120 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] |
| 20146 | 'Luck' is the unpredictable and inexplicable intersection of causal chains [Kekes] |
| 7301 | The phenomenalist says that to be is to be perceivable [Cardinal/Hayward/Jones] |
| 7302 | Linguistic phenomenalism says we can eliminate talk of physical objects [Cardinal/Hayward/Jones] |
| 7303 | If we lack enough sense-data, are we to say that parts of reality are 'indeterminate'? [Cardinal/Hayward/Jones] |
| 7299 | Primary qualities can be described mathematically, unlike secondary qualities [Cardinal/Hayward/Jones] |
| 7300 | An object cannot remain an object without its primary qualities [Cardinal/Hayward/Jones] |
| 23101 | Intuitions don't prove things; they just receptivity to interpretations [Kekes] |
| 7297 | My justifications might be very coherent, but totally unconnected to the world [Cardinal/Hayward/Jones] |
| 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] |
| 9145 | We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor] |
| 20169 | An action may be intended under one description, but not under another [Kekes] |
| 20149 | To control our actions better, make them result from our attitudes, not from circumstances [Kekes] |
| 23086 | Liberals say we are only responsible for fully autonomous actions [Kekes] |
| 23100 | Collective responsibility conflicts with responsibility's requirement of authonomy [Kekes] |
| 19738 | Values are an attempt to achieve well-being by bringing contingencies under control [Kekes] |
| 20145 | Values help us to control life, by connecting it to what is stable and manageable [Kekes] |
| 20170 | Responsibility is unprovoked foreseeable harm, against society, arising from vicious character [Kekes] |
| 23093 | Moral and causal responsibility are not clearly distinct [Kekes] |
| 23096 | Morality should aim to prevent all evil actions, not just autonomous ones [Kekes] |
| 23087 | Much human evil is not autonomous, so moral responsibility need not be autonomous [Kekes] |
| 23098 | Effects show the existence of moral responsibility, and mental states show the degree [Kekes] |
| 23089 | Evil people may not be autonomously aware, if they misjudge the situation [Kekes] |
| 23094 | Ought implies can means moral responsibility needs autonomy [Kekes] |
| 23095 | Why should moral responsibility depend on autonomy, rather than social role or experience? [Kekes] |
| 20165 | Reason and morality do not coincide; immorality can be reasonable, with an ideology [Kekes] |
| 20171 | Practical reason is not universal and impersonal, because it depends on what success is [Kekes] |
| 20175 | If morality has to be rational, then moral conflicts need us to be irrational and immoral [Kekes] |
| 23090 | Liberals assume people are naturally free, equal, rational, and morally good [Kekes] |
| 20174 | Relativists say all values are relative; pluralists concede much of that, but not 'human' values [Kekes] |
| 20159 | Cultural values are interpretations of humanity, conduct, institutions, and evaluations [Kekes] |
| 20161 | The big value problems are evil (humanity), disenchantment (cultures), and boredom (individuals) [Kekes] |
| 20156 | We are bound to regret some values we never aspired to [Kekes] |
| 20150 | There are far more values than we can pursue, so they are optional possibilities [Kekes] |
| 20158 | Innumerable values arise for us, from our humanity, our culture, and our individuality [Kekes] |
| 20151 | Our attitudes include what possibilities we value, and also what is allowable, and unthinkable [Kekes] |
| 20152 | Unconditional commitments are our most basic convictions, saying what must never be done [Kekes] |
| 20153 | Doing the unthinkable damages ourselves, so it is more basic than any value [Kekes] |
| 23117 | Love should be partial, and discriminate in favour of its object [Kekes] |
| 23119 | Sentimental love distorts its object [Kekes] |
| 23088 | Evil is not deviation from the good, any more than good is a deviation from evil [Kekes] |
| 20162 | Evil isn't explained by nature, by monsters, by uncharacteristic actions, or by society [Kekes] |
| 23097 | What matters for morality is the effects of action, not the psychological causes [Kekes] |
| 20157 | Well-being needs correct attitudes and well-ordered commitments to local values [Kekes] |
| 20154 | Control is the key to well-being [Kekes] |
| 23099 | It is said that if an agent is not autonomous then their evil actions don't reflect on their character [Kekes] |
| 23118 | Awareness of others' suffering doesn't create an obligation to help [Kekes] |
| 20172 | Boredom destroys our ability to evaluate [Kekes] |
| 20173 | Boredom is apathy and restlessness, yearning for something interesting [Kekes] |
| 20155 | Society is alienating if it lacks our values, and its values repel us [Kekes] |
| 23109 | The veil of ignorance is only needed because people have bad motivations [Kekes] |
| 23114 | The chief function of the state is to arbitrate between contending visions of the good life [Kekes] |
| 20164 | The ideal of an ideology is embodied in a text, a role model, a law of history, a dream of the past... [Kekes] |
| 20163 | Ideologies have beliefs about reality, ideals, a gap with actuality, and a program [Kekes] |
| 23116 | Citizenship is easier than parenthood [Kekes] |
| 23103 | Power is meant to be confined to representatives, and subsequent delegation [Kekes] |
| 23107 | Prosperity is a higher social virtue than justice [Kekes] |
| 23081 | Liberal basics are pluralism, freedom, rights, equality, and distributive justice - for autonomy [Kekes] |
| 23085 | The key liberal values are explained by the one core value, which is autonomy [Kekes] |
| 23092 | Agents have little control over the capacities needed for liberal autonomy [Kekes] |
| 23102 | Liberals are egalitarians, but in varying degrees [Kekes] |
| 23084 | Are egalitarians too coercive, or not egalitarian enough, or lax over morality? [Kekes] |
| 23079 | Liberal justice ignores desert, which is the essence of justice [Kekes] |
| 23091 | Why do liberals not see a much wider range of values as basic? [Kekes] |
| 23112 | Liberals ignore contingency, and think people are good and equal, and institutions cause evil [Kekes] |
| 23082 | Liberal distribution cares more about recipients than donors [Kekes] |
| 23106 | To rectify the undeserved equality, we should give men longer and women shorter lives [Kekes] |
| 23121 | It is just a fact that some people are morally better than others [Kekes] |
| 23105 | It is not deplorable that billionaires have more than millionaires [Kekes] |
| 23120 | The problem is basic insufficiency of resources, not their inequality [Kekes] |
| 20148 | Equal distribution is no good in a shortage, because there might be no one satisfied [Kekes] |
| 23108 | Justice combines consistency and desert; treat likes alike, judging likeness by desert [Kekes] |
| 23083 | Liberal welfare focuses on need rather than desert [Kekes] |
| 23113 | Sexual morality doesn't require monogamy, but it needs a group of sensible regulations [Kekes] |
| 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] |