Relative theory of internal sets
type: Item
instance of: scholarly article
title: Relative theory of internal sets
zbMATH Open document ID: 0803.03046
author: Yves Péraire
published in: Osaka Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: The author proposes a new set theory called RIST. The theory is developed in Nelson style. An undefined two-place predicate SR is used. \(x\text{SR}y\) is the abbreviation of ``\(x\) est standard relativement à \(y\)'', which is also called ``\(x\) is \(y\)-standard''. We assume that SR is an equivalence relation. For a set \(\alpha\) and a formula \(F\), the following notations are introduced: \[ \forall^ \alpha x F\quad\text{for }\forall x (x\text{SR}\alpha\Rightarrow F), \quad \forall^{\sim\alpha}x F \quad \text{for }\forall x((\sim(x\text{SR}\alpha))\Rightarrow F), \] \[ \forall^{\alpha\text{ fin}}x F\quad \text{for }\forall x (x\text{ finite}\Rightarrow F), \] \[ \exists^ \alpha x F\quad \text{for }\exists x (x\text{SR}\alpha\cap F),\quad \exists^{\alpha\text{ fin}}x F\quad \text{for }\exists^ \alpha x (x\text{ finite}\cap F). \] The axiom system of RIST is ZFC with three axiom schemes as follows: 1) Transfer: For an internal formula \(F(x,t_ 1,\dots,t_ n)\) with free variables \(x\), \(t_ 1,\dots,t_ n\), \[ \forall^ \alpha t_ 1 \forall^ \alpha t_ 2\cdots \forall^ \alpha t_ n (\forall^ \alpha x F(x,t_ 1,\dots,t_ n)\Rightarrow \forall^ \alpha x F(x,t_ 1,\dots,t_ n)). \] 2) Idealization: Let \(F(x,y)\) be an internal formula with \(x\), \(y\) as free variables and possibly other parameters. If \(\beta\), \(\alpha_ 1,\alpha_ 2,\dots,\alpha_ k\) are fixed sets such that \(\beta\) is not \((\alpha_ 1,\alpha_ 2,\dots,\alpha_ k)\)- standard, then \[ \forall^{\alpha_ 1,\text{fin}} z_ 1\cdots\forall ^{\alpha_ k,\text{fin}} z_ k \exists^ \beta y\;\forall x_ 1\in z_ 1\cdots\forall x_ k\in z_ k F(x_ 1,\dots,x_ k,y) \] \[ \Leftrightarrow \exists^ \beta y\;\forall^{\alpha_ 1}x_ 1\cdots \forall^{\alpha_ k}x_ k F(x_ 1,\dots,x_ k,y). \] 3) Standardization: If \(\alpha\) is a set, and \(F(x,\dots)\) is an \(\alpha\)- external formula, then \[ \forall^ \alpha y \exists^ \alpha z \forall^ \alpha t\;(t\in z\Leftrightarrow (t\in y\cap F(t,\dots)). \] Some direct consequences obtaine…
Mathematics Subject Classification ID: 03H05
Mathematics Subject Classification ID: 03E70
Mathematics Subject Classification ID: 26E35
zbMATH DE Number: 219873
zbMATH Keywords: relative standardness
zbMATH Keywords: extension of ZFC
zbMATH Keywords: relative internal set theory
zbMATH Keywords: uniform convergence
zbMATH Keywords: compactness
zbMATH Keywords: calculus
zbMATH Keywords: superstructures
reviewed by: Kiyoshi Iseki
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 219873