We show that the Proper Forcing Axiom implies the Singular Cardinal Hypothesis. The proof uses the reflection principle MRP introduced by Moore in [11].

We present several results relating the general theory of the stationary tower forcing developed by Woodin with forcing axioms. In particular we show that, in combination with class many Woodin cardinals, the forcing axiom MM++ makes the Π2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_2}$$\end{document}-fragment of the theory of Hℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\aleph_2}}$$\end{document} invariant with respect to stationary set preserving forcings that preserve BMM. We argue that this is a promising generalization to (...) Hℵ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H_{\aleph_2}}$$\end{document} of Woodin’s absoluteness results for L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L}$$\end{document}. In due course of proving this, we shall give a new proof of some of these results of Woodin. Finally we relate our generic absoluteness results with the resurrection axioms introduced by Hamkins and Johnstone and with their unbounded versions introduced by Tsaprounis. (shrink)

We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinal axioms, ranging from supercompactness to rank-to-rank embeddings. The majority of these large cardinal properties can be defined in terms of suitable elementary embeddings j:Vγ→Vλ. One key observation is that such embeddings are uniquely determined by the image structures j[Vγ]≺Vλ. These structures will be the prototypes guessing models. We shall show, (...) using guessing models M, how to prove for the ordinal κM=jM) many of the combinatorial properties that we can prove for the cardinal j) using the structure j[Vγ]≺Vj. κM will always be a regular cardinal, but consistently can be a successor and guessing models M with κM=ℵ2 exist assuming the proper forcing axiom. By means of these models we shall introduce a new structural property of models of PFA: the existence of a “Laver function” f:ℵ2→Hℵ2 sharing the same features of the usual Laver functions f:κ→Hκ provided by a supercompact cardinal κ. Further applications of our analysis will be proofs of the singular cardinal hypothesis and of the failure of the square principle assuming the existence of guessing models. In particular the failure of square shows that the existence of guessing models is a very strong assumption in terms of large cardinal strength. (shrink)

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtained in suitable large cardinals properties.The first example I will treat is the proof that the proper forcing axiom PFA (...) implies the singular cardinal hypothesis SCH, this will easily lead to a new proof of Solovay's theorem that SCH holds above a strongly compact cardinal. I will also outline how some of the ideas involved in these proofs can be used as means to evaluate the “saturation” properties of models of strong forcing axioms like MM or PFA.The second example aims to show that the transfer principle ↠ fails assuming Martin's Maximum MM. Also in this case the result can be translated in a large cardinal property, however this requires a familiarity with a rather large fragment of Shelah's pcf-theory.Only sketchy arguments will be given, the reader is referred to the forthcoming [25] and [38] for a thorough analysis of these problems and for detailed proofs. (shrink)

The resurrection axioms are forcing axioms introduced recently by Hamkins and Johnstone, developing on ideas of Chalons and Veličković. We introduce a stronger form of resurrection axioms for a class of forcings Γ and a given ordinal α), and show that RAω implies generic absoluteness for the first-order theory of Hγ+ with respect to forcings in Γ preserving the axiom, where γ = γΓ is a cardinal which depends on Γ. We also prove that the consistency strength of these axioms (...) is below that of a Mahlo cardinal for most forcing classes, and below that of a stationary limit of supercompact cardinals for the class of stationary set preserving posets. Moreover, we outline that simultaneous generic absoluteness for Hγ0+ with respect to Γ0 and for Hγ1+ with respect to Γ1 with γ0 = γΓ0≠γΓ1 = γ1 is in principle possible, and we present several natural models o... (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher bounded category forcing (...) axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [math]. We also show the existence of many classes [math] with [math] giving rise to pairwise incompatible theories for [math]. (shrink)

We present two results which shed some more light on the deep connection between ZFA and the standard ZF set theory: First of all we refine a result of Forti and Honsell in order to prove that the universe of ZFA can also be obtained as the least fixed point of a continuous operator and not only as the greatest fixed point of the powerset operator. Next we show that it is possible to define a new absolute Gödel operation in (...) addition to the standard ones in order to obtain the “constructible” model of ZFA as the least fixed point of the continuous operator of Gödel closure with respect to the standard and the new Gödel operations. (shrink)

We present recent results on the model companions of set theory, placing them in the context of a current debate in the philosophy of mathematics. We start by describing the dependence of the notion of model companionship on the signature, and then we analyze this dependence in the specific case of set theory. We argue that the most natural model companions of set theory describe (as the signature in which we axiomatize set theory varies) theories of $H_{\kappa ^+}$, as $\kappa (...) $ ranges among the infinite cardinals. We also single out $2^{\aleph _0}=\aleph _2$ as the unique solution of the continuum problem which can (and does) belong to some model companion of set theory (enriched with large cardinal axioms). While doing so we bring to light that set theory enriched by large cardinal axioms in the range of supercompactness has as its model companion (with respect to its first order axiomatization in certain natural signatures) the theory of $H_{\aleph _2}$ as given by a strong form of Woodin’s axiom $(*)$ (which holds assuming $\mathsf {MM}^{++}$ ). Finally this model-theoretic approach to set-theoretic validities is explained and justified in terms of a form of maximality inspired by Hilbert’s axiom of completeness. (shrink)

This is an introductory paper to a series of results linking generic absoluteness results for second and third order number theory to the model theoretic notion of model companionship. Specifically we develop here a general framework linking Woodin’s generic absoluteness results for second order number theory and the theory of universally Baire sets to model companionship and show that (with the required care in details) a $$\Pi _2$$ -property formalized in an appropriate language for second order number theory is forcible (...) from some $$T\supseteq \mathsf {ZFC}+$$ large cardinals if and only if it is consistent with the universal fragment of T if and only if it is realized in the model companion of T. In particular we show that the first order theory of $$H_{\omega _1}$$ 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 $$\Delta _0$$ -properties and for all universally Baire sets of reals. We will extend these results also to the theory of $$H_{\aleph _2}$$ in a follow up of this paper. (shrink)

The notion of subject matter is a key concern of contemporary philosophy of language and logic. A central task for a theory of subject matter is to characterise the notion of sentential subject matter, that is, to assign to each sentence of a given language a subject matter that may count as its subject matter. In this paper, we elaborate upon David Lewis’ account of subject matter. Lewis’ proposal is simple and elegant but lacks a satisfactory characterisation of sentential subject (...) matter. Drawing on linguistic literature on focus and on the question under discussion, we offer a neo-Lewisian account of subject matter, which retains all the virtues of Lewis’ but also includes an attractive characterisation of sentential subject matter. (shrink)

This paper defends a relational view of time based on recent work on quantum gravity. Julian barbour's relational approach to physical theory, in particular, is developed as a basis for a relational, rather than anti-realist, metaphysics of time.

This open access book assesses the prospects of (re)adopting organization as a pivotal concept in biology. It shows how organization can nourish biological thinking and practice, by reconnecting with the idea of biology as the science of organized systems. The book provides a comprehensive state-of-the-art picture of the characterizations and uses of the concept of organization in both biological science and philosophy of biology. It also deals with a variety of themes – including evolution, organogenesis, heredity, cognition and ecology – (...) with respect to which the concept of organization can guide the elaboration of original models and new experimental protocols. It will be of interest to biologists and scholars working in philosophy of science alike. (shrink)

The works of the Anglo-Indian artist Anish Kapoor challenge the intangibility of the real and the reality of the objects, the surrounding space, even the spectator himself. They make it appear as an illusion and point to an invisible reality located beyond or beneath, or even at the very heart of the visible. This essay explores the nature of this hidden realm, which the works allow us to see or at least to foresee. It interrogates also the phenomenological mechanisms at (...) play in the works, which induce the spectator into putting the real in doubt, to confront him or herself to the invisible and to name it. (shrink)

Technology as a system -- The digital technological system -- The technological structures of perception -- The life and death of the virtual -- Digital ontophany -- The (digital) design of experience.

The present paper discusses different approaches to metaphysics and defends a specific, non-deflationary approach that nevertheless qualifies as scientifically-grounded and, consequently, as acceptable from the naturalistic viewpoint. By critically assessing some recent work on science and metaphysics, we argue that such a sophisticated form of naturalism, which preserves the autonomy of metaphysics as an a priori enterprise yet pays due attention to the indications coming from our best science, is not only workable but recommended.

Biological and Cultural Bases of Human Inference addresses the interface between social science and cognitive science. In this volume, Viale and colleagues explore which human social cognitive powers evolve naturally and which are influenced by culture. Updating the debate between innatism and culturalism regarding human cognitive abilities, this book represents a much-needed articulation of these diverse bases of cognition. Chapters throughout the book provide social science and philosophical reflections, in addition to the perspective of evolutionary theory and the central (...) assumptions of cognitive science. The overall approach of the text is based on three complementary levels: adult performance, cognitive development, and cultural history and prehistory. Scholars from several disciplines contribute to this volume, including researchers in cognitive, developmental, social and evolutionary psychology, neuropsychology, cognitive anthropology, epistemology, and philosophy of mind. This contemporary, important collection appeals to researchers in the fields of cognitive, social, developmental, and evolutionary psychology and will prove valuable to researchers in the decision sciences. (shrink)

Cet ouvrage est une réédition numérique d’un livre paru au XXe siècle, désormais indisponible dans son format d’origine.

We put forward a new, ‘coherentist’ account of quantum entanglement, according to which entangled systems are characterized by symmetric relations of ontological dependence among the component particles. We compare this coherentist viewpoint with the two most popular alternatives currently on offer—structuralism and holism—and argue that it is essentially different from, and preferable to, both. In the course of this article, we point out how coherentism might be extended beyond the case of entanglement and further articulated.

We study the EPR-type correlations from the perspective of the relational interpretation of quantum mechanics. We argue that these correlations do not entail any form of “non-locality”, when viewed in the context of this interpretation. The abandonment of strict Einstein realism implied by the relational stance permits to reconcile quantum mechanics, completeness, (operationally defined) separability, and locality.

This paper investigates the reception and discussion of Jakob von Uexküll’s biological theory by two German thinkers of his time, Helmuth Plessner and Kurt Goldstein. It demonstrates how their bio-philosophical perspectives are on the one hand indebted to Uexküll’s theory and, on the other, critical of its tendency to excessively harmonize the relationship between living beings and their environment. This original critical reading of the _Umweltlehre_ is rooted in ambiguities within Uexküll’s own thought - between a dynamic conception of the (...) organism-environment relationship and the idea of "conformity to a plan" -,, which is here examined in the second section. In the third and fourth sections we will then focus on Plessner and Goldstein respectively, demonstrating how for these two authors the harmony between organism and environment is not an original state, but only reveals itself against the background of a tension; as such, it can only be partial, unstable and always changing. The two thinkers avoid the rigid alternative between Darwin’s concept of adaptation (_Anpassung_) and Uexküll’s “fitting into” (_Einpassung_) by theorizing the ideal state of the relationship between organism and environment in terms of “adequacy” (_Adäquatheit_) and “adaptability” (_Adaptiertheit_). Between organism and environment there is neither absolute separation nor perfect harmony, but rather a gap which can never be definitively fixed. (shrink)

As innovations in the field of vascular composite allotransplantation (VCA) progress, whole-eye transplantation (WET) is poised to transition from non-human mammalian models to living human recipients. Present treatment options for vision loss are generally considered suboptimal, and attendant concerns ranging from aesthetics and prosthesis maintenance to social stigma may be mitigated by WET. Potential benefits to WET recipients may also include partial vision restoration, psychosocial benefits related to identity and social integration, improvements in physical comfort and function, and reduced surgical (...) risk associated with a biologic eye compared to a prosthesis. Perioperative and postoperative risks of WET are expected to be comparable to those of facial transplantation (FT), and may be similarly mitigated by immunosuppressive protocols, adequate psychosocial support, and a thorough selection process for both the recipient and donor. To minimize the risks associated with immunosuppressive medications, the first attempts in human recipients will likely be performed in conjunction with a FT. If first-in-human attempts at combined FT-WET prove successful and the biologic eye survives, this opens the door for further advancement in the field of vision restoration by means of a viable surgical option. This analysis integrates recent innovations in WET research with the existing discourse on the ethics of surgical innovation and offers preliminary guidance to VCA programs considering undertaking WET in human recipients. (shrink)

This is the first part of a two-tier overview article on fundamentality in metaphysics and the philosophy of physics. It provides an introduction to the notion of fundamentality in metaphysics, as well as to several related concepts. The key issues in the contemporary debate on the topic are summarised, making systematic reference to the most relevant literature. In particular, various ways in which the fundamental entities and the fundamental structure of reality may be conceived are illustrated and discussed. A final (...) brief section looks at the methodological issue of naturalism, paving the way for the survey of fundamentality in the philosophy of physics, which is carried out in the second part. (shrink)

Boffa non-well-founded set theory allows for several distinct sets equal to their respective singletons, the so-called ‘Quine atoms’. Rieger contends that this theory cannot be a faithful description of set-theoretic reality. He argues that, even after granting that there are non-well-founded sets, ‘the extensional nature of sets’ precludes numerically distinct Quine atoms. In this paper we uncover important similarities between Rieger’s argument and how non-rigid structures are conceived within mathematical structuralism. This opens the way for an objection against Rieger, whilst (...) affording the theoretical resources for a defence of Boffa set theory as a faithful description of set-theoretic reality. (shrink)

The mind-society problem deals with the relations between mental and social phenomena. The problem is crucial in the main methodologies of social sciences. The thesis of hermeneutics is that we can only understand but not explain the relationship between beliefs and social action because mental and social events are not natural events. The thesis of social holism is that social phenomena are emergent and irreducible to mental phenomena. The thesis of rational choice theory is that social phenomena are reducible to (...) mental phenomena and this reduction is explained by unrealistic a priori principles of rationality. These theses depend on their different solutions to the following fundamental philosophical issues: the mind-body identity; the causal nature of social explanation; the realistic goal of science. A positive answer to these issues implies support of a different solution to the mind-society problem: social phenomena are conventional concepts and they can be reduced and explained in terms of realistic concepts describing the causal mechanisms of individual reasoning and decision-making. Cognitive psychology seems to supply us some initial tentative models to explain social action, but research into neuropsychology might be able to generate the proper causal representations of the relation between mind and action. (shrink)