Publications

Journal Articles:

1. "What are Extremal Axioms?". Philosophia Mathematica, 2025. Published version 

Extremal axioms impose a condition of either minimality or maximality on the admissible models of an axiomatic theory. In this paper, I propose an alternative formulation of arithmetic and real analysis based on extremal axioms. Once properly formulated, the second-order extremal axiom restricts the quantifiers of the theory to the minimal or maximal domain of discourse. It is proved that extremal axioms are logically equivalent to standard assumptions of, respectively, second-order Induction and Archimedean Completeness. Finally, I distinguish between internalist and externalist accounts of mathematical structures as characterized by extremal axioms and their corresponding axiomatic theories.

2. "A Reassessment of Cantorian Abstraction based on the ε-operator". Synthese, 2022. Published version (Open Access)

Cantor's abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by arguing for the coherence and plausibility of Cantor's proposal. The defence of Cantorian abstraction will be built upon the set theoretic framework of Bourbaki - called BK - which is formulated in First-order Logic extended with Hilbert's ε-operator. I will introduce first the axiomatic setting of BK and the definition of cardinal numbers by means of the ε-operator. Then I will present Cantor's abstractionist theory, pointing out two leading assumptions in Cantor's work which concern the definition of ordinal and cardinal numbers. I will argue that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo-von Neumann and Frege-Russell.  On the basis of these similarities, I will make use the BK framework in meeting Frege's objections to Cantor's proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the ε-operator in the BK definition of cardinal numbers.

Submitted:

1. "Maximality Axioms and the Principle of Plenitude".