73 ideas
1627 | Any statement can be held true if we make enough adjustment to the rest of the system [Quine] |
22820 | Early Romantics sought a plurality of systems, in a quest for freedom [Hösle] |
1623 | Definition rests on synonymy, rather than explaining it [Quine] |
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] |
9204 | Quine's arguments fail because he naively conflates names with descriptions [Fine,K on Quine] |
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] |
17738 | Quine blurs the difference between knowledge of arithmetic and of physics [Jenkins on Quine] |
19492 | Quine is hopeless circular, deriving ontology from what is literal, and 'literal' from good ontology [Yablo on Quine] |
1628 | If physical objects are a myth, they are useful for making sense of experience [Quine] |
10929 | Aristotelian essence of the object has become the modern essence of meaning [Quine] |
12188 | Contrary to some claims, Quine does not deny logical necessity [Quine, by McFetridge] |
15090 | Quine's attack on the analytic-synthetic distinction undermined necessary truths [Quine, by Shoemaker] |
9383 | Metaphysical analyticity (and linguistic necessity) are hopeless, but epistemic analyticity is a priori [Boghossian on Quine] |
12424 | Quine challenges the claim that analytic truths are knowable a priori [Quine, by Kitcher] |
9337 | Science is empirical, simple and conservative; any belief can hence be abandoned; so no a priori [Quine, by Horwich] |
9338 | Quine's objections to a priori knowledge only work in the domain of science [Horwich on Quine] |
9340 | Logic, arithmetic and geometry are revisable and a posteriori; quantum logic could be right [Horwich on Quine] |
1620 | Empiricism makes a basic distinction between truths based or not based on facts [Quine] |
1629 | Our outer beliefs must match experience, and our inner ones must be simple [Quine] |
19488 | The second dogma is linking every statement to some determinate observations [Quine, by Yablo] |
1625 | Statements about the external world face the tribunal of sense experience as a corporate body [Quine] |
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] |
1626 | It is troublesome nonsense to split statements into a linguistic and a factual component [Quine] |
7317 | 'Renate' and 'cordate' have identical extensions, but are not synonymous [Quine, by Miller,A] |
1621 | Once meaning and reference are separated, meaning ceases to seem important [Quine] |
9371 | Analytic statements are either logical truths (all reinterpretations) or they depend on synonymy [Quine] |
9366 | Quine's attack on analyticity undermined linguistic views of necessity, and analytic views of the a priori [Quine, by Boghossian] |
14473 | Quine attacks the Fregean idea that we can define analyticity through synonyous substitution [Quine, by Thomasson] |
7321 | The last two parts of 'Two Dogmas' are much the best [Miller,A on Quine] |
8803 | Erasing the analytic/synthetic distinction got rid of meanings, and saved philosophy of language [Davidson on Quine] |
17737 | The analytic needs excessively small units of meaning and empirical confirmation [Quine, by Jenkins] |
1624 | If we try to define analyticity by synonymy, that leads back to analyticity [Quine] |
1622 | Did someone ever actually define 'bachelor' as 'unmarried man'? [Quine] |
22819 | In the 18th century history came to be seen as progressive, rather than cyclical [Hösle] |
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] |