58 ideas
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] |
6548 | Physicalism requires the naturalisation or rejection of set theory [Lycan] |
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] |
6531 | Institutions are not reducible as types, but they are as tokens [Lycan] |
6532 | Types cannot be reduced, but levels of reduction are varied groupings of the same tokens [Lycan] |
6534 | One location may contain molecules, a metal strip, a key, an opener of doors, and a human tragedy [Lycan] |
6529 | I see the 'role'/'occupant' distinction as fundamental to metaphysics [Lycan] |
6549 | I think greenness is a complex microphysical property of green objects [Lycan] |
6543 | Intentionality comes in degrees [Lycan] |
6537 | Teleological views allow for false intentional content, unlike causal and nomological theories [Lycan] |
6546 | Pain is composed of urges, desires, impulses etc, at different levels of abstraction [Lycan] |
6547 | The right 'level' for qualia is uncertain, though top (behaviourism) and bottom (particles) are false [Lycan] |
6527 | If energy in the brain disappears into thin air, this breaches physical conservation laws [Lycan] |
6528 | In lower animals, psychology is continuous with chemistry, and humans are continuous with animals [Lycan] |
6554 | Two behaviourists meet. The first says,"You're fine; how am I?" [Lycan] |
6545 | If functionalism focuses on folk psychology, it ignores lower levels of function [Lycan] |
6541 | Functionalism must not be too abstract to allow inverted spectrum, or so structural that it becomes chauvinistic [Lycan] |
6539 | The distinction between software and hardware is not clear in computing [Lycan] |
6533 | Mental types are a subclass of teleological types at a high level of functional abstraction [Lycan] |
6535 | Teleological characterisations shade off smoothly into brutely physical ones [Lycan] |
6544 | Identity theory is functionalism, but located at the lowest level of abstraction [Lycan] |
6530 | We reduce the mind through homuncular groups, described abstractly by purpose [Lycan] |
6536 | Teleological functionalism helps us to understand psycho-biological laws [Lycan] |
6542 | A Martian may exhibit human-like behaviour while having very different sensations [Lycan] |
6538 | We need a notion of teleology that comes in degrees [Lycan] |
6551 | 'Physical' means either figuring in physics descriptions, or just located in space-time [Lycan] |
7417 | God can't have silly perfections, but how do we decide which ones are 'silly'? [Joslin] |