78 ideas
15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
18395 | Sets are mereological sums of the singletons of their members [Lewis, by Armstrong] |
15496 | We can build set theory on singletons: classes are then fusions of subclasses, membership is the singleton [Lewis] |
15500 | Classes divide into subclasses in many ways, but into members in only one way [Lewis] |
15499 | A subclass of a subclass is itself a subclass; a member of a member is not in general a member [Lewis] |
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] |
15503 | We needn't accept this speck of nothingness, this black hole in the fabric of Reality! [Lewis] |
15498 | We can accept the null set, but there is no null class of anything [Lewis] |
15502 | There are four main reasons for asserting that there is an empty set [Lewis] |
15506 | If we don't understand the singleton, then we don't understand classes [Lewis] |
15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
15497 | We can replace the membership relation with the member-singleton relation (plus mereology) [Lewis] |
15511 | If singleton membership is external, why is an object a member of one rather than another? [Lewis] |
15513 | Maybe singletons have a structure, of a thing and a lasso? [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] |
15507 | Set theory has some unofficial axioms, generalisations about how to understand it [Lewis] |
10191 | Set theory reduces to a mereological theory with singletons as the only atoms [Lewis, by MacBride] |
13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
15508 | If singletons are where their members are, then so are all sets [Lewis] |
15514 | A huge part of Reality is only accepted as existing if you have accepted set theory [Lewis] |
15523 | Set theory isn't innocent; it generates infinities from a single thing; but mathematics needs it [Lewis] |
14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
15525 | Plural quantification lacks a complete axiom system [Lewis] |
15518 | I like plural quantification, but am not convinced of its connection with second-order logic [Lewis] |
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] |
15524 | Zermelo's model of arithmetic is distinctive because it rests on a primitive of set theory [Lewis] |
18176 | Pure mathematics is pure set theory [Cantor] |
15517 | Giving up classes means giving up successful mathematics because of dubious philosophy [Lewis] |
15515 | To be a structuralist, you quantify over relations [Lewis] |
8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
15520 | Existence doesn't come in degrees; once asserted, it can't then be qualified [Lewis] |
15501 | We have no idea of a third sort of thing, that isn't an individual, a class, or their mixture [Lewis] |
15504 | Atomless gunk is an individual whose parts all have further proper parts [Lewis] |
15516 | A property is any class of possibilia [Lewis] |
14748 | The many are many and the one is one, so they can't be identical [Lewis] |
6129 | Lewis affirms 'composition as identity' - that an object is no more than its parts [Lewis, by Merricks] |
15512 | In mereology no two things consist of the same atoms [Lewis] |
15519 | Trout-turkeys exist, despite lacking cohesion, natural joints and united causal power [Lewis] |
15521 | Given cats, a fusion of cats adds nothing further to reality [Lewis] |
15522 | The one has different truths from the many; it is one rather than many, one rather than six [Lewis] |
14244 | Lewis only uses fusions to create unities, but fusions notoriously flatten our distinctions [Oliver/Smiley on Lewis] |
10660 | A commitment to cat-fusions is not a further commitment; it is them and they are it [Lewis] |
10566 | Lewis prefers giving up singletons to giving up sums [Lewis, by Fine,K] |
15509 | Some say qualities are parts of things - as repeatable universals, or as particulars [Lewis] |
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] |
23053 | The great interest of the human race is cordial unity and unlimited mutual aid [Owen] |
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] |