72 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] |
| 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] |
| 22121 | The concept of being has only one meaning, whether talking of universals or of God [Duns Scotus, by Dumont] |
| 22122 | Being (not sensation or God) is the primary object of the intellect [Duns Scotus, by Dumont] |
| 16660 | Are things distinct if they are both separate, or if only one of them can be separate? [Duns Scotus, by Pasnau] |
| 16648 | Accidents must have formal being, if they are principles of real action, and of mental action and thought [Duns Scotus] |
| 22125 | Duns Scotus was a realist about universals [Duns Scotus, by Dumont] |
| 15386 | If only the singular exists, science is impossible, as that relies on true generalities [Duns Scotus, by Panaccio] |
| 15387 | If things were singular they would only differ numerically, but horse and tulip differ more than that [Duns Scotus, by Panaccio] |
| 16632 | We distinguish one thing from another by contradiction, because this is, and that is not [Duns Scotus] |
| 22127 | Scotus said a substantial principle of individuation [haecceitas] was needed for an essence [Duns Scotus, by Dumont] |
| 13094 | The haecceity is the featureless thing which gives ultimate individuality to a substance [Duns Scotus, by Cover/O'Leary-Hawthorne] |
| 16650 | 'Unity' is a particularly difficult word, because things can have hidden unity [Duns Scotus] |
| 16770 | It is absurd that there is no difference between a genuinely unified thing, and a mere aggregate [Duns Scotus] |
| 16776 | Substance is an intrinsic thing, so parts of substances can't also be intrinsic things [Duns Scotus] |
| 16626 | Substance is only grasped under the general heading of 'being' [Duns Scotus] |
| 16614 | Matter and form give true unity; subject and accident is just unity 'per accidens' [Duns Scotus] |
| 10919 | What prevents a stone from being divided into parts which are still the stone? [Duns Scotus] |
| 22126 | Avicenna and Duns Scotus say essences have independent and prior existence [Duns Scotus, by Dumont] |
| 16768 | Two things are different if something is true of one and not of the other [Duns Scotus] |
| 8361 | What is true used to be possible, but it may no longer be so [Wright,GHv] |
| 22129 | Certainty comes from the self-evident, from induction, and from self-awareness [Duns Scotus, by Dumont] |
| 22130 | Scotus defended direct 'intuitive cognition', against the abstractive view [Duns Scotus, by Dumont] |
| 22128 | Augustine's 'illumination' theory of knowledge leads to nothing but scepticism [Duns Scotus, by Dumont] |
| 22131 | The will retains its power for opposites, even when it is acting [Duns Scotus, by Dumont] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 8363 | p is a cause and q an effect (not vice versa) if manipulations of p change q [Wright,GHv] |
| 8364 | We can imagine controlling floods by controlling rain, but not vice versa [Wright,GHv] |
| 8366 | The very notion of a cause depends on agency and action [Wright,GHv] |
| 8362 | We give regularities a causal character by subjecting them to experiment [Wright,GHv] |
| 8360 | We must further analyse conditions for causation, into quantifiers or modal concepts [Wright,GHv] |
| 8365 | Some laws are causal (Ohm's Law), but others are conceptual principles (conservation of energy) [Wright,GHv] |
| 22123 | The concept of God is the unique first efficient cause, final cause, and most eminent being [Duns Scotus, by Dumont] |
| 22124 | We can't infer the infinity of God from creation ex nihilo [Duns Scotus, by Dumont] |