Pairs, sets and sequences in first-order theories
type: Item
instance of: scholarly article
title: Pairs, sets and sequences in first-order theories
zbMATH Open document ID: 1158.03037
author: Albert Visser
published in: Archive for Mathematical Logic
publication date: 2008-09-09T00:00:00Z
review text: The paper studies first-order theories (recursively axiomatizable, in a finite language) endowed with so-called containers, such as ordered and unordered pairs, sequences, sets. Formally, a theory possesses certain containers if it supports an unrelativized interpretation with absolute equality (a direct interpretation) of a suitable container theory. The author explores the idea that for certain types of containers, this is equivalent to the ostensibly much stricter condition that the theory is definitionally equivalent to an extension (in the same language) of the container theory. Thus, a theory \(U\) is called adaptive if whenever a theory \(V\) directly interprets \(U\) then \(V\) is definitionally equivalent to an extension of \(U\). The main results of the paper state that the theory of functional non-surjective ordered pairing, and a weak set theory WS (axiomatized by the existence of successors and of an empty set) are both adaptive. In particular, a theory is sequential if and only if it is definitionally equivalent to an extension of WS.
reviewed by: Emil Jeřábek
Mathematics Subject Classification ID: 03F25
Mathematics Subject Classification ID: 03B30
zbMATH DE Number: 5323285
zbMATH Keywords: interpretation
zbMATH Keywords: sequential theory
zbMATH Keywords: weak set theory
MaRDI profile type: Publication
full work available at URL: s00153-008-0087-1
OpenAlex ID: W2052127763
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DOI: 10.1007/S00153-008-0087-1
description: scientific article; zbMATH DE number 5323285