68 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 1564 | True and false statements can use exactly the same words [Anon (Diss)] |
| 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] |
| 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] |
| 1561 | Anything can be acceptable in some circumstances and unacceptable in others [Anon (Diss)] |
| 1559 | Thracians think tattooing adds to a girl's beauty, but elsewhere it is a punishment [Anon (Diss)] |
| 1560 | Lydians prostitute their daughters to raise a dowery, but no Greek would marry such a girl [Anon (Diss)] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 1567 | How could someone who knows everything fail to act correctly? [Anon (Diss)] |
| 23928 | Good art produces exaltation and detachment [Bell,C] |
| 23922 | The word 'beauty' leads to confusion, because it denotes distinct emotions [Bell,C] |
| 23929 | We only see landscapes as artistic if we ignore their instrumental value [Bell,C] |
| 23921 | Our feeling for natural beauty is different from the aesthetic emotion of art [Bell,C] |
| 23923 | Visual form can create a sublime mental state [Bell,C] |
| 23927 | Aestheticism invites artist to create beauty, but with no indication of how to do it [Bell,C] |
| 23932 | Art is the expression of an emotion for ultimate reality [Bell,C] |
| 8115 | Only artists can discern significant form; other people must look to art to find it [Bell,C, by Gardner] |
| 23924 | Maybe significant form gives us a feeling for ultimate reality [Bell,C] |
| 23931 | Significant form is the essence of art, which I believe expresses an emotion about reality [Bell,C] |
| 20434 | 'Form' is visual relations, and it is 'significant' if it moves us aesthetically; art needs both [Bell,C, by Feagin] |
| 23934 | The only expression art could have is the emotion resulting from pure form [Bell,C] |
| 23925 | Mere copies of pictures are not significant - unless the copies are very exact [Bell,C] |
| 23926 | Art is distinguished by its aesthetic emotion, which produces appropriate form [Bell,C] |
| 23935 | Aesthetic experience is an exaltation which increases the possibilities of life [Bell,C] |
| 23933 | Aesthetic contemplation is the best and most intense mental state [Bell,C] |
| 22691 | Only artistic qualities matter in art, because they also have the highest moral value [Bell,C] |
| 1563 | Every apparent crime can be right in certain circumstances [Anon (Diss), by PG] |
| 1562 | It is right to lie to someone, to get them to take medicine they are reluctant to take [Anon (Diss)] |
| 1566 | The first priority in elections is to vote for people who support democracy [Anon (Diss)] |
| 1565 | We learn language, and we don't know who teaches us it [Anon (Diss)] |
| 23930 | Religion sees infinite value in some things, and irrelevance in the rest [Bell,C] |