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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.
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:
2. A Dedekind-infinite set is a set that can be put into a one-to-one correspondence with one of its proper subsets, indicating a type of infinity; a Dedekind-finite set is a set that is not Dedekind-infinite. (These are also spelled without the hyphen, as "Dedekind finite" and "Dedekind infinite".) def The set of definable subsets of a set ...
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.
[18] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets. Cantor's ideas ultimately were largely accepted, strongly supported by David Hilbert, amongst others.
In set theory, an infinite set is not considered to be created by some mathematical process such as "adding one element" that is then carried out "an infinite number of times". Instead, a particular infinite set (such as the set of all natural numbers) is said to already exist, "by fiat", as an assumption or an axiom. Given this infinite set ...
Any set can be given the cofinite topology in which the open sets are the empty set and the sets whose complement is finite. This is the smallest T 1 topology on any infinite set. Any set can be given the cocountable topology, in which a set is defined as open if it is either empty or its complement is countable. When the set is uncountable ...
These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
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