73 ideas
18137 | Impredicative definitions are wrong, because they change the set that is being defined? [Bostock] |
18122 | Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock] |
18114 | There is no single agreed structure for set theory [Bostock] |
18107 | A 'proper class' cannot be a member of anything [Bostock] |
18115 | We could add axioms to make sets either as small or as large as possible [Bostock] |
18139 | The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock] |
18105 | Replacement enforces a 'limitation of size' test for the existence of sets [Bostock] |
18108 | First-order logic is not decidable: there is no test of whether any formula is valid [Bostock] |
18109 | The completeness of first-order logic implies its compactness [Bostock] |
18123 | Substitutional quantification is just standard if all objects in the domain have a name [Bostock] |
18120 | The Deduction Theorem is what licenses a system of natural deduction [Bostock] |
18125 | Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock] |
18101 | Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock] |
18100 | ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock] |
18102 | A cardinal is the earliest ordinal that has that number of predecessors [Bostock] |
18106 | Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock] |
18095 | Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock] |
18099 | The number of reals is the number of subsets of the natural numbers [Bostock] |
18093 | For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock] |
18110 | Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock] |
18156 | Modern axioms of geometry do not need the real numbers [Bostock] |
18097 | The Peano Axioms describe a unique structure [Bostock] |
18148 | Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock] |
18145 | Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock] |
18149 | There are many criteria for the identity of numbers [Bostock] |
18143 | Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock] |
18116 | Numbers can't be positions, if nothing decides what position a given number has [Bostock] |
18117 | Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock] |
18141 | Nominalism about mathematics is either reductionist, or fictionalist [Bostock] |
18157 | Nominalism as based on application of numbers is no good, because there are too many applications [Bostock] |
18150 | Actual measurement could never require the precision of the real numbers [Bostock] |
18158 | Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock] |
18127 | Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock] |
18144 | Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock] |
18147 | Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock] |
18146 | If Hume's Principle is the whole story, that implies structuralism [Bostock] |
18129 | Many crucial logicist definitions are in fact impredicative [Bostock] |
18111 | Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock] |
18159 | Higher cardinalities in sets are just fairy stories [Bostock] |
18155 | A fairy tale may give predictions, but only a true theory can give explanations [Bostock] |
18140 | The best version of conceptualism is predicativism [Bostock] |
18138 | Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock] |
18131 | If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock] |
18134 | Predicativism makes theories of huge cardinals impossible [Bostock] |
18135 | If mathematics rests on science, predicativism may be the best approach [Bostock] |
18136 | If we can only think of what we can describe, predicativism may be implied [Bostock] |
18132 | The predicativity restriction makes a difference with the real numbers [Bostock] |
18133 | The usual definitions of identity and of natural numbers are impredicative [Bostock] |
18121 | In logic a proposition means the same when it is and when it is not asserted [Bostock] |
23369 | Some liberals thinks checks and balances are enough, without virtuous citizens [Kymlicka] |
23370 | Good citizens need civic virtues of loyalty, independence, diligence, respect, etc. [Kymlicka] |
23373 | Liberals accept that people need society, but Aristotelians must show that they need political activity [Kymlicka] |
23375 | Minimal liberal citizenship needs common civility, as well as mere non-interference [Kymlicka] |
23376 | Modern non-discrimination obliges modern citizens to treat each other as equals [Kymlicka] |
23377 | The right wing sees citizenship in terms of responsibility to earn a living, rather than rights [Kymlicka] |
23371 | Modern democratic theory focuses on talk, not votes, because we need consensus or compromise [Kymlicka] |
23390 | In a liberal democracy all subjects of authority have a right to determine the authority [Kymlicka] |
23374 | We have become attached to private life because that has become greatly enriched [Kymlicka] |
23387 | Liberals must avoid an official culture, as well as an official religion [Kymlicka] |
23388 | Liberals need more than freedom; they must build a nation, through a language and institutions [Kymlicka] |
23380 | Some individuals can gain citizenship as part of a group, rather than as mere individuals [Kymlicka] |
23381 | The status hierarchy is independent of the economic hierarchy [Kymlicka] |
23383 | Some multiculturalists defended the rights of cohesive minorities against liberal individualism [Kymlicka] |
23384 | 'Culturalist' liberals say that even liberal individuals may need minority rights [Kymlicka] |
23385 | Multiculturalism may entail men dominating women in minority groups [Kymlicka] |
23386 | Liberals must prefer minority right which are freedoms, not restrictions [Kymlicka] |
23389 | Why shouldn't national minorities have their own right to nation-build? [Kymlicka] |
23391 | Multiculturalism is liberal if it challenges inequality, conservative if it emphasises common good [Kymlicka] |
23379 | Rights are a part of nation-building, to build a common national identity and culture [Kymlicka] |
23382 | Rights derived from group membership are opposed to the idea of state citizenship [Kymlicka] |
23378 | The welfare state helps to integrate the working classes into a national culture [Kymlicka] |
1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |