Arbitrarily large

In mathematics, the phrases arbitrarily large, arbitrarily small, and arbitrarily long are used in statements such as:

"ƒ(x) is non-negative for arbitrarily large x."

which is shorthand for:

"For every real number n, ƒ(x) is non-negative for some values of x greater than n."

"Arbitrarily large" is not equivalent to "sufficiently large". For instance, while it is true that prime numbers can be arbitrarily large since there are an infinite number of them, it is not true that all sufficiently large numbers are prime. "Arbitrarily large" does not mean "infinitely large" because although prime numbers can be arbitrarily large, an infinitely large prime does not exist since all prime numbers (as well as all other integers) are finite.

In some cases, phrases such as "P(x) is true for arbitrarily large x" are used primarily for emphasis, as in "P(x) is true for all x, no matter how large x is." In these cases, the phrase "arbitrarily large" does not have the meaning indicated above but is in fact logically synonymous with "all."

To say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers (there is not), nor that there exists any particular arithmetic progression of prime numbers that is in some sense "arbitrarily long", but rather that no matter how large a number n is, there exists some arithmetic progression of prime numbers of length at least n.

The statement "ƒ(x) is non-negative for arbitrarily large x." could be rewritten as:

∀ n ∈ R ,  ∃ x ∈ R  such that  x > n ∧ f ( x ) ≥ 0

Using "sufficiently large" instead yields:

∃ n ∈ R  such that  ∀ x ∈ R ,  x > n ⇒ f ( x ) ≥ 0