nearstandard

Nearstandard is a term used in nonstandard analysis to describe elements of the nonstandard universe that are infinitesimally close to a standard element. Formally, for a standard real a, a hyperreal x is nearstandard if x ≈ a for some standard a, where ≈ denotes infinitesimal closeness.

The set of all x with x ≈ a is called the monad of a, denoted Mon(a). A hyperreal x is nearstandard if it lies in Mon(a) for some standard a. The standard reals are themselves nearstandard (to themselves). Finite hyperreals are nearstandard to exactly one standard real, their standard part, while infinite or unbounded hyperreals are not nearstandard.

The standard part map, st, assigns to each finite nearstandard x the unique standard real a such that x ≈ a. Thus st(x) is the standard real that x is infinitesimally close to. This map provides a bridge between the nonstandard and standard settings and is central to many arguments in nonstandard analysis.

Example: if x = 3 + ε with ε an infinitesimal, then x is nearstandard and its standard part is 3. More generally, nearstandardness underpins common nonstandard proofs and the transfer principle, enabling reasoning about limits and continuity via infinitesimal closeness. A function f is continuous at a standard point a if, whenever x is nearstandard to a, f(x) is nearstandard to f(a).

The concept extends to broader contexts within nonstandard analysis, including monads around other standard objects and the interplay between internal and external sets.