Non-standard Analysis - Approaches To Non-standard Analysis

Approaches To Non-standard Analysis

There are two very different approaches to non-standard analysis: the semantic or model-theoretic approach and the syntactic approach. Both these approaches apply to other areas of mathematics beyond analysis, including number theory, algebra and topology.

Robinson's original formulation of non-standard 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 models) of a theory. Since Robinson's work first appeared, a simpler semantic approach (due to Elias Zakon) has been developed using purely set-theoretic objects called superstructures. In this approach a model of a theory is replaced by an object called a superstructure V(S) over a set S. Starting from a superstructure V(S) one constructs another object *V(S) using the ultrapower construction together with a mapping V(S) → *V(S) which satisfies the transfer principle. The map * relates formal properties of V(S) and *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 non-standard analysis that he called Internal Set Theory (IST). 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 non-standard 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 common 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.

Another example of the syntactic approach is the Alternative Set Theory introduced by Vopěnka, trying to find set-theory axioms more compatible with the non-standard analysis than the axioms of ZF.