| 9955 | Contextual definitions replace a complete sentence containing the expression |
| 10031 | Impredicative definitions quantify over the thing being defined |
| 10098 | The 'power set' of A is all the subsets of A |
| 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}} |
| 10103 | Grouping by property is common in mathematics, usually using equivalence |
| 10104 | 'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words |
| 10096 | Even the elements of sets in ZFC are sets, resting on the pure empty set |
| 10097 | Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y |
| 10100 | Axiom of Pairing: for all sets x and y, there is a set z containing just x and y |
| 17900 | The Axiom of Reducibility made impredicative definitions possible |
| 10109 | ZFC can prove that there is no set corresponding to the concept 'set' |
| 10108 | As a reduction of arithmetic, set theory is not fully general, and so not logical |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world |
| 10129 | A 'model' is a meaning-assignment which makes all the axioms true |
| 10105 | Differences between isomorphic structures seem unimportant |
| 10119 | Consistency is a purely syntactic property, unlike the semantic property of soundness |
| 10126 | A 'consistent' theory cannot contain both a sentence and its negation |
| 10120 | Soundness is a semantic property, unlike the purely syntactic property of consistency |
| 10127 | A 'complete' theory contains either any sentence or its negation |
| 10106 | Rational numbers give answers to division problems with integers |
| 10102 | The integers are answers to subtraction problems involving natural numbers |
| 10107 | Real numbers provide answers to square root problems |
| 9946 | Logicists say mathematics is applicable because it is totally general |
| 10125 | The classical mathematician believes the real numbers form an actual set |
| 17899 | Second-order induction is stronger as it covers all concepts, not just first-order definable ones |
| 10128 | The Incompleteness proofs use arithmetic to talk about formal arithmetic |
| 17902 | A successor is the union of a set with its singleton |
| 10133 | Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle |
| 10130 | Set theory can prove the Peano Postulates |
| 10089 | Talk of 'abstract entities' is more a label for the problem than a solution to it |
| 10131 | If mathematics is not about particulars, observing particulars must be irrelevant |
| 17901 | Type theory prohibits (oddly) a set containing an individual and a set of individuals |
| 10092 | In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. |
| 10094 | The theory of types seems to rule out harmless sets as well as paradoxical ones. |
| 10095 | Type theory has only finitely many items at each level, which is a problem for mathematics |
| 10134 | Much infinite mathematics can still be justified finitely |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions |
| 10123 | The intuitionists are the idealists of mathematics |
| 10124 | Gödel's First Theorem suggests there are truths which are independent of proof |
| 10110 | Corresponding to every concept there is a class (some of them sets) |