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Concept
Almost all
Summary
In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if is a set, "almost all elements of " means "all elements of but those in a negligible subset of ". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null.
In contrast, "almost no" means "a negligible quantity"; that is, "almost no elements of " means "a negligible quantity of elements of ".
Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) except for finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) except for countably many".
Examples:
Almost all positive integers are greater than 1012.
Almost all prime numbers are odd (2 is the only exception).
Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and the four Kepler–Poinsot polyhedra).
If P is a nonzero polynomial, then P(x) ≠ 0 for almost all x (if not all x).
When speaking about the reals, sometimes "almost all" can mean "all reals except for a null set". Similarly, if S is some set of reals, "almost all numbers in S" can mean "all numbers in S except for those in a null set". The real line can be thought of as a one-dimensional Euclidean space. In the more general case of an n-dimensional space (where n is a positive integer), these definitions can be generalised to "all points except for those in a null set" or "all points in S except for those in a null set" (this time, S is a set of points in the space). Even more generally, "almost all" is sometimes used in the sense of "almost everywhere" in measure theory, or in the closely related sense of "almost surely" in probability theory.
Examples:
In a measure space, such as the real line, countable sets are null. The set of rational numbers is countable, so almost all real numbers are irrational.
Georg Cantor's first set theory article proved that the set of algebraic numbers is countable as well, so almost all reals are transcendental.
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