71 ideas
| 10468 | A metaphysics has an ontology (objects) and an ideology (expressed ideas about them) [Oliver] |
| 10471 | Ockham's Razor has more content if it says believe only in what is causal [Oliver] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 10749 | Necessary truths seem to all have the same truth-maker [Oliver] |
| 10750 | Slingshot Argument: seems to prove that all sentences have the same truth-maker [Oliver] |
| 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 |
| 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] |
| 10747 | Accepting properties by ontological commitment tells you very little about them [Oliver] |
| 10748 | Reference is not the only way for a predicate to have ontological commitment [Oliver] |
| 10719 | There are four conditions defining the relations between particulars and properties [Oliver] |
| 10721 | If properties are sui generis, are they abstract or concrete? [Oliver] |
| 10716 | There are just as many properties as the laws require [Oliver] |
| 10720 | We have four options, depending whether particulars and properties are sui generis or constructions [Oliver] |
| 10714 | The expressions with properties as their meanings are predicates and abstract singular terms [Oliver] |
| 10715 | There are five main semantic theories for properties [Oliver] |
| 10738 | Tropes are not properties, since they can't be instantiated twice [Oliver] |
| 10739 | The property of redness is the maximal set of the tropes of exactly similar redness [Oliver] |
| 10740 | The orthodox view does not allow for uninstantiated tropes [Oliver] |
| 10741 | Maybe concrete particulars are mereological wholes of abstract particulars [Oliver] |
| 10742 | Tropes can overlap, and shouldn't be splittable into parts [Oliver] |
| 10472 | 'Structural universals' methane and butane are made of the same universals, carbon and hydrogen [Oliver] |
| 10724 | Located universals are wholly present in many places, and two can be in the same place [Oliver] |
| 7963 | Aristotle's instantiated universals cannot account for properties of abstract objects [Oliver] |
| 10730 | If universals ground similarities, what about uniquely instantiated universals? [Oliver] |
| 10727 | Uninstantiated universals seem to exist if they themselves have properties [Oliver] |
| 7962 | Uninstantiated properties are useful in philosophy [Oliver] |
| 10722 | Instantiation is set-membership [Oliver] |
| 10744 | Nominalism can reject abstractions, or universals, or sets [Oliver] |
| 10726 | Things can't be fusions of universals, because two things could then be one thing [Oliver] |
| 10725 | Abstract sets of universals can't be bundled to make concrete things [Oliver] |
| 10745 | Science is modally committed, to disposition, causation and law [Oliver] |
| 3061 | Anaxarchus said that he was not even sure that he knew nothing [Anaxarchus, by Diog. Laertius] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 10746 | Conceptual priority is barely intelligible [Oliver] |