numberspecific

Numberspecific is a term used in mathematics and related fields to describe a number that is uniquely determined by a fixed predicate within a given domain. In this sense, a number is numberspecific if it is the sole element that satisfies the predicate.

Formally, let D be a set (often the natural numbers N) and P a decidable predicate on D. A number n in D is numberspecific with respect to P if P(n) holds and for all m in D, P(m) implies m = n. If no such n exists, then no numberspecific number exists for P. If more than one element satisfies P, there is also no numberspecific number for P. This emphasizes the idea of a unique solution singled out by a condition.

Relationship to definability: The concept relates to definable numbers in logic, where a number can be singled out by a finite description. Numberspecific emphasizes uniqueness under a specific predicate rather than a general definability property, and it does not require any particular form of the predicate beyond decidability and the uniqueness condition.

Examples: A) 2 is numberspecific for P(n) = (n^2 = 4) in N. B) 3 is numberspecific for Q(n) = (n^2 = 9) in N. C) 0 is numberspecific for R(n) = (n^2 = 0).

Applications and limitations: The term is informal and used to illustrate how numbers may be singled out by conditions in data modeling, formal reasoning, or teaching. It is not an established standard concept; care should be taken to ensure the predicate and domain yield a unique solution.

See also: definable number, unique solution, predicate, computability.