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In mathematics, the phrase arbitrarily large is used in contexts such as:
- <math>f(x)<math> is true for arbitrarily large <math>x<math>
which is actually shorthand for:
- for every <math>N\in\mathbb{R}<math>, <math>f(x)<math> is true for at least one <math>x\ge N<math>.
This should not be confused with the phrase "sufficiently large". For instance, it is true that prime numbers can be arbitrarily large, but it is not true that all sufficiently large numbers are prime. It is also worth noting that "arbitrarily large" does not mean "infinitely large" - for instance, while prime numbers can be arbitrarily large, there is no such thing as an infinitely large prime.
In some cases, phrases such as "f(x) is true for arbitrarily large x" is used primarily for emphasis, as in "f(x) is true for all x, no matter how large x is." In such cases, the phrase "arbitrarily large" does not have the meaning indicated above, but is in fact logically synonymous with "all."
One can of course define terms such as "arbitrarily small," "arbitrarily long," and others in a similar manner.
See also
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