68 ideas
5278 | Philosophy is no more than abstractions concerning observations of human historical development [Marx/Engels] |
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] |
5287 | Philosophical problems are resolved into empirical facts [Marx/Engels] |
23872 | 'Society determines consciousness' is contradictory; society only exists in minds [Weil on Marx/Engels] |
5277 | Life is not determined by consciousness, but consciousness by life [Marx/Engels] |
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] |
5280 | Language co-exists with consciousness, and makes it social [Marx/Engels] |
5276 | The nature of an individual coincides with what they produce and how they produce it [Marx/Engels] |
5281 | Consciousness is a social product [Marx/Engels] |
5289 | When aristocracy or the bourgeoisie dominate, certain values dominate with them [Marx/Engels] |
5274 | Young Hegelians proposed changing our present consciousness for liberating critical consciousness [Marx/Engels] |
21992 | Producing their own subsistence distinguishes men from animals [Marx/Engels] |
5275 | Men distinguish themselves from animals when they begin to produce their means of subsistence [Marx/Engels] |
5292 | Individuals are mutually hostile unless they group together in competition with other groups [Marx/Engels] |
5293 | Only in community are people able to cultivate their gifts, and therefore be free [Marx/Engels] |
5273 | Young Hegelians think consciousness is chains for men, where old Hegelians think it the bond of society [Marx/Engels] |
5283 | In communist society we are not trapped in one activity, but can act freely [Marx/Engels] |
5282 | If the common interest imposes on the individual, his actions become alienated and enslaving [Marx/Engels] |
5288 | The class controlling material production also controls mental production [Marx/Engels] |
5290 | The revolutionary class is opposed to 'class', and represents all of society [Marx/Engels] |
5294 | To assert themselves as individuals, the proletarians must overthrow the State [Marx/Engels] |
5285 | Slavery cannot be abolished without the steam-engine [Marx/Engels] |
5284 | Communism abolishes private property and dissolves the powerful world market [Marx/Engels] |
5291 | The law says private property is the result of the general will [Marx/Engels] |
13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
5279 | Human history must always be studied in relation to industry and exchange [Marx/Engels] |
5286 | Most historians are trapped in the illusions of their own epoch [Marx/Engels] |
10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |