58 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 in which a∈A and b∈B [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] |
17813 | Löwenheim-Skolem says any theory with a true interpretation has a model in the natural numbers [White,NP] |
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] |
17812 | Finite cardinalities don't need numbers as objects; numerical quantifiers will do [White,NP] |
17518 | Counting 'coin in this box' may have coin as the unit, with 'in this box' merely as the scope [Ayers] |
17516 | If counting needs a sortal, what of things which fall under two sortals? [Ayers] |
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] |
17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals [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] |
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] |
17520 | Events do not have natural boundaries, and we have to set them [Ayers] |
17519 | To express borderline cases of objects, you need the concept of an 'object' [Ayers] |
17511 | Recognising continuity is separate from sortals, and must precede their use [Ayers] |
17510 | Speakers need the very general category of a thing, if they are to think about it [Ayers] |
17522 | We use sortals to classify physical objects by the nature and origin of their unity [Ayers] |
17515 | Seeing caterpillar and moth as the same needs continuity, not identity of sortal concepts [Ayers] |
17517 | Could the same matter have more than one form or principle of unity? [Ayers] |
17513 | If there are two objects, then 'that marble, man-shaped object' is ambiguous [Ayers] |
17523 | Sortals basically apply to individuals [Ayers] |
17521 | You can't have the concept of a 'stage' if you lack the concept of an object [Ayers] |
17514 | Temporal 'parts' cannot be separated or rearranged [Ayers] |
17509 | Some say a 'covering concept' completes identity; others place the concept in the reference [Ayers] |
17512 | If diachronic identities need covering concepts, why not synchronic identities too? [Ayers] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |