Logical Framework
Given any set S, the superstructure over a set S is the set V(S) defined by the conditions
Thus the superstructure over S is obtained by starting from S and iterating the operation of adjoining the power set of S and taking the union of the resulting sequence. The superstructure over the real numbers includes a wealth of mathematical structures: For instance, it contains isomorphic copies of all separable metric spaces and metrizable topological vector spaces. Virtually all of mathematics that interests an analyst goes on within V(R).
The working view of nonstandard analysis is a set *R and a mapping
which satisfies some additional properties.
To formulate these principles we first state some definitions: A formula has bounded quantification if and only if the only quantifiers which occur in the formula have range restricted over sets, that is are all of the form:
For example, the formula
has bounded quantification, the universally quantified variable x ranges over A, the existentially quantified variable y ranges over the powerset of B. On the other hand,
does not have bounded quantification because the quantification of y is unrestricted.
Read more about this topic: Non-standard Analysis
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