55 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] |
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] |
4938 | Prior to language, concepts are universals created by self-mapping of brain activity [Edelman/Tononi] |
4934 | Cultures have a common core of colour naming, based on three axes of colour pairs [Edelman/Tononi] |
4924 | A conscious human being rapidly reunifies its mind after any damage to the brain [Edelman/Tononi] |
4932 | A conscious state endures for about 100 milliseconds, known as the 'specious present' [Edelman/Tononi] |
4931 | Consciousness is a process (of neural interactions), not a location, thing, property, connectivity, or activity [Edelman/Tononi] |
4923 | The three essentials of conscious experience are privateness, unity and informativeness [Edelman/Tononi] |
4941 | Consciousness can create new axioms, but computers can't do that [Edelman/Tononi] |
4930 | Consciousness arises from high speed interactions between clusters of neurons [Edelman/Tononi] |
4929 | Dreams and imagery show the brain can generate awareness and meaning without input [Edelman/Tononi] |
4940 | Physicists see information as a measure of order, but for biologists it is symbolic exchange between animals [Edelman/Tononi] |
4935 | The sensation of red is a point in neural space created by dimensions of neuronal activity [Edelman/Tononi] |
4936 | The self is founded on bodily awareness centred in the brain stem [Edelman/Tononi] |
4939 | A sense of self begins either internally, or externally through language and society [Edelman/Tononi] |
4925 | Brains can initiate free actions before the person is aware of their own decision [Edelman/Tononi] |
4933 | Consciousness is a process, not a thing, as it maintains unity as its composition changes [Edelman/Tononi] |
4928 | Brain complexity balances segregation and integration, like a good team of specialists [Edelman/Tononi] |
4927 | Information-processing views of the brain assume the existence of 'information', and dubious brain codes [Edelman/Tononi] |
4922 | Consciousness involves interaction with persons and the world, as well as brain functions [Edelman/Tononi] |
5793 | Concepts and generalisations result from brain 'global mapping' by 'reentry' [Edelman/Tononi, by Searle] |
4926 | Concepts arise when the brain maps its own activities [Edelman/Tononi] |
4937 | Systems that generate a sense of value are basic to the primitive brain [Edelman/Tononi] |
3029 | Stilpo said if Athena is a daughter of Zeus, then a statue is only the child of a sculptor, and so is not a god [Stilpo, by Diog. Laertius] |