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Analyse relative. (French) Zbl 0785.03046
In 1985 Gui Wallet has expressed the idea to introduce into IST a binary predicate of relative standardness instead of Nelson’s predicate St. Later a theory of relative standardness in the framework of IST was developed by E. I. Gordon [Sib. Mat. Zh. 30, No. 1, 89-95 (1989; Zbl 0697.03037)]. In the paper under review, the author presents an axiomatic set theory called RIST (relative internal set theory). Its language contains a new predicate $x\text{SR}y$ (‘$x$ is $y$-standard’). A formula is called internal if the predicate SR does not occur in it. The axioms of RIST include relativized axioms of ZFC. They also include three axioms for the predicate SR. According to these axioms SR is a linear order relation. Let $\alpha$ be a set and $F$ a formula in the language of RIST. The author writes ${\forall }^{\alpha }xF$ for $\forall x\left(x\text{SR}\alpha ⇒F\right)$, ${\forall }^{¬\alpha }xF$ for $\forall x\left(\left(¬\left(x\text{SR}\alpha \right)\right)⇒F\right)$, and so on. The quantifiers ${\forall }^{\alpha }$, ${\forall }^{¬\alpha }$, ${\exists }^{\alpha }$, ${\exists }^{¬\alpha }$ are called external. Axioms that correspond to the axioms of IST are formulated. All of them contain external quantifiers. A subtheory RISTn is defined in the following way. Let ${\alpha }_{1}$ be a standard set. It follows from the RIST axioms that it is possible to choose ${\alpha }_{2},\cdots ,{\alpha }_{n}$ such that ${\alpha }_{1}\text{SR}{\alpha }_{2},{\alpha }_{2}\text{SR}{\alpha }_{3},\cdots ,{\alpha }_{n-1}\text{SR}{\alpha }_{n}$. For every such $x$, assume ${\text{id}}_{p}x$ if $x\text{SR}{\alpha }_{p}$. As usual, well-formed formulae are defined starting with the predicates $\in$, ${\text{id}}_{1},\cdots ,{\text{id}}_{n}$. At last three theorems of RIST are chosen as axioms of RISTn. In the framework of RIST and RISTn, mathematical analysis is developed. It is called relative analysis. Various classes of topological spaces are characterized in terms of RISTn. In conclusion, geometric characterizations of the almost-periodic and almost-automorphic functions are presented.
Reviewer: G.Pestov (Tomsk)
##### MSC:
 03H05 Nonstandard models in mathematics 26E35 Nonstandard real analysis 03E70 Nonclassical set theories 54J05 Nonstandard topology