32 ideas
21463 | Hamann, Herder and Jacobi were key opponents of the Enlightenment [Gardner] |
21459 | Kant halted rationalism, and forced empiricists to worry about foundations [Gardner] |
21460 | Only Kant and Hegel have united nature, morals, politics, aesthetics and religion [Gardner] |
15924 | Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine] |
21443 | Transcendental proofs derive necessities from possibilities (e.g. possibility of experiencing objects) [Gardner] |
17608 | We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo] |
17607 | Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo] |
10870 | ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg] |
13012 | Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy] |
17609 | Set theory can be reduced to a few definitions and seven independent axioms [Zermelo] |
9565 | Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara] |
3339 | For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn] |
17832 | Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M] |
13017 | Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy] |
13028 | Replacement was added when some advanced theorems seemed to need it [Zermelo, by Maddy] |
13015 | Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy] |
13020 | The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy] |
13486 | Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD] |
17626 | The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo] |
21444 | Modern geoemtry is either 'pure' (and formal), or 'applied' (and a posteriori) [Gardner] |
13487 | In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD] |
15897 | Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine] |
18178 | For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy] |
13027 | Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy] |
9627 | Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR] |
21453 | Leibnizian monads qualify as Kantian noumena [Gardner] |
17613 | We should judge principles by the science, not science by some fixed principles [Zermelo] |
8108 | Aesthetics presupposes a distinctive sort of experience, and a unified essence for art [Gardner] |
8112 | Art works originate in the artist's mind, and appreciation is re-creating this mental object [Gardner] |
8111 | Aesthetic objectivists must explain pleasure being essential, but not in the object [Gardner] |
8109 | Aesthetic judgements necessarily require first-hand experience, unlike moral judgements [Gardner] |
3031 | The greatest good is not the achievement of desire, but to desire what is proper [Menedemus, by Diog. Laertius] |