90 ideas
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 15186 | In the tenseless view, all times are equally real, so statements of the future have truth-values [Le Poidevin] |
| 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 |
| 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] |
| 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] |
| 22919 | A thing which makes no difference seems unlikely to exist [Le Poidevin] |
| 15207 | We want illuminating theories, rather than coherent theories [Le Poidevin] |
| 22926 | In addition to causal explanations, they can also be inferential, or definitional, or purposive [Le Poidevin] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 22932 | We don't just describe a time as 'now' from a private viewpoint, but as a fact about the world [Le Poidevin] |
| 6866 | It is disturbing if we become unreal when we die, but if time is unreal, then we remain real after death [Le Poidevin] |
| 15190 | Evil can't be an illusion, because then the illusion that there is evil would be evil [Le Poidevin] |
| 6867 | Existentialism focuses on freedom and self-making, and insertion into the world [Le Poidevin] |
| 22927 | The logical properties of causation are asymmetry, transitivity and irreflexivity [Le Poidevin] |
| 22922 | We can identify unoccupied points in space, so they must exist [Le Poidevin] |
| 22924 | If spatial points exist, then they must be stationary, by definition [Le Poidevin] |
| 22923 | Absolute space explains actual and potential positions, and geometrical truths [Le Poidevin] |
| 22928 | For relationists moving an object beyond the edge of space creates new space [Le Poidevin] |
| 22931 | We distinguish time from space, because it passes, and it has a unique present moment [Le Poidevin] |
| 22921 | Temporal vacuums would be unexperienced, unmeasured, and unending [Le Poidevin] |
| 22917 | Since nothing occurs in a temporal vacuum, there is no way to measure its length [Le Poidevin] |
| 15195 | If the future is not real, we don't seem to have any obligation to future individuals [Le Poidevin] |
| 15188 | If things don't persist through time, then change makes no sense [Le Poidevin] |
| 22934 | Time can't speed up or slow down, so it doesn't seem to be a 'process' [Le Poidevin] |
| 15191 | At the very least, minds themselves seem to be tensed [Le Poidevin] |
| 15197 | Fiction seems to lack a tensed perspective, and offers an example of tenseless language [Le Poidevin] |
| 15206 | It is the view of the future that really decides between tensed and tenseless views of time [Le Poidevin] |
| 15198 | In the B-series, time-positions are unchanging; in the A-series they change (from future to present to past) [Le Poidevin] |
| 15189 | Things which have ceased change their A-series position; things that persist change their B-series position [Le Poidevin] |
| 6865 | A-theory says past, present, future and flow exist; B-theory says this just reports our perspective [Le Poidevin] |
| 15187 | It is claimed that the tense view entails the unreality of both future and past [Le Poidevin] |
| 15205 | Tensed theorists typically try to reduce the tenseless to the tensed [Le Poidevin] |
| 15192 | We share a common now, but not a common here [Le Poidevin] |
| 22939 | The B-series doesn't seem to allow change [Le Poidevin] |
| 22940 | If the B-universe is eternal, why am I trapped in a changing moment of it? [Le Poidevin] |
| 15193 | The new tenseless theory offers indexical truth-conditions, instead of a reductive analysis [Le Poidevin] |
| 22938 | To say that the past causes the present needs them both to be equally real [Le Poidevin] |
| 22947 | An ordered series can be undirected, but time favours moving from earlier to later [Le Poidevin] |
| 22952 | If time's arrow is causal, how can there be non-simultaneous events that are causally unconnected? [Le Poidevin] |
| 22953 | Time's arrow is not causal if there is no temporal gap between cause and effect [Le Poidevin] |
| 22951 | If time's arrow is psychological then different minds can impose different orders on events [Le Poidevin] |
| 22948 | There are Thermodynamic, Psychological and Causal arrows of time [Le Poidevin] |
| 22949 | Presumably if time's arrow is thermodynamic then time ends when entropy is complete [Le Poidevin] |
| 22950 | If time is thermodynamic then entropy is necessary - but the theory says it is probable [Le Poidevin] |
| 22943 | Instantaneous motion is an intrinsic disposition to be elsewhere [Le Poidevin] |
| 22945 | The dynamic view of motion says it is primitive, and not reducible to objects, properties and times [Le Poidevin] |
| 22937 | If the present could have diverse pasts, then past truths can't have present truthmakers [Le Poidevin] |
| 22925 | The present is the past/future boundary, so the first moment of time was not present [Le Poidevin] |
| 22944 | The primitive parts of time are intervals, not instants [Le Poidevin] |
| 22942 | If time is infinitely divisible, then the present must be infinitely short [Le Poidevin] |
| 22946 | The multiverse is distinct time-series, as well as spaces [Le Poidevin] |
| 15196 | God being inside or outside of time both raise a group of difficult problems [Le Poidevin] |
| 22941 | How could a timeless God know what time it is? So could God be both timeless and omniscient? [Le Poidevin] |
| 1513 | The Egyptians were the first to say the soul is immortal and reincarnated [Herodotus] |