Map Index Random Help Group

Group: sets

topics > Group: mathematics

Topic:
collection class
Topic:
set construction
Topic:
set data type
Topic:
set definition by extension or intension
Topic:
set operations
Topic:
set-oriented languages
Topic:
tuples
Group:
data
Group:
philosophy of mathematics
Group:
sequences

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

Summary

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

 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

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

Subtopic: set theory

 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

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

Subtopic: extension

 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

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

Subtopic: set vs. collection

 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

 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

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

Subtopic: universal set

 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

 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

 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

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

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