Editing Nonstandard analysis (section)

== Logical framework ==
Given any set {{mvar|S}}, the ''superstructure'' over a set {{mvar|S}} is the set {{math|''V''(''S'')}} defined by the conditions

:<math>V_0(S) = S, </math>
:<math>V_{n+1}(S) = V_{n}(S) \cup \wp (V_{n}(S)), </math>
:<math>V(S) = \bigcup_{n \in \mathbf{N}} V_{n}(S).</math>

Thus the superstructure over {{mvar|S}} is obtained by starting from {{mvar|S}} and iterating the operation of adjoining the [[power set]] of {{mvar|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 space|separable]] [[metric space]]s and [[metrizable topological vector space]]s. Virtually all of mathematics that interests an analyst goes on within {{math|''V''('''R''')}}.

The working view of nonstandard analysis is a set {{math|*'''R'''}} and a mapping {{math|* : ''V''('''R''') → ''V''(*'''R''')}} that satisfies some additional properties. To formulate these principles we first state some definitions.

A [[Well-formed formula|formula]] has ''[[bounded quantification]]'' if and only if the only quantifiers that occur in the formula have range restricted over sets, that is are all of the form:

:<math> \forall x \in A, \Phi(x, \alpha_1, \ldots, \alpha_n) </math>
:<math> \exists x \in A, \Phi(x, \alpha_1, \ldots, \alpha_n) </math>

For example, the formula

:<math> \forall x \in A, \ \exists y \in 2^B, \quad x \in y </math>

has bounded quantification, the [[Universal quantification|universally quantified]] variable {{mvar|x}} ranges over {{mvar|A}}, the [[Existential quantification|existentially quantified]] variable {{mvar|y}} ranges over the powerset of {{mvar|B}}. On the other hand,

:<math> \forall x \in A, \ \exists y, \quad x \in y </math>

does not have bounded quantification because the quantification of ''y'' is unrestricted.