In set theory, an infinite set is a set that is not a finite set. Infinite sets may be countable or uncountable. Some examples are: Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... In mathematics, a countable set is a set with the same cardinality (i. ... In mathematics, an uncountable or nondenumerable set is a set which is not countable. ...
the set of all integers, {..., -1, 0, 1, 2, ...}, is a countably infinite set; and
the set of all real numbers is an uncountably infinite set.
Counter-example: The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
the set of natural numbers less than four, i.e. {0, 1, 2, 3}, is a finite set, not an infinite set.
An infiniteset can simply be defined as one having the same size as at least one of its "proper" parts; this notion of infinity is called Dedekind infinite.
Generalizing finite and the ordinary infinite sequences which are maps from the positive integers leads to mappings from ordinal numbers, and transfinite sequences.
Likewise, perpetual motion machines theoretically generate infinite energy by attaining 100% efficiency or greater, and emulate every conceivable open system; the impossible problem follows of knowing that the output is actually infinite when the source or mechanism exceeds any known and understood system.
This definition of "infiniteset" should be compared and contrasted to the usual definition: a set A is finite if A is empty, or if there is a positive integer n such that A is equinumerous to the set {1, 2, 3,..., n}.
Since every infinite, well-ordered set is Dedekind-infinite, and since the AC is equivalent to the well-ordering theorem stating that every set can be well-ordered, clearly the general AC implies that every infiniteset is Dedekind-infinite.
It is notable that this definition was the first definition of "infinite" which did not rely on the definition of the natural numbers.
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