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Semantics for naive set theory in many-valued logics: technique and historical account

par Libert, Thierry
Editeur scientifique Van Benthem, Johan;Heinzmann, Gerhard ;Rebuschi, Manuel;Visser, Henk
Référence The age of alternative logics, assessing philosophy of logic and mathematical today, Springer Netherlands, Dordrecht, page (121-136)
Publication Publié, 2006

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Résumé :

It must be admitted that mathematical investigations in providing alternative semantics have carried innovative ideas, and if all have not led to further developments and applications, they have often led to a better understanding of the topic considered. Even within a well-established framework, the use of alternative semantics has proved its fruitfulness. As an example, for independence results in ZF, one may quote the Boolean-valued version of forcing due to Scott and Solovay, in which a set is conceived of as a function which takes its values into a given complete Boolean algebra, no more into the 2-valued one. This concerns classical logic and perhaps this would remind the reader of the primal use of many-valued semantics for proving the independence of axioms in propositional logic. Note that there is no need to be interested in a possible meaning of the additional "truth values" to do that, we would rather say that the understanding is in the application. That said, one may legitimately ask whether the use of many-valued semantics could not also benefit our understanding of the set-theoretical paradoxes themselves. In any case, to know which logic(s) can support the full comprehension scheme, or some maximal fragments of it, is an interesting question in itself, at least not devoid of mathematical curiosity. So we shall review connection with Skolem's pioneering work in Lłukasiewicz's logics. On the way we will also point out and distinguish two ways of formalizing set theory, namely comprehension and abstraction, which have been considered by several authors. Our purpose is to stress the use of fixed-point arguments in semantical consistency proofs and thus the role of continuity in avoiding the paradoxes. Then it will become apparent how these investigations were actually close to other contemporary ones, namely Kripke's work on the liar paradox and Scott's on models for the untyped λ-calculus. To make the links between our different sources clearer, we have included in the appendix on page 134 a diagram of some selected references. Throughout the paper, such references are conveniently marked with the symbol '*'.