70 ideas
| 21474 | Metaphysics studies the inexplicable ends of explanation [Schopenhauer] |
| 10528 | Definitions concern how we should speak, not how things are [Fine,K] |
| 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] |
| 10529 | If Hume's Principle can define numbers, we needn't worry about its truth [Fine,K] |
| 10530 | Hume's Principle is either adequate for number but fails to define properly, or vice versa [Fine,K] |
| 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] |
| 21470 | For me the objective thing-in-itself is the will [Schopenhauer] |
| 21479 | Knowledge is not power! Ignorant people possess supreme authority [Schopenhauer] |
| 21476 | A priori propositions are those we could never be seriously motivated to challenge [Schopenhauer] |
| 21473 | All knowledge and explanation rests on the inexplicable [Schopenhauer] |
| 21478 | Half our thinking is unconscious, and we reach conclusions while unaware of premises [Schopenhauer] |
| 21477 | We don't control our own thinking [Schopenhauer] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 21475 | All of our concepts are borrowed from perceptual knowledge [Schopenhauer] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 10527 | An abstraction principle should not 'inflate', producing more abstractions than objects [Fine,K] |
| 21372 | Aesthetics concerns how we can take pleasure in an object, with no reference to the will [Schopenhauer] |
| 21488 | The beautiful is a perception of Plato's Forms, which eliminates the will [Schopenhauer] |
| 21483 | Man is essentially a dreadful wild animal [Schopenhauer] |
| 21466 | Pleasure is weaker, and pain stronger, than we expect [Schopenhauer] |
| 21484 | A man's character can be learned from a single characteristic action [Schopenhauer] |
| 21481 | Buddhists wisely start with the cardinal vices [Schopenhauer] |
| 21482 | The five Chinese virtues: pity, justice, politeness, wisdom, honesty [Schopenhauer] |
| 21469 | Human life is a mistake, shown by boredom, which is direct awareness of the fact [Schopenhauer] |
| 21480 | Boredom is only felt by those clever enough to need activity [Schopenhauer] |
| 21485 | The state only exists to defend citizens, from exterior threats, and from one another [Schopenhauer] |
| 21486 | Poverty and slavery are virtually two words for the same thing [Schopenhauer] |
| 21487 | The freedom of the press to sell poison outweighs its usefulness [Schopenhauer] |
| 21471 | If suicide was quick and easy, most people would have done it by now [Schopenhauer] |
| 21467 | Would humanity still exist if sex wasn't both desired and pleasurable? [Schopenhauer] |
| 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] |
| 21472 | Only religion introduces serious issues to uneducated people [Schopenhauer] |
| 21468 | The Creator created the possibilities for worlds, so should have made a better one than this possible [Schopenhauer] |