Compared study of ontological commitment of classes and sets theories

• Grant number: 16/10497-8
• Support Opportunities: Scholarships in Brazil - Doctorate (Direct)
• Start date: September 01, 2016
• End date: March 31, 2019
• Field of knowledge: Humanities - Philosophy - Logic
• Principal Investigator: Walter Alexandre Carnielli
• Grantee: Alfredo Roque de Oliveira Freire Filho
• Host Institution: Centro de Lógica, Epistemologia e História da Ciência (CLE). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated scholarship(s): 17/21020-0 - On the conditions for a comparison of ontological commitment between theories, BE.EP.DD

Abstract

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. Nevertheless 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 establish 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 relationship 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 reveal 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. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

FREIRE, ALFREDO ROQUE. The Significance of the New Logic. AXIOMS, v. 8, n. 2, p. 3-pg., 2019-06-01. (16/10497-8)