74 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] |
| 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] |
| 7871 | Perceptual concepts can't just refer to what causes classification [Papineau] |
| 7852 | The only serious mind-brain theories now are identity, token identity, realization and supervenience [Papineau] |
| 7864 | Maybe mind and body do overdetermine acts, but are linked (for some reason) [Papineau] |
| 7873 | Young children can see that other individuals sometimes have false beliefs [Papineau] |
| 7874 | Do we understand other minds by simulation-theory, or by theory-theory? [Papineau] |
| 7882 | Researching phenomenal consciousness is peculiar, because the concepts involved are peculiar [Papineau] |
| 7854 | Whether octopuses feel pain is unclear, because our phenomenal concepts are too vague [Papineau] |
| 7889 | Our concept of consciousness is crude, and lacks theoretical articulation [Papineau] |
| 7891 | We can’t decide what 'conscious' means, so it is undecidable whether cats are conscious [Papineau] |
| 7890 | Maybe a creature is conscious if its mental states represent things in a distinct way [Papineau] |
| 7885 | The 'actualist' HOT theory says consciousness comes from actual higher judgements of mental states [Papineau] |
| 7886 | Actualist HOT theories imply that a non-conscious mental event could become conscious when remembered [Papineau] |
| 7887 | States are conscious if they could be the subject of higher-order mental judgements [Papineau] |
| 7888 | Higher-order judgements may be possible where the subject denies having been conscious [Papineau] |
| 7860 | The epiphenomenal relation of mind and brain is a 'causal dangler', unlike anything else [Papineau] |
| 7862 | Maybe minds do not cause actions, but do cause us to report our decisions [Papineau] |
| 7870 | Role concepts either name the realising property, or the higher property constituting the role [Papineau] |
| 7858 | If causes are basic particulars, this doesn't make conscious and physical properties identical [Papineau] |
| 7865 | Supervenience can be replaced by identifying mind with higher-order or disjunctional properties [Papineau] |
| 7892 | The completeness of physics is needed for mind-brain identity [Papineau] |
| 7879 | Mind-brain reduction is less explanatory, because phenomenal concepts lack causal roles [Papineau] |
| 20971 | Weak reduction of mind is to physical causes; strong reduction is also to physical laws [Papineau] |
| 7856 | It is absurd to think that physical effects are caused twice, so conscious causes must be physical [Papineau] |
| 7881 | Accept ontological monism, but conceptual dualism; we think in a different way about phenomenal thought [Papineau] |
| 7866 | Mary acquires new concepts; she previously thought about the same property using material concepts [Papineau] |
| 7850 | Thinking about a thing doesn't require activating it [Papineau] |
| 7851 | Consciousness affects bodily movement, so thoughts must be material states [Papineau] |
| 7884 | Most reductive accounts of representation imply broad content [Papineau] |
| 7863 | If content hinges on matters outside of you, how can it causally influence your actions? [Papineau] |
| 7883 | Verificationists tend to infer indefinite answers from undecidable questions [Papineau] |
| 7872 | Teleosemantics equates meaning with the item the concept is intended to track [Papineau] |
| 7869 | Truth conditions in possible worlds can't handle statements about impossibilities [Papineau] |
| 7868 | Thought content is possible worlds that make the thought true; if that includes the actual world, it's true [Papineau] |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| 7853 | Causation is based on either events, or facts, or states of affairs [Papineau] |
| 7857 | Causes are instantiations of properties by particulars, or they are themselves basic particulars [Papineau] |
| 20976 | The completeness of physics cannot be proved [Papineau] |
| 20970 | Determinism is possible without a complete physics, if mental forces play a role [Papineau] |
| 20974 | Modern biological research, especially into the cell, has revealed no special new natural forces [Papineau] |
| 20975 | Quantum 'wave collapses' seem to violate conservation of energy [Papineau] |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |