70 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] |
10245 | One geometry cannot be more true than another [Poincaré] |
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] |
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] |
22849 | Rawls's theory cannot justify liberalism, since it presupposes free and equal participants [Charvet] |
22848 | People with strong prior beliefs would have nothing to do with a veil of ignorance [Charvet] |
22838 | Societies need shared values, so conservatism is right if rational discussion of values is impossible [Charvet] |
22846 | The universalism of utilitarianism implies a world state [Charvet] |
22835 | Liberals value freedom and equality, but the society itself must decide on its values [Charvet] |
22831 | Modern libertarian societies still provide education and some housing [Charvet] |
22839 | Liberalism needs people to either have equal autonomy, or everyone to have enough autonomy [Charvet] |
22847 | Kant places a higher value on the universal rational will than on the people asserting it [Charvet] |
22821 | Liberalism asserts maximum freedom, but that must be equal for all participants [Charvet] |
22834 | Egalitarian liberals prefer equality (either of input or outcome) to liberty [Charvet] |
22822 | Liberals promote community and well-being - because all good societies need them [Charvet] |
22841 | Identity multiculturalism emerges from communitarianism, preferring community to humanity [Charvet] |
22842 | For communitarians it seems that you must accept the culture you are born into [Charvet] |
22830 | Give by ability and receive by need, rather than a free labour market [Charvet] |
22829 | Allowing defamatory speech is against society's interests, by blurring which people are trustworthy [Charvet] |
22836 | 'Freedom from' is an empty idea, if the freedom is not from impediments to my desires [Charvet] |
22837 | Positive freedom can lead to coercion, if you are forced to do what you chose to do [Charvet] |
22844 | First level autonomy is application of personal values; second level is criticising them [Charvet] |
22840 | Mere equality, as in two trees being the same height, has no value at all [Charvet] |
22843 | Inequalities are worse if they seem to be your fault, rather than social facts [Charvet] |
22845 | Money allows unlimited inequalities, and we obviously all agree to money [Charvet] |
22828 | The rule of law mainly benefits those with property and liberties [Charvet] |
22825 | The 1689 Bill of Rights denied the monarch new courts, or the right to sit as judge [Charvet] |
22823 | The rule of law is mainly to restrict governments [Charvet] |
22827 | Justice superior to the rule of law is claimed on behalf of the workers, or the will of the nation [Charvet] |
22826 | From 1701 only parliament could remove judges, whose decisions could not be discussed [Charvet] |
22832 | Welfare is needed if citizens are to accept the obligations of a liberal state [Charvet] |
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] |