Extended use of IST
In 1977 Internal Set Theory (IST) was introduced by \textit{E. Nelson} [Bull. Am. Math. Soc. 83, 1165-1198 (1977; Zbl 0373.02040)] as a formalization of nonstandard analysis. The core of IST consists of three axiom schemes, Transfer (T), Idealization (I) and Standardization (S), where the schemes (T) and (I) are applied on internal formulas only and the scheme (S) concerns only the standard elements of a set. In the present paper the author shows that within IST these axiom schemes may be applied also in new, more general situations. He proves that in the scheme (I) instead of internal formulas halic formulas (i.e. formulas of the type \((\forall^{st})A(x)\) with \(A\) internal) may be used (Theorem 2.1) and in the scheme (S) under some restriction the nonstandard elements of a set may be used too (Theorem 3.1). In Chapter 4 the author generalizes the transfer principle (within IST), i.e. he investigates situations where a property may be transferred from a given set to a bigger set. He presents at first two new transfer principles (transfer for images and transfer for graphs) (Theorem 4.1) and then he proves (Theorem 4.4) that for a prehalo \(H\) and an absolute property \(P(x)\), if \(P(x)\) holds on \(H\), then \(P\) also holds on the monad \(M(H)\) of \(H\). The author shows a lot of interesting examples of applications of these extended axiom schemes.
scientific article; zbMATH DE number 64031
scientific article