83 ideas
| 24084 | Seeing with other eyes is more egoism, but exploring other perspectives leads to objectivity [Nietzsche] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 24092 | I tell the truth, even if it is repulsive [Nietzsche] |
| 24114 | The pain in truth is when it destroys a belief [Nietzsche] |
| 24104 | We don't create logic, time and space! The mind obeys laws because they are true [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] |
| 24112 | To think about being we must have an opinion about what it is [Nietzsche] |
| 24089 | Essences are fictions needed for beings who represent things [Nietzsche] |
| 24115 | There is no proof that we forget things - only that we can't recall [Nietzsche] |
| 24090 | Our inclinations would not conflict if we were a unity; we imagine unity for our multiplicity [Nietzsche] |
| 24099 | We contain many minds, which fight for the 'I' of the mind [Nietzsche] |
| 24102 | Thoughts are signs (just as words are) [Nietzsche] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 24120 | Great orators lead their arguments, rather than following them [Nietzsche] |
| 24097 | The pragmatics of language is more comprehensible than the meaning [Nietzsche] |
| 24108 | Actions are just a release of force. They seize on something, which becomes the purpose [Nietzsche] |
| 24105 | Drives make us feel non-feelings; Will is the effect of those feelings [Nietzsche] |
| 24117 | We need lower and higher drives, but they must be under firm control [Nietzsche] |
| 24113 | Our motives don't explain our actions [Nietzsche] |
| 24087 | People who miss beauty seek the sublime, where even the ugly shows its 'beauty' [Nietzsche] |
| 24091 | The sublimity of nature which dwarfs us was a human creation [Nietzsche] |
| 24093 | We can aspire to greatness by creating new functions for ourselves [Nietzsche] |
| 24121 | Greeks might see modern analysis of what is human as impious [Nietzsche] |
| 24107 | Once a drive controls the intellect, it rules, and sets the goals [Nietzsche] |
| 24085 | For absolute morality a goal for mankind is needed [Nietzsche] |
| 24101 | We always assign values, but we may not value those values [Nietzsche] |
| 24094 | Humans are vividly aware of short-term effects, and almost ignorant of the long-term ones [Nietzsche] |
| 24111 | Happiness is the active equilibrium of our drives [Nietzsche] |
| 24109 | Actual morality is more complicated and subtle than theory (which gets paralysed) [Nietzsche] |
| 541 | Virtue comes more from habit than character [Critias] |
| 24110 | Some things we would never do, even for the highest ideals [Nietzsche] |
| 24103 | You should not want too many virtues; one is enough [Nietzsche] |
| 24106 | Talk of 'utility' presupposes that what is useful to people has been defined [Nietzsche] |
| 24086 | The goal is to settle human beings, like other animals, but humans are still changeable [Nietzsche] |
| 24123 | My eternal recurrence is opposed to feeling fragmented and imperfect [Nietzsche] |
| 24088 | See our present lives as eternal! Religions see it as fleeting, and aim at some different life [Nietzsche] |
| 24119 | The eternal return of wastefulness is a terrible thought [Nietzsche] |
| 24116 | Justice says people are not equal, and should become increasingly unequal [Nietzsche] |
| 24098 | Reasons that justify punishment can also justify the crime [Nietzsche] |
| 24118 | Do away with punishment. Counter-retribution is as bad as the crime [Nietzsche] |
| 24100 | If you don't want war, remove your borders; but you set up borders because you want war [Nietzsche] |
| 24095 | Our growth is too subtle to perceive, and long events are too slow for us to grasp [Nietzsche] |
| 24096 | Unlike time, space is subjective. Empty space was assumed, but it doesn't exist [Nietzsche] |
| 24122 | Life is forces conjoined by nutrition, to produce resistance, arrangement and value [Nietzsche] |
| 542 | Fear of the gods was invented to discourage secret sin [Critias] |