In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear the fact that an object is large, small, or long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers (and its subsets thereof), though its meaning can differ from that of "sufficiently" and "infinitely".
The statement
"
f(x)
x
is a shorthand for:
"For every real number
n
f(x)
x
n
In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers. For example, 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". Rather, the phrase is used to refer to the fact that no matter how large a number
n
n
Similar to arbitrarily large, one can also define the phrase "
P(x)
\forall\epsilon\inR+,\existsx\inR:|x|<\epsilon\landP(x)
In other words:
However small a number, there will be a number
x
P(x)
While similar, "arbitrarily large" is not equivalent to "sufficiently large". For instance, while it is true that prime numbers can be arbitrarily large (since there are infinitely many of them due to Euclid's theorem), it is not true that all sufficiently large numbers are prime.
As another example, the statement "
f(x)
x
\foralln\inR,\existsx\inRsuchthatx>n\landf(x)\ge0
However, using "sufficiently large", the same phrase becomes:
\existsn\inRsuchthat\forallx\inR,x>n ⇒ f(x)\ge0
Furthermore, "arbitrarily large" also does not mean "infinitely large". For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers (as well as all other integers) are finite.
In some cases, phrases such as "the proposition
P(x)
x
P(x)
x
x
P(x)