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set theory Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, when dealing with sets of infinite size, the term almost or nearly is used to refer to all but a negligible amount of elements in the set. The notion of "negligible" depends on the context, and may mean "of measure zero" (in a measure space), "finite" (when infinite sets are involved), or "countable" (when uncountably infinite sets are involved). For example: *The set S = \ is almost \mathbb for any k in \mathbb, because only finitely many
natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...
s are less than ''k''. *The set of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s is not almost \mathbb, because there are infinitely many natural numbers that are not prime numbers. *The set of transcendental numbers are almost \mathbb, because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable). *The Cantor set is uncountably infinite, but has Lebesgue measure zero. So almost all real numbers in (0, 1) are members of the complement of the Cantor set.


See also

* Almost periodic function - and Operators * Almost all * Almost surely * Approximation * List of mathematical jargon


References

Mathematical terminology Set theory {{settheory-stub de:Fast alle