54 ideas
| 6253 | Reason is our power of finding out true propositions [Hutcheson] |
| 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] |
| 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] |
| 6248 | Reason is too slow and doubtful to guide all actions, which need external and moral senses [Hutcheson] |
| 6238 | We approve of actions by a superior moral sense [Hutcheson] |
| 6239 | We dislike a traitor, even if they give us great benefit [Hutcheson] |
| 6240 | The moral sense is not an innate idea, but an ability to approve or disapprove in a disinterested way [Hutcheson] |
| 6242 | We cannot choose our moral feelings, otherwise bribery could affect them [Hutcheson] |
| 6247 | Everyone feels uneasy when seeing others in pain, unless the others are evil [Hutcheson] |
| 6256 | Can't the moral sense make mistakes, as the other senses do? [Hutcheson] |
| 6244 | Human nature seems incapable of universal malice, except what results from self-love [Hutcheson] |
| 6243 | As death approaches, why do we still care about family, friends or country? [Hutcheson] |
| 6246 | My action is not made good by a good effect, if I did not foresee and intend it [Hutcheson] |
| 6252 | Happiness is a pleasant sensation, or continued state of such sensations [Hutcheson] |
| 6241 | Contempt of danger is just madness if it is not in some worthy cause [Hutcheson] |
| 6257 | You can't form moral rules without an end, which needs feelings and a moral sense [Hutcheson] |
| 6245 | That action is best, which procures the greatest happiness for the greatest number [Hutcheson] |
| 6251 | The loss of perfect rights causes misery, but the loss of imperfect rights reduces social good [Hutcheson] |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| 6254 | We are asked to follow God's ends because he is our benefactor, but why must we do that? [Hutcheson] |
| 6255 | Why may God not have a superior moral sense very similar to ours? [Hutcheson] |
| 6250 | We say God is good if we think everything he does aims at the happiness of his creatures [Hutcheson] |
| 6249 | If goodness is constituted by God's will, it is a tautology to say God's will is good [Hutcheson] |