49 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] |
| 4986 | A weaker kind of reductionism than direct translation is the use of 'bridge laws' [Kirk,R] |
| 5001 | Maybe we should see intentionality and consciousness as a single problem, not two [Kirk,R] |
| 4993 | If a bird captures a worm, we could say its behaviour is 'about' the worm [Kirk,R] |
| 5000 | Behaviourism says intentionality is an external relation; language of thought says it's internal [Kirk,R] |
| 4982 | Dualism implies some brain events with no physical cause, and others with no physical effect [Kirk,R] |
| 4991 | Behaviourism seems a good theory for intentional states, but bad for phenomenal ones [Kirk,R] |
| 4994 | Behaviourism offers a good alternative to simplistic unitary accounts of mental relationships [Kirk,R] |
| 4992 | In 'holistic' behaviourism we say a mental state is a complex of many dispositions [Kirk,R] |
| 4990 | The inverted spectrum idea is often regarded as an objection to behaviourism [Kirk,R] |
| 4984 | All meaningful psychological statements can be translated into physics [Kirk,R] |
| 4998 | Instead of representation by sentences, it can be by a distribution of connectionist strengths [Kirk,R] |
| 4985 | If mental states are multiply realisable, they could not be translated into physical terms [Kirk,R] |
| 4997 | It seems unlikely that most concepts are innate, if a theory must be understood to grasp them [Kirk,R] |
| 4999 | For behaviourists language is just a special kind of behaviour [Kirk,R] |
| 4995 | Behaviourists doubt whether reference is a single type of relation [Kirk,R] |
| 20166 | A man is a responsible agent to the extent he has an intention, and knows what he is doing [Hampshire] |