61 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] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [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] |
13007 | Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz] |
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] |
22849 | Rawls's theory cannot justify liberalism, since it presupposes free and equal participants [Charvet] |
22848 | People with strong prior beliefs would have nothing to do with a veil of ignorance [Charvet] |
22838 | Societies need shared values, so conservatism is right if rational discussion of values is impossible [Charvet] |
22846 | The universalism of utilitarianism implies a world state [Charvet] |
22835 | Liberals value freedom and equality, but the society itself must decide on its values [Charvet] |
22831 | Modern libertarian societies still provide education and some housing [Charvet] |
22839 | Liberalism needs people to either have equal autonomy, or everyone to have enough autonomy [Charvet] |
22847 | Kant places a higher value on the universal rational will than on the people asserting it [Charvet] |
22821 | Liberalism asserts maximum freedom, but that must be equal for all participants [Charvet] |
22834 | Egalitarian liberals prefer equality (either of input or outcome) to liberty [Charvet] |
22822 | Liberals promote community and well-being - because all good societies need them [Charvet] |
22841 | Identity multiculturalism emerges from communitarianism, preferring community to humanity [Charvet] |
22842 | For communitarians it seems that you must accept the culture you are born into [Charvet] |
22830 | Give by ability and receive by need, rather than a free labour market [Charvet] |
22829 | Allowing defamatory speech is against society's interests, by blurring which people are trustworthy [Charvet] |
22836 | 'Freedom from' is an empty idea, if the freedom is not from impediments to my desires [Charvet] |
22837 | Positive freedom can lead to coercion, if you are forced to do what you chose to do [Charvet] |
22844 | First level autonomy is application of personal values; second level is criticising them [Charvet] |
22840 | Mere equality, as in two trees being the same height, has no value at all [Charvet] |
22843 | Inequalities are worse if they seem to be your fault, rather than social facts [Charvet] |
22845 | Money allows unlimited inequalities, and we obviously all agree to money [Charvet] |
22828 | The rule of law mainly benefits those with property and liberties [Charvet] |
22825 | The 1689 Bill of Rights denied the monarch new courts, or the right to sit as judge [Charvet] |
22823 | The rule of law is mainly to restrict governments [Charvet] |
22827 | Justice superior to the rule of law is claimed on behalf of the workers, or the will of the nation [Charvet] |
22826 | From 1701 only parliament could remove judges, whose decisions could not be discussed [Charvet] |
22832 | Welfare is needed if citizens are to accept the obligations of a liberal state [Charvet] |