53 ideas
| 9786 | Philosophers working like teams of scientists is absurd, yet isolation is hard [Cartwright,R] |
| 9784 | A false proposition isn't truer because it is part of a coherent system [Cartwright,R] |
| 13941 | Are the truth-bearers sentences, utterances, ideas, beliefs, judgements, propositions or statements? [Cartwright,R] |
| 13942 | Logicians take sentences to be truth-bearers for rigour, rather than for philosophical reasons [Cartwright,R] |
| 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] |
| 9783 | While no two classes coincide in membership, there are distinct but coextensive attributes [Cartwright,R] |
| 14961 | Clearly a pipe can survive being taken apart [Cartwright,R] |
| 14962 | Bodies don't becomes scattered by losing small or minor parts [Cartwright,R] |
| 13952 | Essentialism says some of a thing's properties are necessary, and could not be absent [Cartwright,R] |
| 13954 | The difficulty in essentialism is deciding the grounds for rating an attribute as essential [Cartwright,R] |
| 13955 | Essentialism is said to be unintelligible, because relative, if necessary truths are all analytic [Cartwright,R] |
| 13953 | An act of ostension doesn't seem to need a 'sort' of thing, even of a very broad kind [Cartwright,R] |
| 13945 | A token isn't a unique occurrence, as the case of a word or a number shows [Cartwright,R] |
| 13948 | For any statement, there is no one meaning which any sentence asserting it must have [Cartwright,R] |
| 13950 | People don't assert the meaning of the words they utter [Cartwright,R] |
| 13944 | We can pull apart assertion from utterance, and the action, the event and the subject-matter for each [Cartwright,R] |
| 13947 | 'It's raining' makes a different assertion on different occasions, but its meaning remains the same [Cartwright,R] |
| 13943 | We can attribute 'true' and 'false' to whatever it was that was said [Cartwright,R] |
| 13946 | To assert that p, it is neither necessary nor sufficient to utter some particular words [Cartwright,R] |
| 13951 | Assertions, unlike sentence meanings, can be accurate, probable, exaggerated, false.... [Cartwright,R] |
| 6017 | Nomos is king [Pindar] |