In this article, we analyse the ontological import of adding classes to set theories. We assume that this increment is well represented by going from ZF system to NBG. We thus consider the standard techniques of reducing one system to the other. Novak proved that from a model of ZF we can build a model of NBG (and vice versa), while Shoenfield have shown that from a proof in NBG of a set-sentence we can generate a proof in ZF of (...) the same formula. We argue that the first makes use of a too strong metatheory. Although meaningful, this symmetrical reduction does not equate the ontological content of the theories. The strong metatheory levels the two theories. Moreover, we will modernize Shoenfields proof, emphasizing its relation to Herbrands theorem and that it can only be seen as a partial type of reduction. In contrast with symmetrical reductions, we believe that asymmetrical relations are powerful tools for comparing ontological content. In virtue of this, we prove that there is no interpretation of NBG in ZF, while NBG trivially interprets ZF. This challenges the standard view that the two systems have the same ontological content. (shrink)

Interpretations are generally regarded as the formal representation of the concept of translation.We do not subscribe to this view. A translation method must indeed establish relative consistency or have some uniformity. These are requirements of a translation. Yet, one can both be more strict or more flexible than interpretations are. In this article, we will define a general scheme translation. It should incorporate interpretations but also be compatible with more flexible methods. By doing so, we want to account for methods (...) that seem to imply a sense of translation but are not reducible to interpretations. The main example will be the relative consistent proof between ZF and NBG given by Novak (1950). Further, we will explore a way of combining interpretations. This should account for truth conditions discarded by interpretations in translated theories. (shrink)

Often ZF practice includes the use of the meta-theoretical notion of classes as shorthand expressions or in order to simplify the understanding of conceptual resources. NBG theory expresses formally the internalization of this feature in set theory; in this case, classes, before used metatheoretically, will also be captured by quantifiers of the first order theory. Never- theless there is a widespread opinion that this internalization of classes is harmless. In this context, it is common to refer to the conservativeness of (...) NBG in relation to ZF as a sufficient condition to understand those theories as “equivalent”, attributing a sense of virtuality to the use of classes quantified in NBG. We believe, however, that a technique used to estab- lish relationships between theories is not necessarily neutral in relation to its results - so a conservativeness established through models have different meaning and depth of that rela- tionship established by finitary interpretations. We believe, therefore, that the way in which relationships between theories are established influences the analysis result. In the case of the relationship between NBG and ZF, since NBG is finitely axiomatizible and ZF not, we believe that we have sufficient reasons to assert that the use of different analysis tools may re- veal differences such as expressiveness, ontological commitment and logical conservativeness. Therefore, this project aims to clarify the relationship between these two theories through triangulations between them and the different analysis tools. The use of finitary techniques, in this case, may prove greater expressiveness and ontological commitment of NBG in relation to ZF - relation obscured by an infinitary approach. We believe that, through this research, we can contribute to the debate on the basis of mathematics, denaturalizing the supposedly “equivalent” use of NBG and ZF for this purpose. (shrink)