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In ZF, a set is infinite if and only if the power set of its power set is a Dedekind-infinite set, having a proper subset equinumerous to itself. [4] If the axiom of choice is also true, then infinite sets are precisely the Dedekind-infinite sets. If an infinite set is a well-orderable set, then it has many well-orderings which are non-isomorphic.
The best known example of an uncountable set is the set of all real numbers; Cantor's diagonal argument shows that this set is uncountable. The diagonalization proof technique can also be used to show that several other sets are uncountable, such as the set of all infinite sequences of natural numbers , and the set of all subsets of the set of natural numbers.
An infinite set is a set with an infinite number of elements. If the pattern of its elements is obvious, an infinite set can be given in roster notation, with an ellipsis placed at the end of the list, or at both ends, to indicate that the list continues forever. For example, the set of nonnegative integers is
is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set. For example, the set of all positive integers is infinite:
For example, one infinity—the one most people are familiar with—is an infinite set of natural numbers: 1, 2, 3, and so on. However, there’s also an infinite set of real numbers, which ...
This definition of "infinite set" should be compared with the usual definition: a set A is infinite when it cannot be put in bijection with a finite ordinal, namely a set of the form {0, 1, 2, ..., n−1} for some natural number n – an infinite set is one that is literally "not finite", in the sense of bijection.
the set of all binary strings of finite length, and; the set of all finite subsets of any given countably infinite set. These infinite ordinals: ω, ω + 1, ω⋅2, ω 2 are among the countably infinite sets. [6] For example, the sequence (with ordinality ω⋅2) of all positive odd integers followed by all positive even integers
In the formal language of the Zermelo–Fraenkel axioms, the axiom is expressed as follows: [2] ( ( ()) ( ( (( =))))). In technical language, this formal expression is interpreted as "there exists a set 𝐼 (the set that is postulated to be infinite) such that the empty set is an element of it and, for every element of 𝐼, there exists an element of 𝐼 consisting of just the elements of ...