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Arbitrariness and genericity: or on how to speak of the unspeakable

Abstract

With this project, I propose to investigate the notion of arbitrary object using historical, philosophical and logical methodologies. The privileged perspective from which the analysis will be conducted is that of set theory. The project is divided in three complementary axes. First, we propose a general reconstruction of the theorization about arbitrary objects, stressing the role that concepts and their extensions played in shaping axiomatic set theory. Moreover, we plan to reconstruct the origin, and to understand the meaning, of the use of generic objects in the more abstract fields of contemporary Mathematical practice. Later, from a more philosophical perspective, we plan to to confront the concept of arbitrary set with the notions of set based either on Goedel's iterative conception or on Cantor's principle of limitation of size. From a more intensional perspective on the theory of sets, we plan to study whether arbitrary objects or arbitrary reference may clarify the ontological or semantical problems of a process of abstraction à la Frege. To this aim, we propose a philosophy of Mathematical language inspired by Searle's theory of speech acts. Finally, we plan to analyze more formally the notion of arbitrariness by means of that of genericity, on which the technique of Forcing rests. We propose to axiomatize genericity in an abstract setting. Moreover we intent to apply modal logic (RI-logics) in order to capture what is invariant under Forcing (Omega-logic), in the attempt to obtain insights in the solution of the Omega-conjecture. Finally, we propose to use Forcing itself in order to study the relative or absolute character of the notion of genericity. (AU)

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Scientific publications (6)
(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)
JOCKWICH MARTINEZ, S.; VENTURI, G. Non-classical Models of ZF. STUDIA LOGICA, JUL 2020. Web of Science Citations: 0.
RUFFINO, MARCO; SAN MAURO, LUCA; VENTURI, GIORGIO. Speech acts in mathematics. SYNTHESE, JUL 2020. Web of Science Citations: 0.
RUFFINO, MARCO; SAN MAURO, LUCA; VENTURI, GIORGIO. At least one black sheep: Pragmatics and mathematical language. JOURNAL OF PRAGMATICS, v. 160, p. 114-119, APR 2020. Web of Science Citations: 0.
VENTURI, GIORGIO. Infinite Forcing and the Generic Multiverse. STUDIA LOGICA, v. 108, n. 2, p. 277-290, APR 2020. Web of Science Citations: 0.
KUBYSHKINA, EKATERINA; PETROLO, MATTIA. A logic for factive ignorance. SYNTHESE, OCT 2019. Web of Science Citations: 0.
VENTURI, GIORGIO. GENERICITY AND ARBITRARINESS. LOGIQUE ET ANALYSE, n. 248, p. 435-452, 2019. Web of Science Citations: 0.

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