Randomizing a model
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 \(V\).
axiomatization of the first-order theory of randomizations
scientific article; zbMATH DE number 1286377
scientific article