86 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 |
| 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] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 4669 | Persons are conscious, they relate, they think, they feel, and they are self-aware [Glover] |
| 10110 | Corresponding to every concept there is a class (some of them sets) [George/Velleman] |
| 4656 | A problem arises in any moral system that allows more than one absolute right [Glover] |
| 4657 | Double Effect: no bad acts with good consequences, but possibly good acts despite bad consequences [Glover] |
| 4658 | Acts and Omissions: bad consequences are morally better if they result from an omission rather than an act [Glover] |
| 4659 | It doesn't seem worse to switch off a life-support machine than to forget to switch it on [Glover] |
| 4660 | Harmful omissions are unavoidable, while most harmful acts can be avoided [Glover] |
| 4661 | What matters is not intrinsic value of life or rights, but worthwhile and desired life, and avoidance of pain [Glover] |
| 4648 | 'Death' is best seen as irreversible loss of consciousness, since this is why we care about brain function [Glover] |
| 3785 | You can't separate acts from the people performing them [Glover] |
| 3786 | Aggression in defence may be beneficial but morally corrupting [Glover] |
| 4650 | The quality of a life is not altogether independent of its length [Glover] |
| 3784 | Duty prohibits some acts, whatever their consequences [Glover] |
| 3782 | Satisfaction of desires is not at all the same as achieving happiness [Glover, by PG] |
| 3787 | Rule-utilitarianism is either act-utilitarianism, or not really utilitarian [Glover] |
| 3783 | How can utilitarianism decide the ideal population size? [Glover] |
| 4675 | The sanctity of life doctrine implies a serious increase of abnormality among the population [Glover] |
| 4654 | Autonomy favours present opinions over future ones, and says nothing about the interests of potential people [Glover] |
| 4655 | If a whole community did not mind death, respect for autonomy suggests that you could kill them all [Glover] |
| 4680 | Autonomy seems to acquire greater weight when the decision is more important to a person [Glover] |
| 4670 | Being alive is not intrinsically good, and there is no 'right to life' [Glover] |
| 4668 | You can't have a right to something you can't desire, so a foetus has no 'right' to life [Glover] |
| 4651 | Utilitarians object to killing directly (pain, and lost happiness), and to side-effects (loss to others, and precedents) [Glover] |
| 4671 | What is wrong with killing someone, if another equally worthwhile life is substituted? [Glover] |
| 4676 | The 'no trade-off' position: killing is only justified if it prevents other deaths [Glover] |
| 4649 | If someone's life is 'worth living', that gives one direct reason not to kill him [Glover] |
| 4685 | Societies spend a lot to save known persons, but very little to reduce fatal accidents [Glover] |
| 4683 | Involuntary euthanasia is wrong because it violates autonomy, and it has appalling side-effects [Glover] |
| 4682 | Euthanasia is voluntary (patient's wish), or involuntary (ignore wish), or non-voluntary (no wish possible) [Glover] |
| 4684 | Maybe extreme treatment is not saving life, but prolonging the act of dying [Glover] |
| 4681 | The Nazi mass murders seem to have originated in their euthanasia programme [Glover] |
| 4665 | Conception isn't the fixed boundary for a person's beginning, because twins are possible within two weeks [Glover] |
| 4652 | If killing is wrong because it destroys future happiness, not conceiving a happy child is also wrong [Glover] |
| 4662 | Defenders of abortion focus on early pregnancy, while opponents focus on later stages [Glover] |
| 4663 | If abortion is wrong, it is because a foetus is a human being or a person (or potentially so) [Glover] |
| 4664 | If abortion is wrong because of the 'potential' person, that makes contraception wrong too [Glover] |
| 4673 | Abortion differs morally from deliberate non-conception only in its side-effects [Glover] |
| 4666 | If viability is a test or boundary at the beginning of life, it should also be so for frail old people [Glover] |
| 4672 | Apart from side effects, it seems best to replace an inadequate foetus with one which has a better chance [Glover] |
| 4674 | It is always right for a qualified person to perform an abortion when requested by the mother [Glover] |
| 4667 | How would we judge abortion if mothers had transparent wombs? [Glover] |
| 4679 | One test for a worthwhile life is to assess the amount of life for which you would rather be unconscious [Glover] |