Intensionality in mathematics
This is a nice introduction to a form of intensional set-theory in which the identity of classes is determined not by their memberships, but by their explicit presentation in the expressions which define them. These so-called constructive theories, developed by Bishop and others, thus reject the classical principle of extensionality, (\(\forall x)(x\in A\leftrightarrow x\in B)\to A=B\). To facilitate a point by point comparison with classical theory, the author lays out the Bernays-Gödel theory of sets and classes and shows how it provides a foundation for mathematics. He then defines the constructive theories and shows how they accomplish the same end. He himself, however, prefers an intermediate approach, a theory which contains a full classical logic, including the law of the excluded middle and its first-order counterparts, but adhering to the strict constructivists' method of definition for sets and classes. Underlying acceptance of one form of system or another are some philosophical questions which the author, although certainly aware of them, does not try to answer.
scientific article; zbMATH DE number 3889497
scientific article