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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 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 ${\displaystyle S=\{n\in \mathbb {N} \,|\,n\geq k\}}$ 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 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

1. "Almost All Real Numbers are Transcendental - ProofWiki". proofwiki.org. Retrieved 2019-11-16.
2. "Theorem 36: the Cantor set is an uncountable set with zero measure". Theorem of the week. 2010-09-30. Retrieved 2019-11-16.

v t e Set theory
Overview	Set (mathematics)
Axioms	Adjunction Choice countable dependent global Constructibility (V=L) Determinacy projective Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom Axiom schema replacement specification
Operations	Cartesian product Complement (i.e. set difference) De Morgan's laws Disjoint union Identities Intersection Power set Symmetric difference Union
Concepts Methods	Almost Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Set-builder notation Transfinite induction Venn diagram
Set types	Amorphous Countable Empty Finite (hereditarily) Filter base subbase Ultrafilter Fuzzy Infinite (Dedekind-infinite) Recursive Singleton Subset · Superset Transitive Uncountable Universal
Theories	Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck
Paradoxes Problems	Russell's paradox Suslin's problem Burali-Forti paradox
Set theorists	Paul Bernays Georg Cantor Paul Cohen Richard Dedekind Abraham Fraenkel Kurt Gödel Thomas Jech John von Neumann Willard Quine Bertrand Russell Thoralf Skolem Ernst Zermelo

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