71 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] |
10373 | Logical form can't dictate metaphysics, as it may propose an undesirable property [Schaffer,J] |
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] |
10367 | There is only one fact - the True [Schaffer,J] |
3158 | Theories of intentionality presuppose rationality, so can't explain it [Dennett] |
3159 | Beliefs and desires aren't real; they are prediction techniques [Dennett] |
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] |
10359 | In causation there are three problems of relata, and three metaphysical problems [Schaffer,J] |
10372 | Causation may not be transitive; the last event may follow from the first, but not be caused by it [Schaffer,J] |
10374 | There are at least ten theories about causal connections [Schaffer,J] |
10366 | Causation transcends nature, because absences can cause things [Schaffer,J] |
10377 | Causation may not be a process, if a crucial part of the process is 'disconnected' [Schaffer,J] |
10378 | A causal process needs to be connected to the effect in the right way [Schaffer,J] |
10382 | Causation can't be a process, because a process needs causation as a primitive [Schaffer,J] |
10375 | At least four rivals have challenged the view that causal direction is time direction [Schaffer,J] |
10389 | Causal order must be temporal, or else causes could be blocked, and time couldn't be explained [Schaffer,J] |
10390 | Causal order is not temporal, because of time travel, and simultanous, joint or backward causes [Schaffer,J] |
10380 | Causation is primitive; it is too intractable and central to be reduced; all explanations require it [Schaffer,J] |
10385 | If causation is just observables, or part of common sense, or vacuous, it can't be primitive [Schaffer,J] |
10388 | Causation is utterly essential for numerous philosophical explanations [Schaffer,J] |
10387 | The notion of causation allows understanding of science, without appearing in equations [Schaffer,J] |
10384 | If two different causes are possible in one set of circumstances, causation is primitive [Schaffer,J] |
10386 | If causation is primitive, it can be experienced in ourselves, or inferred as best explanation [Schaffer,J] |
10361 | Events are fairly course-grained (just saying 'hello'), unlike facts (like saying 'hello' loudly) [Schaffer,J] |
10360 | Causal relata are events - or facts, features, tropes, states, situations or aspects [Schaffer,J] |
10362 | One may defend three or four causal relata, as in 'c causes e rather than e*' [Schaffer,J] |
10368 | If causal relata must be in nature and fine-grained, neither facts nor events will do [Schaffer,J] |
10383 | The relata of causation (such as events) need properties as explanation, which need causation! [Schaffer,J] |
10393 | Our selection of 'the' cause is very predictable, so must have a basis [Schaffer,J] |
10394 | Selecting 'the' cause must have a basis; there is no causation without such a selection [Schaffer,J] |
10376 | The actual cause may make an event less likely than a possible more effective cause [Schaffer,J] |
10381 | All four probability versions of causation may need causation to be primitive [Schaffer,J] |
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] |