Randomizing a model
type: Item
instance of: scholarly article
title: Randomizing a model
zbMATH Open document ID: 0924.03065
author: H. Jerome Keisler
published in: Advances in Mathematics
publication date: 1999-11-15T00:00:00Z
review text: A randomization of a structure \(\mathcal M\) is a new structure \(\mathcal K\), whose universe is a set of random elements of \(\mathcal M\). Boolean powers and bounded Boolean powers are typical examples for randomizations. The author starts by giving an axiomatization of the first-order theory \(T\) of randomizations. This is a three-sorted theory having variables for random elements, for events and for scalars. The axioms say that the elements of second sort build a Boolean algebra and the elements of the third sort are the domain of a real closed field. An event mapping assigns to each formula of the language of \(\mathcal M\) a Boolean value. A transfer axiom ensures that this value is one for the sentences of \(\text{Th}({\mathcal M})\). The main result of the paper states that \(T\) admits the elimination of quantifiers. This is used to show that \(T\) is complete. It is shown that the theory \(T\) without the transfer axiom also allows quantifier elimination. The author investigates the class of substructures of models of \(T\) and shows that there is a natural set \(V\) of universal sentences of \(T\) which describes this class. This implies that \(T\) is the model completion of \(V\).
reviewed by: Martin Weese
Mathematics Subject Classification ID: 03C90
Mathematics Subject Classification ID: 03C10
zbMATH DE Number: 1286377
zbMATH Keywords: axiomatization of the first-order theory of randomizations
zbMATH Keywords: randomization of a structure
zbMATH Keywords: Boolean powers
zbMATH Keywords: quantifier elimination
zbMATH Keywords: model completion
MaRDI profile type: Publication
OpenAlex ID: W1978403934
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DOI: 10.1006/AIMA.1998.1793
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description: scientific article; zbMATH DE number 1286377