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13Hilbert, completeness and geometryRivista Italiana di Filosofia Analitica Junior 2 (2): 80-102. 2011.This paper aims to show how the mathematical content of Hilbert's Axiom of Completeness consists in an attempt to solve the more general problem of the relationship between intuition and formalization. Hilbert found the accordance between these two sides of mathematical knowledge at a logical level, clarifying the necessary and sufficient conditions for a good formalization of geometry. We will tackle the problem of what is, for Hilbert, the definition of geometry. The solution of this problem w…Read more
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25Foundation of Mathematics between Theory and PracticePhilosophia Scientiae 18 (1): 45-80. 2014.In this article I propose to look at set theory not only as a foundation of mathematics in a traditional sense, but as a foundation for mathematical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some…Read more
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29A direct proof of the five element basis theoremMathematical Logic Quarterly 63 (3-4): 289-298. 2017.We present a direct proof of the consistency of the existence of a five element basis for the uncountable linear orders. Our argument is based on the approach of König, Larson, Moore and Veličković and simplifies the original proof of Moore.
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7A note on the introduction of Hilbert’s Grundlagen der GeometrieManuscrito 40 (2): 5-17. 2017.ABSTRACT We present and discuss a change in the introduction of Hilbert’s Grundlagen der Geometrie between the first and the subsequent editions: the disappearance of the reference to the independence of the axioms. We briefly outline the theoretical relevance of the notion of independence in Hilbert’s work and we suggest that a possible reason for this disappearance is the discovery that Hilbert’s axioms were not, in fact, independent. In the end we show how this change gives textual evidence f…Read more
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23Preservation of suslin trees and side conditionsJournal of Symbolic Logic 81 (2): 483-492. 2016.
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27Hilbert between the formal and the informal side of mathematicsManuscrito 38 (2): 5-38. 2015.: In this article we analyze the key concept of Hilbert's axiomatic method, namely that of axiom. We will find two different concepts: the first one from the period of Hilbert's foundation of geometry and the second one at the time of the development of his proof theory. Both conceptions are linked to two different notions of intuition and show how Hilbert's ideas are far from a purely formalist conception of mathematics. The principal thesis of this article is that one of the main problems that…Read more
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24Foundation of Mathematics between Theory and PracticePhilosophia Scientiae 18 45-80. 2014.In this article I propose to look at set theory not only as a foundation of mathematics in a traditional sense, but as a foundation for mathematical practice. For this purpose I distinguish between a standard, ontological, set theoretical foundation that aims to find a set theoretical surrogate to every mathematical object, and a practical one that tries to explain mathematical phenomena, giving necessary and sufficient conditions for the proof of mathematical propositions. I will present some…Read more