20 research outputs found
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-\`a-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
Gödelâs Cantorianism
Gödelâs philosophical conceptions bear striking similarities to Cantorâs. Although there is no
conclusive evidence that Gödel deliberately used or adhered to Cantorâs views, one can successfully
reconstruct and see his âCantorianismâ at work in many parts of his thought. In this paper, I aim to
describe the most prominent conceptual intersections between Cantorâs and Gödelâs thought, particularly on such matters as the nature and existence of mathematical entities (sets), concepts, Platonism, the Absolute Infinite, the progress and inexhaustibility of mathematics
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
Fallacious Analogical Reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI)
In this article, we address fallacious analogical reasoning and the Metaphoric Fallacy to a Deductive Inference (MFDI), recently discussed by B. Lightbody and M. Berman (2010). We claim that the authorsâ proposal to introduce a new fallacy is only partly justified. We also argue that, in some relevant cases, fallacious analogical reasoning involving metaphors is only affected by the use of quaternio terminorum
Ipotesi del Continuo
LâIpotesi del Continuo, formulata da Cantor nel 1878, Ăš una delle congetture piĂč note della teoria degli insiemi. Il Problema del Continuo, che ad essa Ăš collegato, fu collocato da Hilbert, nel 1900, fra i principali problemi insoluti della matematica. A seguito della dimostrazione di indipendenza dellâIpotesi del Continuo dagli assiomi della teoria degli insiemi, lo status attuale del problema Ăš controverso. In anni piĂč recenti, la ricerca di una soluzione del Problema del Continuo Ăš stata anche una delle ragioni fondamentali per la ricerca di nuovi assiomi in matematica. Lâarticolo fornisce un quadro generale dei risultati matematici fondamentali, e unâanalisi di alcune delle questioni filosofiche connesse al Problema del Continuo
On Forms of Justification in Set Theory
In the contemporary philosophy of set theory, discussion of new axiomsthat purport to resolve independence necessitates an explanation of howthey come to bejustified. Ordinarily, justification is divided into two broadkinds:intrinsicjustification relates to how âintuitively plausibleâ an axiomis, whereasextrinsicjustification supports an axiom by identifying certainâdesirableâ consequences. This paper puts pressure on how this distinctionis formulated and construed. In particular, we argue that the distinction asoften presented is neitherwell-demarcatednor sufficientlyprecise. Instead, wesuggest that the process of justification in set theory should not be thoughtof as neatly divisible in this way, but should rather be understood as a con-ceptually indivisible notion linked to the goal ofexplanation
Steel's Programme: Evidential Framework, the Core and Ultimate-_L_
We address Steelâs Programme to identify a âpreferredâ universe of set theory and the best axioms extending ZFC by using his multiverse axioms MV and the âcore hypothesisâ. In the first part, we examine the evidential framework for MV, in particular the use of large cardinals and of âworldsâ obtained through forcing to ârepresentâ 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 MVâs evidential framework
Maximality Principles in the Hyperuniverse Programme
In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows how optimal principles,
among those available, may be selected in a justifiable way
On Forms of Justification in 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