Many structures in mathematics are incomplete in one or more ways. For instance, the field of rationals $latex {{\bf Q}}&fg=000000$ or the r
Many structures in mathematics are incomplete in one or more ways. ...
... A fourth type of incompleteness, which is slightly less well known than the above three [algebraic, metric, logical], is what I will call elementary incompleteness (and which model theorists call the failure of the countable saturation property). It applies to any structure that is describable by a first-order language, such as a field, a metric space, or a universe of sets. For instance, in the language of ordered real fields, the real line R is elementarily incomplete, because there exists a sequence of statements (such as the statements 0 < x < 1/n for natural numbers n=1,2,...) in this language which are potentially simultaneously satisfiable (in the sense that any finite number of these statements can be satisfied by some real number x) but are not actually simultaneously satisfiable in this theory.
...[I]f one starts with an arbitrary structure U, one can form an elementary completion *U of it, which is a significantly larger structure which contains U as a substructure.... Furthermore, *U is elementarily complete; any sequence of statements that are potentially simultaneously satisfiable in *U (in the sense that any finite number of statements in this collection are simultaneously satisfiable), will actually be simultaneously satisfiable. ... If U is the standard universe of all the standard objects one considers in mathematics, then its elementary completion *U is known as the nonstandard universe, and is the setting for nonstandard analysis.
