Psychology Dictionary of Arguments

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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.

Author Concept Summary/Quotes Sources

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.
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!)
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)
Die 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)
Die 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
Die 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
Handlung, 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

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

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