Group: sets

topics > Group: mathematics

collection class
set construction
set data type
set definition by extension or intension
set operations
set-oriented languages
philosophy of mathematics

data as a named set of data objects
data type as a set of values and a set of operations
group names
infinity and infinitesimal
meaning vs. reference
number as a named set of numbers
number as the extension of a class of equinumerous classes
objects as a set of attributes
what is a number


A set is a collection of things. It is closely related to the concepts of number, computation, and relationship. A set assumes a universe out of which its elements are selected. Without such a universe, one gets paradoxes and the inability to recognize what is in a set. Scott's type theory defines a particularly rich collection of sets.

The Zermelo axioms for set theory start with the notion of extensionality--that a set is defined by its members. The axiom of elementary sets defines the null set, unit sets, and pairs. The axiom of separation allows one to define a subset by a predicate, while the axioms of power set and union define larger sets. The axiom of choice is a curious one. It allows one to create a set by choosing a single element for each member of a set of sets. It is related to Zorn's lemma. The axiom of infinity constructs an infinite set that includes the natural numbers.

Frege emphasized that number is about concepts and not about objects. The same can be said about sets. Extension, i.e., the members of a set, are what make sets a valuable construct. (cbb 4/94)

Subtopic: set as fundamental concept up

Quote: sets are classes; the notion of class is so fundamental to thought that we cannot hope to define it [»quinWV_1969]
Quote: set theory investigates the fundamental notions of number, order, and function [»zermE19_1908]
Quote: a set can be object that is related to its members [»kentW_1978]
Quote: a set is the barest notion of data storage, i.e., mere existence [»millHD2_1986]
Quote: in paper solutions, sets or subsets used for 95% of statements on multiple objects; 5% loops or iteration [»paneJF2_2001]
Quote: prefer set and subset expressions over iterative operations; avoids loop counters, terminating flags, and current object [»paneJF2_2001]

Subtopic: data as set up

Quote: XML does not distinguish between data and sets of data [»boswA10_2005]

Subtopic: set theory up

Quote: Zermelo's axioms for set theory: extensionality, elementary sets, separation, union, choice, infinity [»zermE19_1908]
Quote: the set theory of Cantor and Dedekind is reduced to a seven axioms and a few definitions; it avoids the antinomies discovered so far [»zermE19_1908]
Quote: axiom of choice--given a non-empty class of disjoint non-empty sets, can form a set with one element from each set [»simmGF_1963]
Quote: can not prove Zorn's lemma: if every chain of a partially ordered set is bounded then the set has a maximal element [»simmGF_1963]
Quote: Zorn's lemma is equivalent to the axiom of choice

Subtopic: set as geometry or topology up

Quote: situs--parts are ordered in a line or a set in a vicinity [»leibGW_1666]

Subtopic: extension up

Quote: Frege assumes that the extension of a concept is known; basis for recognizing identity
Quote: there must be generative process whereby a class is created before it is named [»bateG_1979]

Subtopic: intension vs. prototype up

Quote: can represent a set by its essential properties or by a prototypical example [»liebH11_1986]

Subtopic: set vs. collection 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]

Subtopic: relation up

Quote: a relation is a subset of the Cartesian product of a set of domains; i.e., n-tuples of elements from each domain [»coddEF6_1970]
Quote: a mathematical relation is an unordered set of tuples all of the same type [»coddEF_1990]
Quote: relations are not tables; e.g., they do not allow duplicate rows [»coddEF_1990]
Quote: Kleisli uses the nested relational calculus (NRC); lambda calculus plus records, and sets [»wongL9_2000]
Quote: nested relational calculus decomposes sets using a restricted form of structural recursion
Quote: duplicate rows do not represent distinct objects; a fact is a fact; avoid redundancy [»coddEF_1990]

Subtopic: empty set up

Quote: return a zero-length array instead of a null; nulls require special code [»blocJ_2001]

Subtopic: universal set up

Quote: resolve logical difficulties with sets by assuming a fixed, universal set [»simmGF_1963]
Quote: the universal set is the intersection of an empty class of sets [»simmGF_1963]
Quote: Nothing and Universe are the limits of class extension; interpret 0 as Nothing and 1 as Universe [»boolG_1854, OK]
Quote: a class is defined by a genus and those attributes that distinguish members from other members of the genus [»maclBJ12_1983]
Quote: if types define subsets of a universal type, then their unions and intersections are well-defined [»reynJC9_1983]

Subtopic: universal domain up

Quote: Scott proved the existence of a universal domain that contains sub-domains including function spaces [»straC3_1973]
Quote: Scott's type theory defines types on a universal set while avoiding Russell's paradox
Quote: in type-free, lambda calculus, can not define a universal domain that includes all functions [»straC3_1973]

Subtopic: problems with set up

Quote: while a collection of integers appears to make sense, a collection that can't be listed at all does not [»benaP_1983]
Quote: because of self-reference, could not construct a set-theoretic model of the .lambda.-calculus

Group: sets up

Topic: collection class (11 items)
Topic: set construction (20 items)
Topic: set data type (16 items)
Topic: set definition by extension or intension (18 items)
Topic: set operations (12 items)
Topic: set-oriented languages (20 items)
Topic: tuples
(17 items)

Related Topics up

Group: data   (140 topics, 3126 quotes)
Group: philosophy of mathematics   (11 topics, 330 quotes)
Group: sequences   (7 topics, 97 quotes)

Topic: abstraction (62 items)
Topic: arrays (58 items)
Topic: classification (65 items)
Topic: data as a named set of data objects (22 items)
Topic: data type as a set of values and a set of operations (16 items)
Topic: entities (20 items)
Topic: group names (16 items)
Topic: infinity and infinitesimal (37 items)
Topic: meaning vs. reference (49 items)
Topic: number as a named set of numbers (15 items)
Topic: number as the extension of a class of equinumerous classes (23 items)
Topic: objects as a set of attributes (39 items)
Topic: topology (29 items)
Topic: what is a number
(55 items)

Updated barberCB 5/05
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