A set is simply a well defined collection of distinct objects, considered together, but the idea, developed rigorously, turned out able to serve as a common foundation underneath nearly the whole of mathematics; a number, a function, a geometric point, can each be reconstructed as a particular kind of set. Georg Cantor created set theory from 1874 onward while investigating the size of infinite collections, and his own naive version of the theory was shown in the early twentieth century to permit genuine logical paradoxes (Bertrand Russell's paradox, concerning the set of all sets that do not contain themselves, is the most famous); the axiomatic version developed afterward, chiefly by Ernst Zermelo and Abraham Fraenkel, repairs this and remains the standard foundation used today.
Facts
Connections
Associated With
Source Algebraic Structure (Wikipedia)
The archetypal Boolean algebra is the algebra of subsets of a set under union, intersection and complement, so its structure is built directly on the set concept.
Additional Source Boolean Algebra (Wikipedia)Lead section
The theorem compares the cardinality of a set to its own power set, so its subject is the set concept itself.
Additional Source Cantor's Theorem (Wikipedia)lead
Cardinality, Concepts Cardinality is defined as an inherent property of sets, so the set concept is what the property is measured on.
Additional Source Cardinality (Wikipedia)Lead section
Countable Set, Concepts Countability is a classification of sets by whether they can be put in one to one correspondence with the natural numbers, so the concept is a property of the set concept.
Additional Source Countable Set (Wikipedia)Lead section
Created set theory from 1874 onward.
Source The Stanford Encyclopedia of Philosophy
Source Partition of a Set (Wikipedia)
Source Well-Ordering Principle (Wikipedia)
Attributed To
Source Wikipedia: Set (Mathematics)
In Branch
Source The Stanford Encyclopedia of Philosophy
Kind Of Object Of
Sources
1. The Stanford Encyclopedia of Philosophy
Center for the Study of Language and Information, Stanford University
Wikipedia: Set (Mathematics)
Wikimedia FoundationBasic notions
In mathematics, a set is a collection of different things, called elements or members of the set.
Attributed To: Ernst Zermelo, Lead
Roster or enumeration notation is a notation introduced by Ernst Zermelo in 1908 that specifies a set by listing its elements between braces, separated by commas
View the Source Cantor's Theorem (Wikipedia)
Wikimedia FoundationAssociated With: Cantor's Theorem, leadQuote, Associated With: Cantor's Theorem, lead
Every set is smaller than its power set
Cardinality (Wikipedia)
Wikimedia FoundationAssociated With: Cardinality, Lead sectionQuote, Associated With: Cardinality, Lead section
cardinality is an inherent property of sets, roughly meaning the number of individual objects they contain, which may be infinite
Countable Set (Wikipedia)
Wikimedia FoundationAssociated With: Countable Set, Lead sectionQuote, Associated With: Countable Set, Lead section
A mathematical set is countable if either it is finite or it can be put in one to one correspondence with the set of natural numbers
Boolean Algebra (Wikipedia)
Wikimedia FoundationAssociated With: Boolean Algebra, Lead sectionQuote, Associated With: Boolean Algebra, Lead section
In mathematics and mathematical logic, Boolean algebra is a branch of algebra.
Well-Ordering Principle (Wikipedia)
Wikimedia FoundationAssociated With: Well-Ordering Principle, Lead section, first sentenceQuote, Associated With: Well-Ordering Principle, Lead section, first sentence
the well-ordering principle states that every non-empty subset of nonnegative integers contains a least element
Partition of a Set (Wikipedia)
Wikimedia FoundationAssociated With: Partition of a Set, Lead section, first sentenceQuote, Associated With: Partition of a Set, Lead section, first sentence
a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset
Algebraic Structure (Wikipedia)
Wikimedia FoundationAssociated With: Algebraic Structure, Lead section, first paragraphQuote, Associated With: Algebraic Structure, Lead section, first paragraph
an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities (known as axioms) that these operations must satisfy.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.