87 ideas
9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
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 in which a∈A and b∈B [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] |
23308 | Reasoning relates to understanding as time does to eternity [Boethius, by Sorabji] |
5771 | Knowledge of present events doesn't make them necessary, so future events are no different [Boethius] |
5767 | Rational natures require free will, in order to have power of judgement [Boethius] |
5768 | God's universal foreknowledge seems opposed to free will [Boethius] |
5769 | Does foreknowledge cause necessity, or necessity cause foreknowledge? [Boethius] |
10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
5762 | The wicked want goodness, so they would not be wicked if they obtained it [Boethius] |
5770 | Rewards and punishments are not deserved if they don't arise from free movement of the mind [Boethius] |
7222 | It is a crime for someone with a violent disposition to get drunk [Mill] |
5764 | When people fall into wickedness they lose their human nature [Boethius] |
5756 | Happiness is a good which once obtained leaves nothing more to be desired [Boethius] |
5763 | The bad seek the good through desire, but the good through virtue, which is more natural [Boethius] |
5759 | Varied aims cannot be good because they differ, but only become good when they unify [Boethius] |
7214 | Ethics rests on utility, which is the permanent progressive interests of people [Mill] |
7212 | Individuals have sovereignty over their own bodies and minds [Mill] |
7210 | The will of the people is that of the largest or most active part of the people [Mill] |
7227 | It is evil to give a government any more power than is necessary [Mill] |
7228 | Individuals often do things better than governments [Mill] |
7230 | Aim for the maximum dissemination of power consistent with efficiency [Mill] |
20515 | Maximise happiness by an area of strict privacy, and an area of utilitarian interventions [Mill, by Wolff,J] |
7229 | People who transact their own business will also have the initiative to control their government [Mill] |
7211 | Prevention of harm to others is the only justification for exercising power over people [Mill] |
7231 | The worth of a State, in the long run, is the worth of the individuals composing it [Mill] |
7217 | The main argument for freedom is that interference with it is usually misguided [Mill] |
5754 | You can't control someone's free mind, only their body and possessions [Boethius] |
7213 | Liberty arises at the point where people can freely and equally discuss things [Mill] |
20517 | Utilitarianism values liberty, but guides us on which ones we should have or not have [Mill, by Wolff,J] |
20516 | Mill defends freedom as increasing happiness, but maybe it is an intrinsic good [Wolff,J on Mill] |
7215 | True freedom is pursuing our own good, while not impeding others [Mill] |
7218 | Individuals are not accountable for actions which only concern themselves [Mill] |
7221 | Blocking entry to an unsafe bridge does not infringe liberty, since no one wants unsafe bridges [Mill] |
7223 | Pimping and running a gambling-house are on the border between toleration and restraint [Mill] |
7220 | Restraint for its own sake is an evil [Mill] |
7219 | Society can punish actions which it believes to be prejudicial to others [Mill] |
7226 | Benefits performed by individuals, not by government, help also to educate them [Mill] |
7224 | We need individual opinions and conduct, and State education is a means to prevent that [Mill] |
7225 | It is a crime to create a being who lacks the ordinary chances of a desirable existence [Mill] |
16692 | Divine eternity is the all-at-once and complete possession of unending life [Boethius] |
5752 | Where does evil come from if there is a god; where does good come from if there isn't? [Boethius] |
5758 | God is the good [Boethius] |
5757 | God is the supreme good, so no source of goodness could take precedence over God [Boethius] |
5760 | The power through which creation remains in existence and motion I call 'God' [Boethius] |
5753 | The regular events of this life could never be due to chance [Boethius] |
7216 | The ethics of the Gospel has been supplemented by barbarous Old Testament values [Mill] |
5765 | The reward of the good is to become gods [Boethius] |
5761 | God can do anything, but he cannot do evil, so evil must be nothing [Boethius] |
5766 | If you could see the plan of Providence, you would not think there was evil anywhere [Boethius] |