88 ideas
| 15901 | Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine] |
| 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] |
| 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] |
| 6653 | Syntactical methods of proof need only structure, where semantic methods (truth-tables) need truth [Lowe] |
| 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] |
| 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] |
| 8733 | The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro] |
| 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] |
| 18174 | Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy] |
| 15893 | Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine] |
| 18173 | Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy] |
| 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] |
| 6618 | A 'substance' is a thing that remains the same when its properties change [Lowe] |
| 6635 | Causal theories of belief make all beliefs true, and can't explain belief about the future [Lowe] |
| 6619 | Perhaps 'I' no more refers than the 'it' in 'it is raining' [Lowe] |
| 6643 | 'Ecological' approaches say we don't infer information, but pick it up directly from reality [Lowe] |
| 6638 | One must be able to visually recognise a table, as well as knowing its form [Lowe] |
| 6644 | Computationalists object that the 'ecological' approach can't tell us how we get the information [Lowe] |
| 6647 | Comparing shapes is proportional in time to the angle of rotation [Lowe] |
| 6639 | The 'disjunctive' theory of perception says true perceptions and hallucinations need have nothing in common [Lowe] |
| 6640 | A causal theorist can be a direct realist, if all objects of perception are external [Lowe] |
| 6645 | If blindsight shows we don't need perceptual experiences, the causal theory is wrong [Lowe] |
| 6637 | How could one paraphrase very complex sense-data reports adverbially? [Lowe] |
| 6667 | There are memories of facts, memories of practical skills, and autobiographical memory [Lowe] |
| 6642 | Psychologists say illusions only occur in unnatural and passive situations [Lowe] |
| 6641 | Externalists say minds depend on environment for their very existence and identity [Lowe] |
| 6617 | The main questions are: is mind distinct from body, and does it have unique properties? [Lowe] |
| 6626 | 'Phenomenal' consciousness is of qualities; 'apperceptive' consciousness includes beliefs and desires [Lowe] |
| 6646 | The brain may have two systems for vision, with only the older one intact in blindsight [Lowe] |
| 6665 | Persons are selves - subjects of experience, with reflexive self-knowledge [Lowe] |
| 6670 | If my brain could survive on its own, I cannot be identical with my whole body [Lowe] |
| 6671 | It seems impossible to get generally applicable mental concepts from self-observation [Lowe] |
| 6666 | All human languages have an equivalent of the word 'I' [Lowe] |
| 6625 | If qualia are causally inert, how can we even know about them? [Lowe] |
| 6621 | You can only identify behaviour by ascribing belief, so the behaviour can't explain the belief [Lowe] |
| 6654 | A computer program is equivalent to the person AND the manual [Lowe] |
| 6629 | Functionalism commits us to bizarre possibilities, such as 'zombies' [Lowe] |
| 6623 | Functionalism can't distinguish our experiences in spectrum inversion [Lowe] |
| 6628 | Functionalism only discusses relational properties of mental states, not intrinsic properties [Lowe] |
| 6622 | Non-reductive physicalism accepts token-token identity (not type-type) and asserts 'supervenience' of mind and brain [Lowe] |
| 6634 | Physicalists must believe in narrow content (because thoughts are merely the brain states) [Lowe] |
| 6630 | Eliminativism is incoherent if it eliminates reason and truth as well as propositional attitudes [Lowe] |
| 6648 | Some behaviourists believe thought is just suppressed speech [Lowe] |
| 6651 | People are wildly inaccurate in estimating probabilities about an observed event [Lowe] |
| 6652 | 'Base rate neglect' makes people favour the evidence over its background [Lowe] |
| 16383 | Puzzled Pierre has two mental files about the same object [Recanati on Kripke] |
| 6655 | The 'Frame Problem' is how to program the appropriate application of general knowledge [Lowe] |
| 6657 | Computers can't be rational, because they lack motivation and curiosity [Lowe] |
| 6656 | The Turing test is too behaviourist, and too verbal in its methods [Lowe] |
| 6636 | The naturalistic views of how content is created are the causal theory and the teleological theory [Lowe] |
| 6633 | Twin Earth cases imply that even beliefs about kinds of stuff are indexical [Lowe] |
| 8715 | Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend] |
| 13454 | Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor] |
| 6632 | The same proposition provides contents for the that-clause of an utterance and a belief [Lowe] |
| 6631 | If propositions are abstract entities, how can minds depend on their causal powers? [Lowe] |
| 6659 | The three main theories of action involve the will, or belief-plus-desire, or an agent [Lowe] |
| 6661 | Libet gives empirical support for the will, as a kind of 'executive' mental operation [Lowe] |
| 6662 | We feel belief and desire as reasons for choice, not causes of choice [Lowe] |
| 6663 | People's actions are explained either by their motives, or their reasons, or the causes [Lowe] |
| 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] |