of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context. almost all A shorthand term for "all except for a set of measure zero", when there is a measure to speak of. For example, "almost all real numbers are transcendental" because the algebraic real numbers form a countable subset of the real numbers with measure zero. One can also speak of "almost all" integers having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are odd". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with generic, below. arbitrarily large Notions which arise mostly in the context of limits, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicate P is satisfied by arbitrarily large values, can be expressed in more formal notation by . See also frequently. The statement that quantity f(x) depending on x "can be made" arbitrarily large, corresponds to . arbitrary A shorthand for the universal quantifier. An arbitrary choice is one which is made unrestrictedly, or alternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set. Also much in general-language use among mathematicians: "Of course, this problem can be arbitrarily complicated". eventuallyIn the context of limits, this is shorthand meaning for sufficiently large arguments; the relevant argument(s) are implicit in the context. As an example, the function log(log(x)) eventually becomes larger than 100"; in this context, "eventually" means "for sufficiently large x." factor through A term in category theory referring to composition of morphisms. If for three objects A, B, and C a map can be written as a composition with and , then f is said to factor through any (and all) of , , and . finite "Not infinite". For example, if the variance of a random variable is said to be finite, this implies it is a non-negative real number. frequently In the context of limits, this is shorthand for arbitrarily large arguments and its relatives; as with eventually, the intended variant is implicit. As an example, the sequence is frequently in