da Costa, Newton C. A. and Krause, Décio
(2020)
Suppes predicate for classes of structures and the notion of transportability.
[Preprint]
Abstract
Patrick Suppes’ maxim “to axiomatize a theory is to define a set- theoretical predicate” is usually taking as entailing that the formula that defines the predicate needs to be transportable in the sense of Bourbaki. We argue that this holds for theories, where we need to cope with all structures (the models) satisfying the predicate. For in- stance, in axiomatizing the theory of groups, we need to grasp all groups. But we may be interested in catching not all structures of a species, but just some of them. In this case, the formula that defines the predicate doesn’t need to be transportable. The study of this ques- tion has lead us to a careful consideration of Bourbaki’s definition of transportability, usually not found in the literature. In this paper we discuss this topic with examples, recall the notion of transportable formulas and show that we can have significant set-theoretical pred- icates for classes of structures defined by non transportable formulas as well.
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