arXiv subjects

Claudio Ternullo

Publications and source records attributed to Claudio Ternullo.

4 recordsLinked to original sources

arXiv2023

Intrinsic Justification for Large Cardinals and Structural Reflection

We deal with the complex issue of whether large cardinals are intrinsically justified principles of set theory (we call this the Intrinsicness Issue). In order to do this, we review, in a systematic fashion, (1.) the abstract principles that have been formulated to motivate them, as well as (2.) their mathematical expressions, and assess the justifiability of both on the grounds of the (iterative) concept of set. A parallel, but closely linked, issue is whether there exist mathematical principles able to yield all known large cardinals (we call this the Universality Issue), and we also test principles for their responses to this issue. Finally, we discuss the first author's Structural Reflection Principles (SRPs), and their response to Intrinsicness and Universality. We conclude the paper with some considerations on the global justifiability of SRPs, and on alternative construals of the concept of set also potentially able to intrinsically justify large cardinals.

math.LO↗

arXiv2023

Higher-Order Platonism and Multiversism

Joel Hamkins has described his multiverse position as being one of `higher-order realism -- Platonism about universes', whereby one takes models of set theory to be actually existing objects (vis-à-vis `first-order realism', which takes only sets to be actually existing objects). My goal in this paper is to make sense of the view in the very context of Hamkins' own multiversism. To this end, I will explain what may be considered the central features of higher-order platonism, and then will focus on Zalta and Linsky's Object Theory, which, I will argue, is able to faithfully express Hamkins' conception. I will then show how the embedding of higher-order platonism into Object Theory may help the Hamkinsian multiversist to respond to salient criticisms of the multiverse conception, especially those relating to its articulation, skeptical attitude, and relationship with set-theoretic practice.

math.LO↗

arXiv2022

Peano's Conception of a Single Infinite Cardinality

While Peano's negative attitude towards infinitesimals, in particular, geometric infinitesimals, is widely documented, his conception of a single infinite cardinality and, more generally, his views on the infinite, are a lot less known. The paper reconstructs the evolution of Peano's ideas on these questions, and formulates several hypotheses about their underlying motivations.

math.HO↗

arXiv2021

Steel's Programme: Evidential Framework, the Core and Ultimate-L

We address Steel's Programme to identify a 'preferred' universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the 'core hypothesis'. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of 'worlds' obtained through forcing to 'represent' alternative extensions of ZFC. In the second part, we address the existence and the possible features of the core of MV_T (where T is ZFC+Large Cardinals). In the last part, we discuss the hypothesis that the core is Ultimate-L, and examine whether and how, based on this fact, the Core Universist can justify V=Ultimate-L as the best (and ultimate) extension of ZFC. To this end, we take into account several strategies, and assess their prospects in the light of MV's evidential framework.

math.LO↗