75 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 18996 | A statement S is 'partly true' if it has some wholly true parts [Yablo] |
| 19006 | An 'enthymeme' is an argument with an indispensable unstated assumption [Yablo] |
| 8859 | The main modal logics disagree over three key formulae [Yablo] |
| 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] |
| 18999 | y is only a proper part of x if there is a z which 'makes up the difference' between them [Yablo] |
| 10111 | Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman] |
| 19001 | 'Pegasus doesn't exist' is false without Pegasus, yet the absence of Pegasus is its truthmaker [Yablo] |
| 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] |
| 9138 | An infinite series of sentences asserting falsehood produces the paradox without self-reference [Yablo, by Sorensen] |
| 8865 | If 'the number of Democrats is on the rise', does that mean that 50 million is on the rise? [Yablo] |
| 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] |
| 19002 | A nominalist can assert statements about mathematical objects, as being partly true [Yablo] |
| 8863 | We must treat numbers as existing in order to express ourselves about the arrangement of planets [Yablo] |
| 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] |
| 10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
| 10114 | Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman] |
| 10134 | Much infinite mathematics can still be justified finitely [George/Velleman] |
| 10579 | Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo] |
| 8862 | Platonic objects are really created as existential metaphors [Yablo] |
| 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] |
| 10577 | Concrete objects have few essential properties, but properties of abstractions are mostly essential [Yablo] |
| 10578 | We are thought to know concreta a posteriori, and many abstracta a priori [Yablo] |
| 19489 | For me, fictions are internally true, without a significant internal or external truth-value [Yablo] |
| 19490 | Make-believe can help us to reason about facts and scientific procedures [Yablo] |
| 19491 | 'The clouds are angry' can only mean '...if one were attributing emotions to clouds' [Yablo] |
| 8864 | We quantify over events, worlds, etc. in order to make logical possibilities clearer [Yablo] |
| 19494 | Fictionalism allows that simulated beliefs may be tracking real facts [Yablo] |
| 8858 | Philosophers keep finding unexpected objects, like models, worlds, functions, numbers, events, sets, properties [Yablo] |
| 14381 | A statue is essentially the statue, but its lump is not essentially a statue, so statue isn't lump [Yablo, by Rocca] |
| 18998 | Parthood lacks the restriction of kind which most relations have [Yablo] |
| 19493 | Governing possible worlds theory is the fiction that if something is possible, it happens in a world [Yablo] |
| 19004 | Gettier says you don't know if you are confused about how it is true [Yablo] |
| 19007 | A theory need not be true to be good; it should just be true about its physical aspects [Yablo] |
| 18993 | If sentences point to different evidence, they must have different subject-matter [Yablo] |
| 19003 | Most people say nonblack nonravens do confirm 'all ravens are black', but only a tiny bit [Yablo] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 10805 | A sentence should be recarved to reveal its content or implication relations [Yablo] |
| 18992 | Sentence-meaning is the truth-conditions - plus factors responsible for them [Yablo] |
| 18994 | The content of an assertion can be quite different from compositional content [Yablo] |
| 18997 | Truth-conditions as subject-matter has problems of relevance, short cut, and reversal [Yablo] |
| 19005 | Not-A is too strong to just erase an improper assertion, because it actually reverses A [Yablo] |
| 8861 | Hardly a word in the language is devoid of metaphorical potential [Yablo] |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |