65 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 9470 | Modal logic is not an extensional language [Parsons,C] |
| 10098 | The 'power set' of A is all the subsets of A [George/Velleman] |
| 10101 |
Cartesian Product A x B: the set of all ordered pairs |
| 10099 | The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [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] |
| 13418 | The old problems with the axiom of choice are probably better ascribed to the law of excluded middle [Parsons,C] |
| 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] |
| 9469 | Substitutional existential quantifier may explain the existence of linguistic entities [Parsons,C] |
| 9468 | On the substitutional interpretation, '(∃x) Fx' is true iff a closed term 't' makes Ft true [Parsons,C] |
| 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] |
| 17447 | Parsons says counting is tagging as first, second, third..., and converting the last to a cardinal [Parsons,C, by Heck] |
| 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] |
| 18201 | General principles can be obvious in mathematics, but bold speculations in empirical science [Parsons,C] |
| 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] |
| 13419 | If functions are transfinite objects, finitists can have no conception of them [Parsons,C] |
| 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] |
| 13417 | If a mathematical structure is rejected from a physical theory, it retains its mathematical status [Parsons,C] |
| 20339 | Classes rarely share properties with their members - unlike universals and types [Wollheim] |
| 20338 | We often treat a type as if it were a sort of token [Wollheim] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 20342 | Interpretation is performance for some arts, and critical for all arts [Wollheim] |
| 20343 | A love of nature must precede a love of art [Wollheim] |
| 20348 | A criterion of identity for works of art would be easier than a definition [Wollheim] |
| 20347 | If beauty needs organisation, then totally simple things can't be beautiful [Wollheim] |
| 20331 | It is claimed that the expressive properties of artworks are non-physical [Wollheim] |
| 20345 | Some say art must have verbalisable expression, and others say the opposite! [Wollheim] |
| 20336 | Style can't be seen directly within a work, but appreciation needs a grasp of style [Wollheim] |
| 20337 | The traditional view is that knowledge of its genre to essential to appreciating literature [Wollheim] |
| 20333 | If artworks are not physical objects, they are either ideal entities, or collections of phenomena [Wollheim] |
| 20334 | The ideal theory says art is an intuition, shaped by a particular process, and presented in public [Wollheim] |
| 20335 | The ideal theory of art neglects both the audience and the medium employed [Wollheim] |
| 20340 | A musical performance has virtually the same features as the piece of music [Wollheim] |
| 20341 | An interpretation adds further properties to the generic piece of music [Wollheim] |
| 20332 | A drawing only represents Napoleon if the artist intended it to [Wollheim] |