Topic: set definition by extension or intension

topics > mathematics > Group: sets

data type
philosophy of mathematics

abstraction by common attributes
analytic truth
denoting phrases and definite descriptions
definition by example
meaning vs. reference
set construction


A fundamental distinction about sets is intension vs. extension. The intension of a set is its description or defining properties, i.e., what is true about members of a set. The extension of a set is its members or contents. The intension of a set may appear to be more important than the extension. The extension of a set may change without changing its intension, and a given set of members may satisfy many intensions. But intensions are hard to pin down, so extensional definitions are often preferred. In mathematics, a set is its extension.

Types are a form of intentional definition. A type defines the possible extensions of a set. We think of sufficiently different types as being incompatible. But in traditional set theory, every set induces union and intersection sets. This indicates that the extensional definition of set doesn't quite capture the idea of a collection or set of things.

Frege's distinction between sense (meaning) and reference, and Mill's distinction between connotation and denotation are similar. Intensions and extensions are also related to the distinction between analytic and synthetic statements. An analytic statement follows from intensions while a synthetic statement is verified by extensions.

Kripke specifies a partially defined statement by two sets of members: its extension and its anti-extension. The anti-extension is those members that clearly do not satisfy the predicate. (cbb 4/94)

Subtopic: meaning intension vs. extension up

Quote: each symbol has two kinds of meaning: sense (connotation, intension) and denotation (reference, extension) [»kimuTD7_1985]
Quote: the intension of a word is meaning in semantic memory while its extension is the corresponding set of things [»sowaJF_1984]
Quote: the intension or meaning of a term is different than its extension
Quote: the extension of a term is those entities for which the term is true
Quote: an analytic statement follows from intensions; a synthetic statement is verified with extensions [»sowaJF_1984]
Quote: all concrete names are connotative because the denoted subjects are defined by attributes; e.g., man is corporal, animal, rational and human [»millJS_1843, OK]

Subtopic: type intension up

Quote: a type is the abstract idea of a set (its intension), not its contents [»kentW_1978]
Quote: shouldn't there be types sufficiently different that they do not induce union or intersection types?
Quote: traditional set theory defines sets by their contents -- not relevant to types [»kentW_1978]

Subtopic: class intension vs. extension up

Quote: classes are abstract entities, universals, but not aggregates or collections; e.g., a heap of stones is not a class, What is its size? [»quinWV2_1947]
Quote: can define a class by extension (enumerate its members) or intension (a defining property); can reduce extension to intension [»russB_1919, OK]
Quote: the difference between a class and its definition is that a class has many defining characteristics but only one set of members [»russB_1919, OK]
Quote: there must be generative process whereby a class is created before it is named [»bateG_1979]

Subtopic: set intension vs. extension up

Quote: an intensional set is a carrier object, i.e., an object with a persistent identity or name and a mutable content (cargo or extension) [»kentW6_1991]
Quote: databases use extension (database entities) and intension (database constraints) [»sowaJF_1984]
Quote: deal with all elements of a set via the set's definition and not the elements themselves [»dijkEW12_1989]

Subtopic: predicate up

Quote: handle a partially defined predicate by specifying its extension (true for these) and anti-extension (false for these); e.g., three-valued logic [»kripS_1975]
Quote: an open (parameterized) sentence defines a class of things for which the sentence is true; but not required for classes [»quinWV_1969]

Related Topics up

Group: data type   (34 topics, 730 quotes)
Group: philosophy of mathematics   (11 topics, 330 quotes)

Topic: abstraction by common attributes (19 items)
Topic: analytic truth (51 items)
Topic: classification (65 items)
Topic: denoting phrases and definite descriptions (21 items)
Topic: definition by example (26 items)
Topic: elements (22 items)
Topic: entities (20 items)
Topic: logic (84 items)
Topic: meaning vs. reference (49 items)
Topic: set construction
(20 items)

Updated barberCB 1/05
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