64 ideas
| 14721 | Metaphysical enquiry can survive if its conclusions are tentative [Sider] |
| 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] |
| 14760 | Four-dimensionalism sees things and processes as belonging in the same category [Sider] |
| 14194 | Proper ontology should only use categorical (actual) properties, not hypothetical ones [Sider] |
| 14745 | If sortal terms fix the kind and the persistence conditions, we need to know what kinds there are [Sider] |
| 14740 | If Tib is all of Tibbles bar her tail, when Tibbles loses her tail, two different things become one [Sider] |
| 14752 | Artists 'create' statues because they are essentially statues, and so lack identity with the lump of clay [Sider] |
| 14743 | The stage view of objects is best for dealing with coincident entities [Sider] |
| 14747 | 'Composition as identity' says that an object just is the objects which compose it [Sider] |
| 14757 | Mereological essentialism says an object's parts are necessary for its existence [Sider] |
| 14727 | Three-dimensionalists assert 'enduring', being wholly present at each moment, and deny 'temporal parts' [Sider] |
| 14738 | Some might say that its inconsistency with time travel is a reason to favour three-dimensionalism [Sider] |
| 14726 | Four-dimensionalists assert 'temporal parts', 'perduring', and being spread out over time [Sider] |
| 14728 | 4D says intrinsic change is difference between successive parts [Sider] |
| 14729 | 4D says each spatiotemporal object must have a temporal part at every moment at which it exists [Sider] |
| 14730 | Temporal parts exist, but are not prior building blocks for objects [Sider] |
| 14731 | Temporal parts are instantaneous [Sider] |
| 14758 | How can an instantaneous stage believe anything, if beliefs take time? [Sider] |
| 14762 | Four-dimensionalism says temporal parts are caused (through laws of motion) by previous temporal parts [Sider] |
| 14741 | The ship undergoes 'asymmetric' fission, where one candidate is seen as stronger [Sider] |
| 14754 | If you say Leibniz's Law doesn't apply to 'timebound' properties, you are no longer discussing identity [Sider] |
| 14763 | Counterparts rest on similarity, so there are many such relations in different contexts [Sider] |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| 14725 | Maybe motion is a dynamical quantity intrinsic to a thing at a particular time [Sider] |
| 14735 | Space is 3D and lacks a direction; time seems connected to causation [Sider] |
| 14722 | Between presentism and eternalism is the 'growing block' view - the past is real, the future is not [Sider] |
| 14756 | For Presentists there must always be a temporal vantage point for any description [Sider] |
| 14724 | Presentists must deny truths about multiple times [Sider] |
| 14723 | Talk using tenses can be eliminated, by reducing it to indexical connections for an utterance [Sider] |
| 14736 | The B-theory is adequate, except that it omits to say which time is present [Sider] |
| 14734 | The B-series involves eternalism, and the reduction of tense [Sider] |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |