almost

{{short description|Term in set theory}}

{{Other uses}}

{{More citations needed|date=January 2021}}

In set theory, 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 $S\; =\; \backslash \{\; n\; \backslash in\; \backslash mathbb\{N\}\backslash ,|\backslash ,\; n\; \backslash ge\; k\; \backslash \}$ is almost $\backslash mathbb\{N\}$ for any $k$ in $\backslash mathbb\{N\}$, because only finitely many natural numbers are less than $k$.
• The set of prime numbers is not almost $\backslash mathbb\{N\}$, because there are infinitely many natural numbers that are not prime numbers.
• The set of transcendental numbers are almost $\backslash mathbb\{R\}$, because the algebraic real numbers form a countable subset of the set of real numbers (which is uncountable).{{Cite web|url=https://proofwiki.org/wiki/Almost_All_Real_Numbers_are_Transcendental|title=Almost All Real Numbers are Transcendental - ProofWiki|website=proofwiki.org|access-date=2019-11-16}}
• The Cantor set is uncountably infinite, but has Lebesgue measure zero.{{Cite web|url=https://theoremoftheweek.wordpress.com/2010/09/30/theorem-36-the-cantor-set-is-an-uncountable-set-with-zero-measure/|title=Theorem 36: the Cantor set is an uncountable set with zero measure|date=2010-09-30|website=Theorem of the week|language=en|access-date=2019-11-16}} So almost all real numbers in (0, 1) are members of the complement of the Cantor set.