64 ideas
| 7914 | To try to be wise all on one's own is folly [Rochefoucauld] |
| 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] |
| 15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
| 15947 | The infinite is extrapolation from the experience of indefinitely large size [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] |
| 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] |
| 9240 | Love creates a necessity concerning what to care about [Frankfurt] |
| 7118 | La Rochefoucauld's idea of disguised self-love implies an unconscious mind [Rochefoucauld, by Sartre] |
| 9264 | Persons are distinguished by a capacity for second-order desires [Frankfurt] |
| 9266 | A person essentially has second-order volitions, and not just second-order desires [Frankfurt] |
| 9267 | Free will is the capacity to choose what sort of will you have [Frankfurt] |
| 9265 | The will is the effective desire which actually leads to an action [Frankfurt] |
| 20015 | Freedom of action needs the agent to identify with their reason for acting [Frankfurt, by Wilson/Schpall] |
| 9228 | Ranking order of desires reveals nothing, because none of them may be considered important [Frankfurt] |
| 9270 | A 'wanton' is not a person, because they lack second-order volitions [Frankfurt] |
| 9269 | A person may be morally responsible without free will [Frankfurt] |
| 9238 | Morality isn't based on reason; moral indignation is quite unlike disapproval of irrationality [Frankfurt] |
| 9232 | It is by caring about things that we infuse the world with importance [Frankfurt] |
| 9234 | If you don't care about at least one thing, you can't find reasons to care about anything [Frankfurt] |
| 9229 | What is worthwhile for its own sake alone may be worth very little [Frankfurt] |
| 9233 | Our criteria for evaluating how to live offer an answer to the problem [Frankfurt] |
| 7912 | Judging by effects, love looks more like hatred than friendship [Rochefoucauld] |
| 9235 | Rather than loving things because we value them, I think we value things because we love them [Frankfurt] |
| 9236 | Love can be cool, and it may not involve liking its object [Frankfurt] |
| 9237 | The paradigm case of pure love is not romantic, but that between parents and infants [Frankfurt] |
| 9239 | I value my children for their sake, but I also value my love for them for its own sake [Frankfurt] |
| 7915 | Supreme cleverness is knowledge of the real value of things [Rochefoucauld] |
| 7917 | Realising our future misery is a kind of happiness [Rochefoucauld] |
| 9227 | We might not choose a very moral life, if the character or constitution was deficient [Frankfurt] |
| 9230 | People want to fulfill their desires, but also for their desires to be sustained [Frankfurt] |
| 9241 | Loving oneself is not a failing, but is essential to a successful life [Frankfurt] |
| 7913 | Virtue doesn't go far without the support of vanity [Rochefoucauld] |
| 7916 | True friendship is even rarer than true love [Rochefoucauld] |
| 9300 | Boredom is serious, not just uncomfortable; it threatens our psychic survival [Frankfurt] |
| 9299 | We are bored by people to whom we ourselves are boring [Rochefoucauld] |
| 9231 | Freedom needs autonomy (rather than causal independence) - embracing our own desires and choices [Frankfurt] |