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] |
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 in which a∈A and b∈B [George/Velleman] |
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] |
10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [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] |
458 | Nothing could come out of nothing, and existence could never completely cease [Empedocles] |
5112 | Empedocles says things are at rest, unless love unites them, or hatred splits them [Empedocles, by Aristotle] |
13209 | There is no coming-to-be of anything, but only mixing and separating [Empedocles, by Aristotle] |
457 | Substance is not created or destroyed in mortals, but there is only mixing and exchange [Empedocles] |
462 | One vision is produced by both eyes [Empedocles] |
22765 | Wisdom and thought are shared by all things [Empedocles] |
1524 | For Empedocles thinking is almost identical to perception [Empedocles, by Theophrastus] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
589 | 'Nature' is just a word invented by people [Empedocles] |
21823 | The principle of 'Friendship' in Empedocles is the One, and is bodiless [Empedocles, by Plotinus] |
2680 | Empedocles said that there are four material elements, and two further creative elements [Empedocles, by Aristotle] |
6002 | Empedocles says bone is water, fire and earth in ratio 2:4:2 [Empedocles, by Inwood] |
13207 | Fire, Water, Air and Earth are elements, being simple as well as homoeomerous [Empedocles, by Aristotle] |
13218 | The elements combine in coming-to-be, but how do the elements themselves come-to-be? [Aristotle on Empedocles] |
459 | All change is unity through love or division through hate [Empedocles] |
13225 | Love and Strife only explain movement if their effects are distinctive [Aristotle on Empedocles] |
460 | If the one Being ever diminishes it would no longer exist, and what could ever increase it? [Empedocles] |
5090 | Maybe bodies are designed by accident, and the creatures that don't work are destroyed [Empedocles, by Aristotle] |
466 | God is pure mind permeating the universe [Empedocles] |
461 | God is a pure, solitary, and eternal sphere [Empedocles] |
1719 | In Empedocles' theory God is ignorant because, unlike humans, he doesn't know one of the elements (strife) [Aristotle on Empedocles] |
21277 | Even an imperfect machine can exhibit obvious design [Paley] |
21275 | Unlike a stone, the parts of a watch are obviously assembled in order to show the time [Paley] |
21276 | From the obvious purpose and structure of a watch we must infer that it was designed [Paley] |
21357 | No organ shows purpose more obviously than the eyelid [Paley] |
21278 | All the signs of design found in a watch are also found in nature [Paley] |
1522 | It is wretched not to want to think clearly about the gods [Empedocles] |
552 | Empedocles said good and evil were the basic principles [Empedocles, by Aristotle] |