64 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 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] |
| 20457 | Zeno assumes collecting an infinity of things makes an infinite thing [Rovelli] |
| 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] |
| 20468 | Quantum mechanics deals with processes, rather than with things [Rovelli] |
| 7949 | Varied descriptions of an event will explain varied behaviour relating to it [Davidson, by Macdonald,C] |
| 20467 | Quantum mechanics describes the world entirely as events [Rovelli] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 20020 | If one action leads directly to another, they are all one action [Davidson, by Wilson/Schpall] |
| 20072 | We explain an intention by giving an account of acting with an intention [Davidson, by Stout,R] |
| 20045 | Acting for a reason is a combination of a pro attitude, and a belief that the action is appropriate [Davidson] |
| 23734 | The best explanation of reasons as purposes for actions is that they are causal [Davidson, by Smith,M] |
| 23737 | Reasons can give purposes to actions, without actually causing them [Smith,M on Davidson] |
| 20075 | Early Davidson says intentional action is caused by reasons [Davidson, by Stout,R] |
| 6664 | Reasons must be causes when agents act 'for' reasons [Davidson, by Lowe] |
| 3395 | Davidson claims that what causes an action is the reason for doing it [Davidson, by Kim] |
| 20469 | There are probably no infinities, and 'infinite' names what we do not yet know [Rovelli] |
| 20461 | The basic ideas of fields and particles are merged in quantum mechanics [Rovelli] |
| 20462 | Because it is quantised, a field behaves like a set of packets of energy [Rovelli] |
| 20463 | There are about fifteen particles fields, plus a few force fields [Rovelli] |
| 20464 | The world consists of quantum fields, with elementary events happening in spacetime [Rovelli] |
| 20459 | Electrons only exist when they interact, and their being is their combination of quantum leaps [Rovelli] |
| 20460 | Electrons are not waves, because their collisions are at a point, and not spread out [Rovelli] |
| 20466 | Quantum Theory describes events and possible interactions - not how things are [Rovelli] |
| 20465 | Nature has three aspects: granularity, indeterminacy, and relations [Rovelli] |
| 20458 | The world is just particles plus fields; space is the gravitational field [Rovelli] |
| 20470 | Only heat distinguishes past from future [Rovelli] |