the Peano Postulates [George/Velleman] |
10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman] |
10131 | If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman] |
10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman] |
10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman] |
10095 | Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman] |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
10123 | The intuitionists are the idealists of mathematics [George/Velleman] |
10124 | Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman] |
18422 | Prioprioception focuses on your body parts, not on your self, or indexicality [Cappelen/Dever] |
18425 | We can acquire self-knowledge with mirrors, not just with proprioception and introspection [Cappelen/Dever] |
18421 | Proprioception is only immune from error if you are certain that it represents the agent [Cappelen/Dever] |
18419 | Folk Functionalism is a Ramsification of our folk psychology [Cappelen/Dever] |
18404 | It is assumed that indexical content is needed to represent the perspective of perception [Cappelen/Dever] |
18426 | All information is objective, and purely indexical information is not much use [Cappelen/Dever] |
18427 | If some of our thought is tied to its context, it will be hard to communicate it [Cappelen/Dever] |
18428 | You don't remember your house interior just from an experienced viewpoint [Cappelen/Dever] |
18429 | Our beliefs and desires are not organised around ourselves, but around the world [Cappelen/Dever] |
18407 | Indexicality is not significantly connected to agency [Cappelen/Dever] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
18413 | Fregeans can't agree on what 'senses' are [Cappelen/Dever] |
18417 | Possible worlds accounts of content are notoriously coarse-grained [Cappelen/Dever] |
18408 | Indexicals are just non-constant in meaning, and don't involve any special concepts [Cappelen/Dever] |
18414 | Fregeans say 'I' differs in reference, so it must also differ in sense [Cappelen/Dever] |
18423 | All indexicals can be expressed non-indexically [Cappelen/Dever] |
18406 | The basic Kaplan view is that there is truth-conditional content, and contextual character [Cappelen/Dever] |
18411 | It is proposed that a huge range of linguistic items are context-sensitive [Cappelen/Dever] |
18420 | We deny that action involves some special class of beliefs [Cappelen/Dever] |
6004 | The cardinal virtues are theoretical (based on knowledge), and others are 'non-theoretical' [Hecato, by Dorandi] |