infinite set

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infinite set

[′in·fə·nət ′set]
(mathematics)
A set with more elements than any fixed integer; such a set can be put into a one to one correspondence with a proper subset of itself.

infinite set

(mathematics)
A set with an infinite number of elements. There are several possible definitions, e.g.

(i) ("Dedekind infinite") A set X is infinite if there exists a bijection (one-to-one mapping) between X and some proper subset of X.

(ii) A set X is infinite if there exists an injection from N (the set of natural numbers) to X.

In the presence of the Axiom of Choice all such definitions are equivalent.
References in periodicals archive ?
Writing in an admirably succinct and clear style, he covers logic, sets, functions, counting infinite sets, infinite cardinals, well-ordered sets, inductions and numbers, prime numbers, and logic and meta-mathematics.
plates and napkins on a table) to establish they're the same size to using the operation to establish when infinite sets have the same size.
As students of mathematics we invariably confront infinite sets.
1] can be expressed as a disjoint union of a class C of countably infinite sets, the cardinality of C being [[omega].
This includes a problem list compile by the speakers that reflects some of the most important questions in various areas of logic, and the presenters cover such topics as a stationary-tower-free proof of the derives model theorem, a proof of an absoluteness theorem, a simple inductive measure analysis of cardinals under the axiom of determinacy, the complexity of index set and Ehrenfeucht theories, computable structures in familiar classes, the classes of separating sets, voting rules for infinite sets and Boolean algebras, "very mad families," Borel boundedness and the lattice surrounding property, and Steinhous sets and Jackson sets.
Of course, the counting numbers can be paired with even "smaller" infinite sets such as the prime numbers.
Infinite wisdom A mathematician proposed a new approach to resolving a longstanding question about infinite sets of numbers (164: 139).
The topic chosen for this purpose, "Comparing Infinite Sets within the Cantorian Set Theory", is part of an advanced mathematics course.
Cantor was able to show that it is possible to generate, 'bigger and bigger infinite sets from ones that we already have.
Infinite sets were identified as sets bijective to a proper subset of themselves.
describes the uses of infinity as he explains elementary set theory, functions (including inverse functions), counting infinite sets (including Hilbert's Infinite Hotel), infinite Cardinals (including the two infinities), well-ordered sets (such as the arithmetic of ordinals and cardinals as ordinals), inductions and numbers, including number theory, prime numbers, including the Riemann Zeta function.
For one thing, there's no way to simply count the elements of two infinite sets and determine which set has more.