# Almost - Wikipedia Mobile

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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 in the context, and may mean "of measure zero" (in a measure space), "countable" (when uncountably infinite sets are involved), or "finite" (when infinite sets are involved).[1]

For example:

• The set ${\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}}$ is almost ${\displaystyle \mathbb {N} }$ for any ${\displaystyle k}$ in ${\displaystyle \mathbb {N} }$, because only finitely many natural numbers are less than ${\displaystyle k}$.
• The set of prime numbers is not almost ${\displaystyle \mathbb {N} }$, because there are infinitely many natural numbers that are not prime numbers.
• The set of transcendental numbers are almost ${\displaystyle \mathbb {R} }$, because the algebraic real numbers form a countable subset of the set of real number (the latter of which is uncountable).[2]
• The Cantor set is uncountably infinite, but has Lebesgue measure zero.[3] So almost all real numbers in (0, 1) are member of the complement of the Cantor set.

## References

1. ^ "The Definitive Glossary of Higher Mathematical Jargon — Almost". Math Vault. 2019-08-01. Retrieved 2019-11-16.
2. ^ "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.
3. ^ "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.