72 ideas
21422 | Moral self-knowledge is the beginning of all human wisdom [Kant] |
21408 | For any subject, its system of non-experiential concepts needs a metaphysics [Kant] |
21416 | Philosophers should not offer multiple proofs - suggesting the weakness of each of them [Kant] |
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] |
10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
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] |
17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
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] |
21410 | That a concept is not self-contradictory does not make what it represents possible [Kant] |
21421 | Within nature man is unimportant, but as moral person he is above any price [Kant] |
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] |
21415 | Duty is impossible without prior moral feeling, conscience, love and self-respect [Kant] |
21409 | Moral principles do not involve feelings [Kant] |
21431 | The love of man is required in order to present the world as a beautiful and perfect moral whole [Kant] |
21437 | All morality directs the will to love of others' ends, and respect for others' rights [Kant] |
21429 | The duty of love is to makes the ends of others one's own [Kant] |
21411 | A duty of virtue is a duty which is also an end [Kant] |
21413 | Virtue is strong maxims for duty [Kant] |
21414 | The supreme principle of virtue is to find universal laws for ends [Kant] |
21436 | We are obliged to show the social virtues, but at least they make a virtuous disposition fashionable [Kant] |
21419 | If virtue becomes a habit, that is a loss of the freedom needed for adopting maxims [Kant] |
21417 | How do we distinguish a mean? The extremes can involve quite different maxims [Kant] |
21420 | If virtue is the mean between vices, then virtue is just the vanishing of vice [Kant] |
21418 | There is one principle of virtues; the virtues are distinguished by their objects [Kant] |
20239 | Unlike us, the early Greeks thought envy was a good thing, and hope a bad thing [Hesiod, by Nietzsche] |
21425 | We can love without respect, and show respect without love [Kant] |
21430 | Disrespect is using a person as a mere means to my own ends [Kant] |
21427 | Respect is limiting our self-esteem by attending to the human dignity of other persons [Kant] |
21426 | Love urges us to get closer to people, but respect to keep our distance [Kant] |
21428 | Respect is purely negative (of not exalting oneself over others), and is thus a duty of Right [Kant] |
21434 | We must respect the humanity even in a vicious criminal [Kant] |
21412 | Humans are distinguished from animals by their capacity to set themselves any sort of end [Kant] |
21435 | Man is both social, and unsociable [Kant] |
21433 | Violation of rights deserves punishment, which is vengeance, rather than restitution [Kant] |
21423 | Men can only have duties to those who qualify as persons [Kant] |
21424 | Cruelty to animals is bad because it dulls our empathy for pain in humans [Kant] |
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] |