51 ideas
| 1765 | Diogenes said avoidance of philosophy is the lack of a desire to live properly [Diogenes of Sin., by Diog. Laertius] |
| 15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
| 15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
| 15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
| 15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
| 15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
| 15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
| 15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
| 15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
| 15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
| 15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
| 15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
| 15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
| 15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
| 15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
| 15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
| 15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
| 15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
| 15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
| 15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
| 15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
| 18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
| 15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
| 15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
| 15940 | The intuitionist endorses only the potential infinite [Lavine] |
| 15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
| 15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
| 15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
| 15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
| 15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
| 15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
| 15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
| 19712 | Maybe there is plain 'animal' knowledge, and clearly justified 'reflective' knowledge [Vahid] |
| 19703 | Epistemic is normally marked out from moral or pragmatic justifications by its truth-goal [Vahid] |
| 19705 | 'Mentalist' internalism seems to miss the main point, if it might not involve an agent's access [Vahid] |
| 19706 | Strong access internalism needs actual awareness; weak versions need possibility of access [Vahid] |
| 19707 | Maybe we need access to our justification, and also to know why it justifies [Vahid] |
| 19709 | Internalism in epistemology over-emphasises deliberation about beliefs [Vahid] |
| 19704 | Externalism may imply that identical mental states might go with different justifications [Vahid] |
| 19710 | With a counterfactual account of the causal theory, we get knowledge as tracking or sensitive to truth [Vahid] |
| 19711 | Externalism makes the acquisition of knowledge too easy? [Vahid] |
| 1762 | When someone denied motion, Diogenes got up and walked away [Diogenes of Sin., by Diog. Laertius] |
| 7813 | Cynicism was open to anyone, and needed neither education nor sophistication [Diogenes of Sin., by Grayling] |
| 1763 | Diogenes said a plucked chicken fits Plato's definition of man [Diogenes of Sin., by Diog. Laertius] |
| 5071 | The Cynics rejected what is conventional as irrational, and aimed to live by nature [Taylor,R on Diogenes of Sin.] |
| 7812 | For peace of mind, you need self-government, indifference and independence [Diogenes of Sin.] |
| 1764 | Diogenes said he was a citizen of the world [Diogenes of Sin., by Diog. Laertius] |
| 5968 | Diogenes masturbated in public, wishing he could get rid of hunger so easily [Diogenes of Sin., by Plutarch] |
| 1766 | Diogenes said that the most excellent thing among men was freedom of speech [Diogenes of Sin., by Diog. Laertius] |