79 ideas
| 7834 | Great philosophies are confessions by the author, growing out of moral intentions [Nietzsche] |
| 7080 | Metaphysics divided the old unified Greek world into two [Nietzsche, by Critchley] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 11090 | Why do we want truth, rather than falsehood or ignorance? The value of truth is a problem [Nietzsche] |
| 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] |
| 7079 | Nietzsche resists nihilism through new values, for a world of becoming, without worship [Nietzsche, by Critchley] |
| 2878 | We see an approximation of a tree, not the full detail [Nietzsche] |
| 20140 | We shouldn't object to a false judgement, if it enhances and preserves life [Nietzsche] |
| 2877 | Morality becomes a problem when we compare many moralities [Nietzsche] |
| 20355 | The ranking of a person's innermost drives reveals their true nature [Nietzsche] |
| 2291 | A thought comes when 'it' wants, not when 'I' want [Nietzsche] |
| 2871 | Wanting 'freedom of will' is wanting to pull oneself into existence out of the swamp of nothingness by one's own hair [Nietzsche] |
| 20381 | It is psychology which reveals the basic problems [Nietzsche] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 2860 | The most boring and dangerous of all errors is Plato's invention of pure spirit and goodness [Nietzsche] |
| 1568 | Nietzsche felt that Plato's views downgraded the human body and its brevity of life [Nietzsche, by Roochnik] |
| 2883 | Noble people see themselves as the determiners of values [Nietzsche] |
| 23440 | Nietzsche's judgement of actions by psychology instead of outcome was poisonous [Foot on Nietzsche] |
| 2875 | That which is done out of love always takes place beyond good and evil [Nietzsche] |
| 2868 | Nature is totally indifferent, so you should try to be different from it, not live by it [Nietzsche] |
| 2882 | Morality originally judged people, and actions only later on [Nietzsche] |
| 2872 | In the earliest phase of human history only consequences mattered [Nietzsche] |
| 2885 | The noble soul has reverence for itself [Nietzsche] |
| 20134 | Moralities extravagantly address themselves to 'all', by falsely generalising [Nietzsche] |
| 2881 | Virtue has been greatly harmed by the boringness of its advocates [Nietzsche] |
| 20382 | The four virtues are courage, insight, sympathy, solitude [Nietzsche] |
| 2879 | In ancient Rome pity was considered neither good nor bad [Nietzsche] |
| 2859 | The idea of the categorical imperative is just that we should all be very obedient [Nietzsche] |
| 2884 | The morality of slaves is the morality of utility [Nietzsche] |
| 2880 | The greatest possibilities in man are still unexhausted [Nietzsche] |
| 2876 | The thought of suicide is a great reassurance on bad nights [Nietzsche] |
| 7078 | The freedom of the subject means the collapse of moral certainty [Nietzsche, by Critchley] |
| 2874 | Man is the animal whose nature has not yet been fixed [Nietzsche] |
| 6869 | Nietzsche thinks the human condition is to overcome and remake itself [Nietzsche, by Ansell Pearson] |
| 20137 | The great person engages wholly with life, and is happy to endlessly relive the life they created [Nietzsche] |
| 20139 | Only aristocratic societies can elevate the human species [Nietzsche] |
| 20373 | A healthy aristocracy has no qualms about using multitudes of men as instruments [Nietzsche] |
| 22394 | Democracy diminishes mankind, making them mediocre and lowering their value [Nietzsche] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| 2867 | Christianity is Platonism for the people [Nietzsche] |