OUTPUTS

In this article we review Cantor’s contribution to the construction of a theory of infinity. We will present and discuss the technical achievements and the philosophical ideas that brought Cantor to the creation of set theory and to its justification.

In this article we compare the notions of genericity and arbitrariness on the basis of the realist import of the method of forcing. We argue that Cohen’s Theorem, similarly to Cantor’s Theorem, can be considered a meta-theoretical argument in favor of the existence of uncountable collections. Then we discuss the effects of this metatheoretical perspective on Skolem’s Paradox. We conclude discussing how the connection between arbitrariness and genericity can be used to argue in favor of Forcing Axioms.

To the best of our knowledge, there are very few results on how Heyting-valued models are affected by the morphisms on the complete Heyting algebras that determine them: the only cases found in the literature are concerning automorphisms of complete Boolean algebras and complete embedding between them (i.e., injective Boolean algebra homomorphisms that preserves arbitrary suprema and arbitrary infima). In the present work, we consider and explore how more general kinds of morphisms between complete Heyting algebras ℍ and ℍ′ induce arrows between V^(ℍ) and V^(ℍ′), and between their corresponding localic toposes Set^(ℍ) (≃Sh^(ℍ)) and Set^(ℍ′) (≃Sh^(ℍ′)). In more details: any geometric morphism f^∗:Set^(ℍ)→Set^(ℍ′), (that automatically came from a unique locale morphism f:ℍ→ℍ′), can be "lifted" to an arrow f̃ :V^(ℍ)→V^(ℍ′). We also provide also some semantic preservation results concerning this arrow f̃ :V^(ℍ)→V^(ℍ′).

In this article we present a technique for selecting models of set theory that are complete in a model-theoretic sense. Specifically, we will apply Robinson infinite forcing to the collections of models of ZFC obtained by Cohen forcing. This technique will be used to suggest a unified perspective on generic absoluteness principles.

In this paper we review the approach to algebra-valued models and we introduce an infinite class of natural models for paraconsistent set theory. We then present several paraconsistent set theories based on Logics of Formal Inconsistency (LFI), known in the literature as LFI-set theories, and we show that there is no natural model that validates these systems. We therefore suggest to abandon these theories.

In this work, we will be interested on the investigation of categorial forms of the Axiom of Choice (AC). The results are intended to befurther results with respect to [3]. We introduce some more new categorialforms of AC, and we discuss a number of categorial versions of the Zorn’s Lemma.

In this article we review the present situation in the foundations of set theory, discussing two programs meant to overcome the phenomenon of independence centered, respectively, on forcing axioms and Woodin’s V = Ultimate-L conjecture. While doing so, we briefly introduce the key notions of set theory.

In the contemporary philosophy of set theory, discussion of new axioms that purport to resolve independence necessitates an explanation of how they come to be justified. Ordinarily, justification is divided into two broad kinds: intrinsic justification relates to how ‘intuitively plausible’ an axiom is, whereas extrinsic justification supports an axiom by identifying certain ‘desirable’ consequences. This paper puts pressure on how this distinction is formulated and construed. In particular, we argue that the distinction as often presented is neither well-demarcated nor sufficiently precise. Instead, we suggest that the process of justification in set theory should not be thought of as neatly divisible in this way, but should rather be understood as a conceptually indivisible notion linked to the goal of explanation.

In this article we analyze the notion of natural axiom in set theory. To this aim we review the intrinsic-extrinsic dichotomy, finding both theoretical and practical difficulties in its use. We will describe and discuss a theoretical framework, that we will call conceptual realism, where the standard justification strategy is usually placed. In outlining our view, we suggest that the extensive use of naturalness calls for a revision of the standard strategy, in favor of a justification that takes into account also the historical process that lead to the formalization of set theory. Specifically we will argue that an axiom can be considered natural when it helps the clarification of the notion of arbitrary set.

We first show that the first order theory of H_ω1 is the model companion of the first order theory of the universe of sets assuming the existence of class many Woodin cardinals, and working in a signature with predicates for all universally Baire sets of reals. We conclude our analysis with some basic conditions granting the model completeness of the first order theory of H_ω2 and of the axiom system ZF + V = L in an appropriate language.