85 ideas
| 6052 | Definitions identify two concepts, so they presuppose identity [McGinn] |
| 6064 | Regresses are only vicious in the context of an explanation [McGinn] |
| 6088 | Truth is a method of deducing facts from propositions [McGinn] |
| 6084 | 'Snow does not fall' corresponds to snow does fall [McGinn] |
| 6085 | The idea of truth is built into the idea of correspondence [McGinn] |
| 6083 | The coherence theory of truth implies idealism, because facts are just coherent beliefs [McGinn] |
| 6086 | Truth is the property of propositions that makes it possible to deduce facts [McGinn] |
| 6087 | Without the disquotation device for truth, you could never form beliefs from others' testimony [McGinn] |
| 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] |
| 6051 | In 'x is F and x is G' we must assume the identity of x in the two statements [McGinn] |
| 6055 | Both non-contradiction and excluded middle need identity in their formulation [McGinn] |
| 6059 | Identity is unitary, indefinable, fundamental and a genuine relation [McGinn] |
| 6042 | The quantifier is overrated as an analytical tool [McGinn] |
| 6067 | Existential quantifiers just express the quantity of things, leaving existence to the predicate 'exists' [McGinn] |
| 6069 | 'Partial quantifier' would be a better name than 'existential quantifier', as no existence would be implied [McGinn] |
| 6068 | We need an Intentional Quantifier ("some of the things we talk about.."), so existence goes into the proposition [McGinn] |
| 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] |
| 6070 | Existence is a primary quality, non-existence a secondary quality [McGinn] |
| 6062 | Existence can't be analysed as instantiating a property, as instantiation requires existence [McGinn] |
| 6065 | We can't analyse the sentence 'something exists' in terms of instantiated properties [McGinn] |
| 6082 | If causal power is the test for reality, that will exclude necessities and possibilities [McGinn] |
| 6075 | Facts are object-plus-extension, or property-plus-set-of-properties, or object-plus-property [McGinn] |
| 6058 | Identity propositions are not always tautological, and have a key epistemic role [McGinn] |
| 6053 | Identity is as basic as any concept could ever be [McGinn] |
| 6043 | Type-identity is close similarity in qualities [McGinn] |
| 6044 | Qualitative identity is really numerical identity of properties [McGinn] |
| 6046 | Qualitative identity can be analysed into numerical identity of the type involved [McGinn] |
| 6045 | It is best to drop types of identity, and speak of 'identity' or 'resemblance' [McGinn] |
| 6066 | Existence is a property of all objects, but less universal than self-identity, which covers even conceivable objects [McGinn] |
| 6054 | Sherlock Holmes does not exist, but he is self-identical [McGinn] |
| 6047 | All identity is necessary, though identity statements can be contingently true [McGinn] |
| 6049 | Leibniz's Law says 'x = y iff for all P, Px iff Py' [McGinn] |
| 6048 | Leibniz's Law is so fundamental that it almost defines the concept of identity [McGinn] |
| 6050 | Leibniz's Law presupposes the notion of property identity [McGinn] |
| 6080 | Modality is not objects or properties, but the type of binding of objects to properties [McGinn] |
| 6079 | If 'possible' is explained as quantification across worlds, there must be possible worlds [McGinn] |
| 6081 | Necessity and possibility are big threats to the empiricist view of knowledge [McGinn] |
| 6071 | Scepticism about reality is possible because existence isn't part of appearances [McGinn] |
| 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] |
| 6077 | Semantics should not be based on set-membership, but on instantiation of properties in objects [McGinn] |
| 6074 | Clearly predicates have extensions (applicable objects), but are the extensions part of their meaning? [McGinn] |
| 22485 | Non-cognitivists give the conditions of use of moral sentences as facts about the speaker [Foot] |
| 22486 | The mistake is to think good grounds aren't enough for moral judgement, which also needs feelings [Foot] |
| 22487 | Moral arguments are grounded in human facts [Foot] |
| 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] |
| 6072 | If Satan is the most imperfect conceivable being, he must have non-existence [McGinn] |
| 6073 | I think the fault of the Ontological Argument is taking the original idea to be well-defined [McGinn] |