76 ideas
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 7401 | Heat and colour don't exist, so cannot mislead about the external world [Galileo, by Tuck] |
| 5454 | Tastes, odours and colours only reside in consciousness, and would disappear with creatures [Galileo] |
| 16560 | Galileo introduced geometrico-mechanical explanation, based on Archimedes [Galileo, by Machamer/Darden/Craver] |
| 4363 | The word 'person' is useless in ethics, because what counts as a good or bad self-conscious being? [Hursthouse] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 4355 | There may be inverse akrasia, where the agent's action is better than their judgement recommends [Hursthouse] |
| 4325 | Must all actions be caused in part by a desire, or can a belief on its own be sufficient? [Hursthouse] |
| 4351 | It is a fantasy that only through the study of philosophy can one become virtuous [Hursthouse] |
| 4340 | You are not a dishonest person if a tragic dilemma forces you to do something dishonest [Hursthouse] |
| 4329 | After a moral dilemma is resolved there is still a 'remainder', requiring (say) regret [Hursthouse] |
| 4330 | Deontologists resolve moral dilemmas by saying the rule conflict is merely apparent [Hursthouse] |
| 4341 | Involuntary actions performed in tragic dilemmas are bad because they mar a good life [Hursthouse] |
| 4358 | Virtue may be neither sufficient nor necessary for eudaimonia [Hursthouse] |
| 4337 | Teenagers are often quite wise about ideals, but rather stupid about consequences [Hursthouse] |
| 4324 | Animals and plants can 'flourish', but only rational beings can have eudaimonia [Hursthouse] |
| 4359 | When it comes to bringing up children, most of us think that the virtues are the best bet [Hursthouse] |
| 4336 | Any strict ranking of virtues or rules gets abandoned when faced with particular cases [Hursthouse] |
| 4334 | Virtue ethics is open to the objection that it fails to show priority among the virtues [Hursthouse] |
| 4361 | Good animals can survive, breed, feel characteristic pleasure and pain, and contribute to the group [Hursthouse] |
| 4349 | Virtuous people may not be fully clear about their reasons for action [Hursthouse] |
| 4352 | Performing an act simply because it is virtuous is sufficient to be 'morally motivated' or 'dutiful' [Hursthouse] |
| 4353 | If moral motivation is an all-or-nothing sense of duty, how can children act morally? [Hursthouse] |
| 4346 | The emotions of sympathy, compassion and love are no guarantee of right action or acting well [Hursthouse] |
| 4339 | According to virtue ethics, two agents may respond differently, and yet both be right [Hursthouse] |
| 4354 | Maybe in a deeply poisoned character none of their milder character traits could ever be a virtue [Hursthouse] |
| 4364 | Being unusually virtuous in some areas may entail being less virtuous in others [Hursthouse] |
| 4356 | We are puzzled by a person who can show an exceptional virtue and also behave very badly [Hursthouse] |
| 4327 | Deontologists do consider consequences, because they reveal when a rule might apply [Hursthouse] |
| 4335 | 'Codifiable' morality give rules for decisions which don't require wisdom [Hursthouse] |
| 4328 | Preference utilitarianism aims to be completely value-free, or empirical [Hursthouse] |
| 4343 | We are torn between utilitarian and deontological views of lying, depending on the examples [Hursthouse] |
| 4338 | Deontologists usually accuse utilitarians of oversimplifying hard cases [Hursthouse] |
| 4365 | We are distinct from other animals in behaving rationally - pursuing something as good, for reasons [Hursthouse] |
| 3645 | To understand the universe mathematics is essential [Galileo] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |
| 4350 | If people are virtuous in obedience to God, would they become wicked if they lost their faith? [Hursthouse] |