58 ideas
| 12585 | Most people can't even define a chair [Peacocke] |
| 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] |
| 16062 | A necessary relation between fact-levels seems to be a further irreducible fact [Lynch/Glasgow] |
| 16061 | If some facts 'logically supervene' on some others, they just redescribe them, adding nothing [Lynch/Glasgow] |
| 16060 | Nonreductive materialism says upper 'levels' depend on lower, but don't 'reduce' [Lynch/Glasgow] |
| 16064 | The hallmark of physicalism is that each causal power has a base causal power under it [Lynch/Glasgow] |
| 12581 | Perceptual concepts causally influence the content of our experiences [Peacocke] |
| 12579 | Perception has proto-propositions, between immediate experience and concepts [Peacocke] |
| 12586 | Consciousness of a belief isn't a belief that one has it [Peacocke] |
| 12608 | Concepts are distinguished by roles in judgement, and are thus tied to rationality [Peacocke] |
| 18568 | Philosophy should merely give necessary and sufficient conditions for concept possession [Peacocke, by Machery] |
| 18571 | Peacocke's account of possession of a concept depends on one view of counterfactuals [Peacocke, by Machery] |
| 18572 | Peacocke's account separates psychology from philosophy, and is very sketchy [Machery on Peacocke] |
| 17722 | The concept 'red' is tied to what actually individuates red things [Peacocke] |
| 11127 | If concepts just are mental representations, what of concepts we may never acquire? [Peacocke] |
| 12577 | Possessing a concept is being able to make judgements which use it [Peacocke] |
| 12578 | A concept is just what it is to possess that concept [Peacocke] |
| 12587 | Employing a concept isn't decided by introspection, but by making judgements using it [Peacocke] |
| 12605 | A sense is individuated by the conditions for reference [Peacocke] |
| 12607 | Fregean concepts have their essence fixed by reference-conditions [Peacocke] |
| 12609 | Concepts have distinctive reasons and norms [Peacocke] |
| 12584 | An analysis of concepts must link them to something unconceptualized [Peacocke] |
| 12604 | Any explanation of a concept must involve reference and truth [Peacocke] |
| 9335 | Concepts are constituted by their role in a group of propositions to which we are committed [Peacocke, by Greco] |
| 9336 | A concept's reference is what makes true the beliefs of its possession conditions [Peacocke, by Horwich] |
| 12610 | Encountering novel sentences shows conclusively that meaning must be compositional [Peacocke] |