59 ideas
| 21360 | Unobservant thinkers tend to dogmatise using insufficient facts [Aristotle] |
| 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] |
| 13212 | Infinity is only potential, never actual [Aristotle] |
| 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] |
| 13221 | Existence is either potential or actual [Aristotle] |
| 16100 | True change is in a thing's logos or its matter, not in its qualities [Aristotle] |
| 16101 | A change in qualities is mere alteration, not true change [Aristotle] |
| 12133 | If the substratum persists, it is 'alteration'; if it doesn't, it is 'coming-to-be' or 'passing-away' [Aristotle] |
| 13213 | All comings-to-be are passings-away, and vice versa [Aristotle] |
| 12134 | Matter is the substratum, which supports both coming-to-be and alteration [Aristotle] |
| 16572 | Does the pure 'this' come to be, or the 'this-such', or 'so-great', or 'somewhere'? [Aristotle] |
| 16573 | Philosophers have worried about coming-to-be from nothing pre-existing [Aristotle] |
| 13214 | The substratum changing to a contrary is the material cause of coming-to-be [Aristotle] |
| 13215 | If a perceptible substratum persists, it is 'alteration'; coming-to-be is a complete change [Aristotle] |
| 16717 | Which of the contrary features of a body are basic to it? [Aristotle] |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| 13216 | Matter is the limit of points and lines, and must always have quality and form [Aristotle] |
| 17994 | The primary matter is the substratum for the contraries like hot and cold [Aristotle] |
| 13224 | There couldn't be just one element, which was both water and air at the same time [Aristotle] |
| 16594 | The Four Elements must change into one another, or else alteration is impossible [Aristotle] |
| 13223 | Fire is hot and dry; Air is hot and moist; Water is cold and moist; Earth is cold and dry [Aristotle] |
| 13220 | Bodies are endlessly divisible [Aristotle] |
| 13210 | Wood is potentially divided through and through, so what is there in the wood besides the division? [Aristotle] |
| 13211 | If a body is endlessly divided, is it reduced to nothing - then reassembled from nothing? [Aristotle] |
| 13228 | There is no time without movement [Aristotle] |
| 16595 | If each thing can cease to be, why hasn't absolutely everything ceased to be long ago? [Aristotle] |
| 13227 | Being is better than not-being [Aristotle] |
| 13226 | An Order controls all things [Aristotle] |