71 ideas
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 15946 | Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine] |
| 9616 | A set is a collection into a whole of distinct objects of our intuition or thought [Cantor] |
| 13444 | Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD] |
| 18098 | Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock] |
| 15505 | If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis] |
| 10701 | Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter] |
| 10865 | The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg] |
| 17831 | Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake] |
| 13016 | The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy] |
| 14199 | Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley] |
| 10082 | There are infinite sets that are not enumerable [Cantor, by Smith,P] |
| 13483 | Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD] |
| 8710 | The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend] |
| 15910 | Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine] |
| 15905 | Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine] |
| 9983 | Cantor took the ordinal numbers to be primary [Cantor, by Tait] |
| 17798 | Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry] |
| 9971 | Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait] |
| 9892 | Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett] |
| 15911 | Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine] |
| 14136 | A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor] |
| 15906 | Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine] |
| 11015 | Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read] |
| 15903 | A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine] |
| 18251 | Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine] |
| 15902 | Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine] |
| 15908 | It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine] |
| 13464 | Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD] |
| 10112 | The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman] |
| 15896 | Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine] |
| 17889 | CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner] |
| 13447 | Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD] |
| 10883 | Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten] |
| 13528 | Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS] |
| 9555 | Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 9992 | The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor] |
| 10232 | Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro] |
| 18176 | Pure mathematics is pure set theory [Cantor] |
| 8631 | Cantor says that maths originates only by abstraction from objects [Cantor, by Frege] |
| 4938 | Prior to language, concepts are universals created by self-mapping of brain activity [Edelman/Tononi] |
| 12887 | A whole must have one characteristic, an internal relation, and a structure [Rescher/Oppenheim] |
| 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] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 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] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| 4937 | Systems that generate a sense of value are basic to the primitive brain [Edelman/Tononi] |
| 10863 | Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg] |
| 13465 | Only God is absolutely infinite [Cantor, by Hart,WD] |