Extended use of IST
type: Item
instance of: scholarly article
title: Extended use of IST
zbMATH Open document ID: 0777.03019
DOI: 10.1016/0168-0072(92)90035-X
author: I. P. Van den Berg
published in: Annals of Pure and Applied Logic
publication date: 1992-09-27T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03H05
zbMATH DE Number: 64031
zbMATH Keywords: transfer for images
zbMATH Keywords: Internal Set Theory
zbMATH Keywords: formalization of nonstandard analysis
zbMATH Keywords: Transfer
zbMATH Keywords: Idealization
zbMATH Keywords: Standardization
zbMATH Keywords: transfer for graphs
zbMATH Keywords: prehalo
zbMATH Keywords: monad
reviewed by: Jiří Močkoř
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 64031