## Psychology Dictionary of ArgumentsHome | |||

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Set Theory: set theory is the system of rules and axioms, which regulates the formation of sets. The elements are exclusively numbers. Sets contain individual objects, that is, numbers as elements. Furthermore, sets contain sub-sets, that is, again sets of elements. The set of all sub-sets of a set is called the power set. Each set contains the empty set as a subset, but not as an element. The size of sets is called the cardinality. Sets containing the same elements are identical. See also comprehension, comprehension axiom, selection axiom, infinity axiom, couple set axiom, extensionality principle._____________ Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. | |||

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David K. Lewis on Set Theory - Dictionary of Arguments Schwarz I 75 Set theory/Mereology/Lewis: (Parts of Classes, 1991) ^{(1)}: are sets simply mereological sums? Set theory proves to be mereologically extended arithmetic, with successor relation, a set relation between thing A and its unit set {A}.Through a structural analysis of this relationship Lewis finally leads the whole mathematics back to the assumption that there are many things. >Mereology/Lewis. Schwarz I 78 Classical set theory/Schwarz: sets form a hierarchical structure (cumulative or iterative). lowest level: things that are not sets "individuals", "primal elements". pure set theory: here the lowest level is empty (no individuals, nothing outside sets, nothing is not a set!) >Individual/Lewis. Omega/ω/Set theory/Schwarz: on ω all sets are located whose elements occur on one of the finite levels. On ω+1 there are sets whose elements are on ω or below etc. up to ω + ω (=ω * ω) followed by ω * 2 + 1 etc. A set that contains itself/Russell's paradox/Schwarz: is excluded by the hierarchy: it must already have occurred at a level below the level at which it occurs for the first time. Then there are also no quantities of all quantities that do not contain themselves, because that would be nothing other than the quantity of all quantities. Cf. >Russellean paradox. Schwarz I 79f Non-naive set theory/Schwarz: here things only form a set if they are not too many, i.e. if they do not correspond one-to-one with all sets. This motivates the selection axiom and the replacement axiom. Schwarz I 79ff Classical Set Theory: set and element (member) are undefined. Schwarz I 80 Set theory/Mereology/Lewis: (Parts of Classes ^{(1)}, Part 1): Thesis: sets and classes are mereological sums. But the parts are not elements but subsets. >Mereological sum/Lewis. Main thesis: (MT): x is subclass of y, gdw. y is a class and x is part of y. (1991 ^{(1)},§1,3)Schwarz I 93 Set theory/Properties/VsLewis/Schwarz: Lewis has a similar problem: according to his set-theoretical structuralism, an expression like "{A,B,C}" does not refer to a particular thing, the class of A, B, and C. Classes are relative to single set relations and single set relations are very numerous. According to Lewis, statements about classes - and thus also about properties - are actually plural quantifications about single set relationships (2002a ^{(4)}, §5, (1986e^{(2)}, 52 Fn 39). Quantification via properties would then be plural quantification via ED. For example that a thing is red: that it is one of the red things. Schwarz I 94 SchwarzVsLewis: does not say how this should work for relations. - - - V 346 "Nominalistic Set Theory" (1970d) ^{(3)}Nominalistic set theory/Lewis: if one assumes the individual calculus and a relation of the neighborhood between atoms as basic concepts, it is possible to define a pseudo element relation between individuals. 1. David Lewis [1991]: Parts of Classes. Oxford: Blackwell 2. David Lewis [1986e]: On the Plurality of Worlds. Malden (Mass.): Blackwell 3. David Lewis [1970d]: “Nominalistic Set Theory”. Nous, 4. 4. David Lewis [2002a]: “Tensing the Copula”. Mind, 111: 1–13 _____________ Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition. |
Lewis I David K. Lewis Die Identität von Körper und Geist Frankfurt 1989 Lewis I (a) David K. Lewis An Argument for the Identity Theory, in: Journal of Philosophy 63 (1966) InDie Identität von Körper und Geist, , Frankfurt/M. 1989 Lewis I (b) David K. Lewis Psychophysical and Theoretical Identifications, in: Australasian Journal of Philosophy 50 (1972) InDie Identität von Körper und Geist, , Frankfurt/M. 1989 Lewis I (c) David K. Lewis Mad Pain and Martian Pain, Readings in Philosophy of Psychology, Vol. 1, Ned Block (ed.) Harvard University Press, 1980 InDie Identität von Körper und Geist, , Frankfurt/M. 1989 Lewis II David K. Lewis "Languages and Language", in: K. Gunderson (Ed.), Minnesota Studies in the Philosophy of Science, Vol. VII, Language, Mind, and Knowledge, Minneapolis 1975, pp. 3-35 InHandlung, Kommunikation, Bedeutung, Georg Meggle, Frankfurt/M. 1979 Lewis IV David K. Lewis Philosophical Papers Bd I New York Oxford 1983 Lewis V David K. Lewis Philosophical Papers Bd II New York Oxford 1986 Lewis VI David K. Lewis Convention. A Philosophical Study, Cambridge/MA 1969 German Edition: Konventionen Berlin 1975 LewisCl Clarence Irving Lewis Collected Papers of Clarence Irving Lewis Stanford 1970 LewisCl I Clarence Irving Lewis Mind and the World Order: Outline of a Theory of Knowledge (Dover Books on Western Philosophy) 1991 Schw I W. Schwarz David Lewis Bielefeld 2005 |

> Counter arguments against **Lewis**

> Counter arguments in relation to **Set Theory**