58 ideas
| 8893 | For any given area, there seem to be a huge number of possible coherent systems of beliefs [Bonjour] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 10911 | Part-whole is the key relation among truth-makers [Mulligan/Simons/Smith] |
| 10909 | Truth-makers cannot be the designata of the sentences they make true [Mulligan/Simons/Smith] |
| 10906 | Moments (objects which cannot exist alone) may serve as truth-makers [Mulligan/Simons/Smith] |
| 10907 | The truth-maker for a sentence may not be unique, or may be a combination, or several separate items [Mulligan/Simons/Smith] |
| 10912 | Despite negative propositions, truthmakers are not logical complexes, but ordinary experiences [Mulligan/Simons/Smith] |
| 10908 | Correspondence has to invoke facts or states of affairs, just to serve as truth-makers [Mulligan/Simons/Smith] |
| 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] |
| 8888 | The concept of knowledge is so confused that it is best avoided [Bonjour] |
| 8887 | It is hard to give the concept of 'self-evident' a clear and defensible characterization [Bonjour] |
| 8897 | The adverbial account will still be needed when a mind apprehends its sense-data [Bonjour] |
| 8896 | Conscious states have built-in awareness of content, so we know if a conceptual description of it is correct [Bonjour] |
| 8891 | My incoherent beliefs about art should not undermine my very coherent beliefs about physics [Bonjour] |
| 8892 | Coherence seems to justify empirical beliefs about externals when there is no external input [Bonjour] |
| 8894 | Coherentists must give a reason why coherent justification is likely to lead to the truth [Bonjour] |
| 8889 | Reliabilists disagree over whether some further requirement is needed to produce knowledge [Bonjour] |
| 8890 | If the reliable facts producing a belief are unknown to me, my belief is not rational or responsible [Bonjour] |
| 8895 | If neither the first-level nor the second-level is itself conscious, there seems to be no consciousness present [Bonjour] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |