71 ideas
| 23027 | Ideals and metaphysics are practical, not imaginative or speculative [Green,TH, by Muirhead] |
| 9955 | Contextual definitions replace a complete sentence containing the expression [George/Velleman] |
| 10031 | Impredicative definitions quantify over the thing being defined [George/Velleman] |
| 23030 | Truth is a relation to a whole of organised knowledge in the collection of rational minds [Green,TH, by Muirhead] |
| 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] |
| 10304 | Very few things in set theory remain valid in intuitionist mathematics [Bernays] |
| 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] |
| 10303 | Restricted Platonism is just an ideal projection of a domain of thought [Bernays] |
| 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] |
| 10306 | Mathematical abstraction just goes in a different direction from logic [Bernays] |
| 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] |
| 23044 | All knowledge rests on a fundamental unity between the knower and what is known [Green,TH, by Muirhead] |
| 23034 | The ultimate test for truth is the systematic interdependence in nature [Green,TH, by Muirhead] |
| 23029 | Knowledge is secured by the relations between its parts, through differences and identities [Green,TH, by Muirhead] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 23035 | The good life aims at perfections, or absolute laws, or what is absolutely desirable [Green,TH] |
| 23032 | What is distinctive of human life is the desire for self-improvement [Green,TH, by Muirhead] |
| 23033 | Hedonism offers no satisfaction, because what we desire is self-betterment [Green,TH, by Muirhead] |
| 23046 | States only have full authority if they heed the claims of human fellowship [Green,TH] |
| 23045 | Politics is compromises, which seem supported by a social contract, but express the will of no one [Green,TH] |
| 23050 | The ideal is a society in which all citizens are ladies and gentlemen [Green,TH] |
| 23052 | Enfranchisement is an end in itself; it makes a person moral, and gives a basis for respect [Green,TH] |
| 23036 | The good is identified by the capacities of its participants [Green,TH, by Muirhead] |
| 23039 | A true state is only unified and stabilised by acknowledging individuality [Green,TH, by Muirhead] |
| 23037 | People are improved by egalitarian institutions and habits [Green,TH] |
| 23051 | Equality also implies liberty, because equality must be of opportunity as well as possessions [Green,TH] |
| 23043 | All talk of the progress of a nation must reduce to the progress of its individual members [Green,TH] |
| 23038 | People only develop their personality through co-operation with the social whole [Green,TH, by Muirhead] |
| 23028 | The highest political efforts express our deeper social spirit [Green,TH, by Muirhead] |
| 23054 | Communism is wrong because it restricts the freedom of individuals to contribute to the community [Green,TH, by Muirhead] |
| 23047 | Original common ownership is securing private property, not denying it [Green,TH, by Muirhead] |
| 23042 | National spirit only exists in the individuals who embody it [Green,TH, by Muirhead] |
| 23048 | The ground of property ownership is not force but the power to use it for social ends [Green,TH, by Muirhead] |
| 23049 | Property is needed by all citizens, to empower them to achieve social goods [Green,TH] |
| 23040 | If something develops, its true nature is embodied in its end [Green,TH] |
| 23031 | God is the ideal end of the mature mind's final development [Green,TH] |
| 23041 | God is the realisation of the possibilities of each man's self [Green,TH] |