45 ideas
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] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [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] |
22668 | Apprehension is a complex intellect grasping the essence of a complex object [Holt,L] |
17471 | Using mechanisms as explanatory schemes began in chemistry [Weisberg/Needham/Hendry] |
17472 | Thick mechanisms map whole reactions, and thin mechanism chart the steps [Weisberg/Needham/Hendry] |
17465 | Lavoisier's elements included four types of earth [Weisberg/Needham/Hendry] |
17468 | Over 100,000,000 compounds have been discovered or synthesised [Weisberg/Needham/Hendry] |
17469 | 'H2O' just gives the element proportions, not the microstructure [Weisberg/Needham/Hendry] |
17470 | Water molecules dissociate, and form large polymers, explaining its properties [Weisberg/Needham/Hendry] |
17473 | It is unlikely that chemistry will ever be reduced to physics [Weisberg/Needham/Hendry] |
17474 | Quantum theory won't tell us which structure a set of atoms will form [Weisberg/Needham/Hendry] |
17475 | For temperature to be mean kinetic energy, a state of equilibrium is also required [Weisberg/Needham/Hendry] |
17467 | Isotopes (such as those of hydrogen) can vary in their rates of chemical reaction [Weisberg/Needham/Hendry] |
17466 | Mendeleev systematised the elements, and also gave an account of their nature [Weisberg/Needham/Hendry] |