65 ideas
21584 | A sense of timelessness is essential to wisdom [Russell] |
21572 | Philosophical disputes are mostly hopeless, because philosophers don't understand each other [Russell] |
21571 | Philosophical systems are interesting, but we now need a more objective scientific philosophy [Russell] |
21574 | Hegel's confusions over 'is' show how vast systems can be built on simple errors [Russell] |
21587 | Philosophers sometimes neglect truth and distort facts to attain a nice system [Russell] |
21582 | Physicists accept particles, points and instants, while pretending they don't do metaphysics [Russell] |
21573 | When problems are analysed properly, they are either logical, or not philosophical at all [Russell] |
6782 | Realism is the only philosophy of science that doesn't make the success of science a miracle [Putnam] |
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
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] |
21588 | Logic gives the method of research in philosophy [Russell] |
21586 | The logical connectives are not objects, but are formal, and need a context [Russell] |
10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
21585 | The tortoise won't win, because infinite instants don't compose an infinitely long time [Russell] |
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] |
11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
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] |
10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
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] |
13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
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] |
21684 | Atomic facts may be inferrable from others, but never from non-atomic facts [Russell] |
22181 | Putnam says anti-realism is a bad explanation of accurate predictions [Putnam, by Okasha] |
22316 | A positive and negative fact have the same constituents; their difference is primitive [Russell] |
21576 | With asymmetrical relations (before/after) the reduction to properties is impossible [Russell] |
21575 | When we attribute a common quality to a group, we can forget the quality and just talk of the group [Russell] |
21580 | Science condemns sense-data and accepts matter, but a logical construction must link them [Russell] |
21583 | When sense-data change, there must be indistinguishable sense-data in the process [Russell] |
21577 | Empirical truths are particular, so general truths need an a priori input of generality [Russell] |
21579 | Objects are treated as real when they connect with other experiences in a normal way [Russell] |
21578 | Global scepticism is irrefutable, but can't replace our other beliefs, and just makes us hesitate [Russell] |
6416 | Other minds seem to exist, because their testimony supports realism about the world [Russell, by Grayling] |
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] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
21581 | We never experience times, but only succession of events [Russell] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |