Editing Nonstandard analysis (section)

== Approaches ==
There are two, main, different approaches to nonstandard analysis: the [[Semantics of logic|semantic]] or [[model-theoretic approach]] and the syntactic approach. Both of these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

Robinson's original formulation of nonstandard analysis falls into the category of the ''semantic approach''. As developed by him in his papers, it is based on studying models (in particular [[saturated model]]s) of a [[Theory (model theory)|theory]]. Since Robinson's work first appeared, a simpler semantic approach (due to Elias Zakon) has been developed using purely set-theoretic objects called [[Universe (mathematics)|superstructures]]. In this approach ''a model of a theory'' is replaced by an object called a ''superstructure'' {{math|''V''(''S'')}} over a set {{mvar|S}}. Starting from a superstructure {{math|''V''(''S'')}} one constructs another object {{math|*''V''(''S'')}} using the [[ultrapower]] construction together with a mapping {{math|''V''(''S'') → *''V''(''S'')}} that satisfies the [[transfer principle]]. The map * relates formal properties of {{math|''V''(''S'')}} and {{math|*''V''(''S'')}}. Moreover, it is possible to consider a simpler form of saturation called [[countable]] saturation. This simplified approach is also more suitable for use by mathematicians who are not specialists in model theory or logic.

The ''syntactic approach'' requires much less logic and model theory to understand and use. This approach was developed in the mid-1970s by the mathematician [[Edward Nelson]]. Nelson introduced an entirely axiomatic formulation of nonstandard analysis that he called [[internal set theory]] (IST).<ref name="Nel">[[Edward Nelson]]: ''Internal Set Theory: A New Approach to Nonstandard Analysis'', Bulletin of the American Mathematical Society, Vol. 83, Number 6, November 1977. A chapter on internal set theory is available at [http://www.math.princeton.edu/~nelson/books/1.pdf http://www.math.princeton.edu/~nelson/books/1.pdf]</ref> IST is an extension of [[Zermelo–Fraenkel set theory]] (ZF) in that alongside the basic binary membership relation ∈, it introduces a new unary predicate ''standard'', which can be applied to elements of the mathematical universe together with some axioms for reasoning with this new predicate.

Syntactic nonstandard analysis requires a great deal of care in applying the principle of set formation (formally known as the [[axiom of comprehension]]), which mathematicians usually take for granted. As Nelson points out, a fallacy in reasoning in IST is that of ''illegal set formation''. For instance, there is no set in IST whose elements are precisely the standard integers (here ''standard'' is understood in the sense of the new predicate). To avoid illegal set formation, one must only use predicates of ZFC to define subsets.<ref name="Nel" />

Another example of the syntactic approach is the [[Vopěnka's alternative set theory]],<ref>Vopěnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.</ref> which tries to find set-theory axioms more compatible with the nonstandard analysis than the axioms of ZF.