63 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] |
| 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] |
| 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] |
| 15148 | Powers give explanations, without being necessary for some class membership [Chakravartty] |
| 15145 | A kind essence is the necessary and sufficient properties for membership of a class [Chakravartty] |
| 15147 | Cluster kinds are explained simply by sharing some properties, not by an 'essence' [Chakravartty] |
| 15144 | Explanation of causal phenomena concerns essential kinds - but also lack of them [Chakravartty] |
| 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] |
| 22673 | Wherever there is a small community, the association of the people is natural [Tocqueville] |
| 22676 | The people are just individuals, and only present themselves as united to foreigners [Tocqueville] |
| 22679 | Vast empires are bad for well-being and freedom, though they may promote glory [Tocqueville] |
| 22680 | People would be much happier and freer in small nations [Tocqueville] |
| 22675 | In American judges rule according to the Constitution, not the law [Tocqueville] |
| 22677 | A monarchical family is always deeply concerned with the interests of the state [Tocqueville] |
| 22683 | Despots like to see their own regulations ignored, by themselves and their agents [Tocqueville] |
| 22669 | Aristocracy is constituted by inherited landed property [Tocqueville] |
| 22674 | In Europe it is thought that local government is best handled centrally [Tocqueville] |
| 22678 | An election, and its lead up time, are always a national crisis [Tocqueville] |
| 22682 | Universal suffrage is no guarantee of wise choices [Tocqueville] |
| 22670 | Slavery undermines the morals and energy of a society [Tocqueville] |
| 22681 | The liberty of the press is more valuable for what it prevents than what it promotes [Tocqueville] |
| 22672 | It is admirable to elevate the humble to the level of the great, but the opposite is depraved [Tocqueville] |
| 22671 | Equality can only be established by equal rights for all (or no rights for anyone) [Tocqueville] |
| 15146 | Some kinds, such as electrons, have essences, but 'cluster kinds' do not [Chakravartty] |
| 15151 | Many causal laws do not refer to kinds, but only to properties [Chakravartty] |
| 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] |