The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One particular motivation for resolving MSC (...) is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. (shrink)

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ and another in which the least strongly compact cardinal is supercompact. (...) If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$$\end{document}. Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH + V = HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results. (shrink)

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the basic (...) axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann, Mostowski, and more recently Gödel, who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model. (shrink)

We extend the theory of “Fine structure and iteration trees” to models having more than one Woodin cardinal.

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, (...) 1.32, 1.33, 1.51, viz.:This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system. (shrink)

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I. The condition of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition and the completeness conditions 1.41, 1.42 in terms of Vm, we may (...) state the definition in the form:3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM’ thus:whereψ is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form and then replacing ‘ϕ bywhich is in turn replaced by, and similarly replacing ‘ϕ’ by ‘ϕ’ by ‘), andThusψ is obtained in primitive notation. (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form (...) of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω + ω, (...) … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture. (shrink)

Let M be a model of ZFAC (ZFC modified to allow a set of atoms), and let N be an inner model with the same set of atoms and the same pure sets (sets with no atoms in their transitive closure) as M. We show that N is a permutation submodel of M if and only if N satisfies the principle SVC (Small Violations of Choice), a weak form of the axiom of choice which says that in some sense, (...) all violations of choice are localized in a set. A special case is considered in which there exists an SVC witness which satisfies a certain homogeneity condition. (shrink)

In this paper it is shown that the global statement that the dominating number for k is less than $2^k $ for all regular k, is internally consistent, given the existence of $0^\# $ . The possible range of values for the dominating number for k and $2^k $ which may be simultaneously true in an inner model is also explored.

We present a characterization of supercompactness measures for ω1 in L(R), and of countable products of such measures, using inner models. We give two applications of this characterization, the first obtaining the consistency of $\delta_3^1 = \omega_2$ with $ZFC+AD^{L(R)}$ , and the second proving the uniqueness of the supercompactness measure over ${\cal P}_{\omega_1} (\lambda)$ in L(R) for $\lambda > \delta_1^2$.

We analyze the hereditarily ordinal definable sets $\operatorname {HOD} $ in $M_n[g]$ for a Turing cone of reals x, where $M_n$ is the canonical inner model with n Woodin cardinals build over x and g is generic over $M_n$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbol \Pi ^1_{n+2}$ -determinacy, for a Turing cone of reals x, $\operatorname {HOD} ^{M_n[g]} = M_n,$ where $\mathcal {M}_{\infty }$ is a direct limit of iterates of (...) $M_{n+1}$, $\delta _{\infty }$ is the least Woodin cardinal in $\mathcal {M}_{\infty }$, $\kappa _{\infty }$ is the least inaccessible cardinal in $\mathcal {M}_{\infty }$ above $\delta _{\infty }$, and $\Lambda $ is a partial iteration strategy for $\mathcal {M}_{\infty }$. It will also be shown that under the same hypothesis $\operatorname {HOD}^{M_n[g]} $ satisfies $\operatorname {GCH} $. (shrink)

An inner model operator is a function M such that given a Turing degree d, M is a countable set of reals, d M, and M has certain closure properties. The notion was introduced by Steel. In the context of AD, we study inner model operators M such that for a.e. d, there is a wellorder of M in L). This is related to the study of mice which are below the minimal inner model with ω Woodin (...) cardinals. As a technical tool, we show that the alternative fine structure theory developed by Mitchell and Steel for mice with Woodin cardinals is equivalent to the traditional fine structure theory developed by Jensen for L. (shrink)

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate this (...) by examining Easton's strengthening of Cohen's result:Theorem 1. There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals. (shrink)

Given an inner model and a regular cardinal κ, we consider two alternatives for adding a subset to κ by forcing: the Cohen poset Add(κ, 1), and the Cohen poset of the inner model. The forcing from W will be at least as strong as the forcing from V (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a (...) sufficient condition is established for the poset from V to fail to be as strong as that from W. The results are generalized to, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model. (shrink)

An elementary embedding [Formula: see text] between two inner models of [Formula: see text] is cardinal preserving if [Formula: see text] and [Formula: see text] correctly compute the class of cardinals. We look at the case [Formula: see text] and show that there is no nontrivial cardinal preserving elementary embedding from [Formula: see text] into [Formula: see text], answering a question of Caicedo.

There is a partial order ${\mathbb{P}}$ preserving stationary subsets of ω 1 and forcing that every partial order in the ground model V that collapses a sufficiently large ordinal to ω 1 over V also collapses ω 1 over ${V^{\mathbb{P}}}$ . The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using one feature of the proof of (...) the above result together with an argument involving the stationary tower it is shown that sometimes, after adding one Cohen real c, there are, for every real a in V[c], sets A and B such that c is Cohen generic over both L[A] and L[B] but a is constructible from A together with B. (shrink)

Journal of Mathematical Logic, Ahead of Print. An elementary embedding [math] between two inner models of [math] is cardinal preserving if [math] and [math] correctly compute the class of cardinals. We look at the case [math] and show that there is no nontrivial cardinal preserving elementary embedding from [math] into [math], answering a question of Caicedo.

Assume $(\exists\kappa) \lbrack\kappa \rightarrow (\kappa)^{ . Then a new inner model H exists and has the following properties: (1) H ≠ HOD; (2) Th(H) = Th(HOD); (3) there is j: H → H; (4) there is a c.u.b. class of indiscernibles for H.

We show that the predicate “x is the power set of y” is ‐definable, if V = L[E] is an extender model constructed from a coherent sequences of extenders, provided that there is no inner model with a Woodin cardinal. Here is a predicate true of just the infinite cardinals. From this we conclude: the validities of second order logic are reducible to, the set of validities of the Härtig quantifier logic. Further we show that if no L[E] model (...) has a cardinal strong up to one of its ℵ‐fixed points, and, the Löwenheim number of this logic, is less than the least weakly inaccessible δ, then (i) is a limit of measurable cardinals of K, and (ii) the Weak Covering Lemma holds at δ. (shrink)

We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". (...) Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

We discuss some of the relationships between the notion of "mutual stationarity" of Foreman and Magidor and measurability in inner models. The general thrust of these is that very general mutual stationarity properties on small cardinals, such as the ℵns, is a large cardinal property. A number of open problems, theorems, and conjectures are stated.

We introduce a natural principleStrong Chang Reflectionstrengthening the classical Chang Conjectures. This principle is between a huge and a two huge cardinal in consistency strength. In this note we prove that it implies the existence of an inner model with a huge cardinal. The technique we explore for building inner models with huge cardinals adapts to show thatdecisiveideals imply the existence of inner models with supercompact cardinals. Proofs for all of these claims can be found (...) in [10].1,2. (shrink)

In several recent essays, Sydney Shoemaker argues that introspective knowledge lacks certain central features which parallel the conditions satisfied by ordinary cases of sense perception. In one influential paper, he discusses and criticizes the “broad perceptual” model of the nature of introspective knowledge of mental states, the view which claims that our introspective awareness of internal facts is analogous to our awareness of facts about the external world. This model may be characterized by its conformance to two conditions of ordinary (...) perceptual awareness which Shoemaker dubs the causation condition and the independence condition. Shoemaker attacks the broad perceptual model by arguing that certain mental facts are “self-intimating”, with the implication being that introspective awareness of mental states does not satisfy the independence condition, and hence its character is not adequately captured by the broad perceptual model. In what follows, I will discuss the main arguments of Shoemaker’s essay. I will argue that a broad perceptual model of introspection can successfully circumvent the central problems he raises; and along the way I will develop some criticisms regarding certain aspects of Shoemaker’s positive proposal.U nekolicini recentnih tekstova, Sydney Shoemaker tvrdi da introspektivnom znanju nedostaju određene ključne centralne značajke koje odgovaraju uvjetima zadovoljenima u običnim slučajevima osjetilne percepcije. U jednom utjecajnom radu on razmatra i kritizira »široki perceptualni « model prirode introspektivnog znanja mentalnih stanja, stajalište koje tvrdi da je naša introspektivna svijest unutrašnjih činjenica analogna našoj svijesti o činjenicama izvanjskoga svijeta. Ovaj model može biti okarakteriziran njegovom sukladnošću dvama uvjetima obične perceptivne svijesti koje Shoemaker naziva uvjet kauzacije i uvjet neovisnosti. Shoemaker napada široki perceptualni model tvrdeći da su određene mentalne činjenice »samozadane«, implicirajući da introspektivna svijest o mentalnim stanjima ne zadovoljava uvjet neovisnosti, te stoga njen karakter nije adekvatno obuhvaćen širokim perceptualnim modelom. U onome što slijedi razmotrit ću glavne argumente Shoemakerovoga rada. Tvrdim da široki perceptualni model introspekcije može uspješno zaobići centralne problem koje on ističe; i usput ću razviti neke kritike vezane uz određene aspekte Shoemakerovog pozitivnog prijedloga.Dans plusieurs essais récents, Sydney Shoemaker affirme que la connaissance introspective est dépourvue de certaines caractéristiques centrales correspondant aux conditions qui, elles, sont remplies dans les cas ordinaires de perception sensorielle. Dans un article influent, il examine et critique le modèle « perceptuel large » de la nature de la connaissance introspective des états mentaux, une position qui affirme que notre conscience introspective des faits internes est analogue à notre conscience des faits du monde externe. Ce modèle peut se définir par sa conformité à deux conditions de la conscience perceptuelle ordinaire que Shoemaker appelle condition de causation et condition d’indépendance. Shoemaker attaque d’abord le modèle perceptuel large en affirmant que certains faits mentaux se « laissent entendre d’eux-mêmes », impliquant que la conscience introspective des états mentaux ne remplit pas la condition d’indépendance et que, par conséquent, son caractère n’est pas saisi de manière adéquate par le modèle perceptuel large. Dans ce qui suit, j’examinerai les principaux arguments de l’essai de Shoemaker. J’affirmerai qu’un modèle perceptuel large de l’introspection peut contourner avec réussite les problèmes centraux qu’il soulève ; ce faisant, je développerai quelques critiques à l’égard de certains aspects de la proposition positive de Shoemaker.In ein paar rezenten Essays postuliert Sydney Shoemaker, das introspektive Wissen ermangele gewisser zentraler Spezifika, die den in herkömmlichen Fällen der Sinneswahrnehmung erfüllten Umständen gleichkommen. In einem wirkungsreichen Paper behandelt und kritisiert er das „breite perzeptive“ Modell der Natur des introspektiven Wissens von Mentalzuständen, eine Betrachtungsweise, die besagt, unser introspektives Bewusstsein der internen Tatsachen sei übereinstimmend mit unserem Bewusstsein der Tatsachen über die Außenwelt. Dieses Modell lässt sich aufgrund seiner Deckungsgleichheit mit zwei Voraussetzungen des ordinären perzeptiven Bewusstseins illustrieren, die Shoemaker Bedingung der Kausalität bzw. Bedingung der Unabhängigkeit nennt. Shoemaker attackiert das breite perzeptive Modell, indem er bekräftigt, bestimmte mentale Fakten seien „selbstimplizierend“; zudem befriedige das introspektive Bewusstsein der Mentalzustände nicht die Bedingung der Unabhängigkeit, weswegen dessen Charakterzug inadäquat vom breiten perzeptiven Modell umfasst werde. Im Folgenden gehe ich die Hauptargumente von Shoemakers Essay durch. Ich stelle die Behauptung auf, das breite perzeptive Modell der Introspektion könne die von ihm aufgeworfenen Kernprobleme erfolgreich umgehen. Nebenher erarbeite ich einige Kritik in puncto gewisser Aspekte von Shoemakers positivem Vorschlag. (shrink)

Pickering & Garrod (P&G) consider the possibility that inner speech might be a product of forward production models. Here I consider the idea of inner speech as a forward model in light of empirical work from the past few decades, concluding that, while forward models could contribute to it, inner speech nonetheless requires activity from the implementers.

From climate change forecasts and pandemic maps to Lego sets and Ancestry algorithms, models encompass our world and our lives. In her thought-provoking new book, Annabel Wharton begins with a definition drawn from the quantitative sciences and the philosophy of science but holds that history and critical cultural theory are essential to a fuller understanding of modeling. Considering changes in the medical body model and the architectural model, from the Middle Ages to the twenty-first century, Wharton demonstrates the ways (...) in which all models are historical and political.0Examining how cadavers have been described, exhibited, and visually rendered, she highlights the historical dimension of the modified body and its depictions. Analyzing the varied reworkings of the Holy Sepulchre in Jerusalem-including by monumental commanderies of the Knights Templar, Alberti's Rucellai Tomb in Florence, Franciscans' olive wood replicas, and video game renderings-she foregrounds the political force of architectural representations. And considering black boxes-instruments whose inputs we control and whose outputs we interpret, but whose inner workings are beyond our comprehension-she surveys the threats posed by such opaque computational models, warning of the dangers that models pose when humans lose control of the means by which they are generated and understood. Engaging and wide-ranging, 'Models and World Making' conjures new ways of seeing and critically evaluating how we make and remake the world in which we live. (shrink)

Abstract.Since the gene splicing debates of the 1980s, the public has been exposed to an ongoing sequence of genetic and reproductive technologies. Many issue areas have outcomes that lose track of people's inner values or engender opposing religious viewpoints defying final resolution. This essay relocates the discussion of what is an acceptable application from the individual to the societal level, examining technologies that stand to address large numbers of people and thus call for policy resolution, rather than individual fiat, (...) in their application. A major source of guidance is the “Genetic Frontiers” series of professional dialogues and conferences held by the National Conference for Community and Justice from 2002 to 2004. Genetic testing, human gene therapy, genetic engineering of plants and animals, and stem cell technology are examined. While differences in perspective on the beginning of life persist, a stepwise approach to the examination of genetic testing reveals areas of general agreement. Stewardship of life, human co‐creativity with the divine, and social justice help define the bounds of application of genetic engineering and therapy; compassionate care plays a major role in establishing stem cell policy. Active, sustained dialogue is a useful resource for enabling sharing of religious values and crystallization of policies. (shrink)

We investigate iterating the construction of $C^{*}$, the L-like inner model constructed using first order logic augmented with the “cofinality $\omega $ ” quantifier. We first show that $\left (C^{*}\right )^{C^{*}}=C^{*}\ne L$ is equiconsistent with $\mathrm {ZFC}$, as well as having finite strictly decreasing sequences of iterated $C^{*}$ s. We then show that in models of the form $L[U]$ we get infinite decreasing sequences of length $\omega $, and that an inner model with a measurable cardinal is (...) required for that. (shrink)

We construct, assuming that there is no inner model with a Woodin cardinal but without any large cardinal assumption, a model Kc which is iterable for set length iterations, which is universal with respect to all weasels with which it can be compared, and is universal with respect to set sized premice.

We put in print a classical result that states that for most purposes, there is no harm in assuming the existence of saturated models in model theory. The presentation is aimed for model theorists with only basic knowledge of axiomatic set theory.

Refining the model for an emergency department‐based mental health nurse practitioner outpatient service The mental health nurse practitioner (MHNP) role based in the emergency department (ED) has emerged in response to an increase in mental health‐related presentations and subsequent concerns over waiting times, co‐ordination of care and therapeutic intervention. The MHNP role also provides scope for the delivery of specialised primary care. Nursing authors are reporting on nurse‐led outpatient clinics as a method of healthcare delivery that allows for enhanced access (...) to health‐care, particularly following hospital discharge. However, due to a lack of in‐depth substantiation, this mode of service delivery requires more thorough investigation. This study describes the refinement phase undertaken before the implementation and pilot evaluation of a formalised and structured MHNP outpatient service in the ED of a large inner‐city hospital in Sydney, Australia. An expert advisory panel (EAP) consisting of key local informants was convened to provide feedback on and refinement to the proposed model. This related to issues such as target population, structure and process considerations, outcome measures and interface within the overall health service. Findings from the EAP meeting are presented and discussed. The importance of linking methods with the appropriate methodology in evaluating a healthcare program is highlighted. (shrink)

We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable (...) sets in these models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets ( $${\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}$$ ) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets ( $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ ) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 $${\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}$$ is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which $$2{\mathfrak {m}} = {\mathfrak {m}}$$ for every infinite cardinal number $${\mathfrak {m}}$$. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false. (shrink)

Two kinds of epistemological sceptical paradox are reviewed and a shared assumption, that warrant to accept a proposition has to be the same thing as having evidence for its truth, is noted. 'Entitlement', as used here, denotes a kind of rational warrant that counterexemplifies that identification. The paper pursues the thought that there are various kinds of entitlement and explores the possibility that the sceptical paradoxes might receive a uniform solution if entitlement can be made to reach sufficiently far. Three (...) kinds of entitlement are characterized and given prima facie support, and a fourth is canvassed. Certain foreseeable limitations of the suggested antisceptical strategy are noted. The discussion is grounded, overall, in a conception of the sceptical paradoxes not as directly challenging our having any warrant for large classes of our beliefs but as crises of intellectual conscience for one who wants to claim that we do. (shrink)

We explore the possibilities for elementary embeddings $j : M \to N$, where M and N are models of ZFC with the same ordinals, $M \subseteq N$, and N has access to large pieces of j. We construct commuting systems of such maps between countable transitive models that are isomorphic to various canonical linear and partial orders, including the real line ${\mathbb R}$.

Prediction Error Minimization theory (PEM) is one of the most promising attempts to model perception in current science of mind, and it has recently been advocated by some prominent philosophers as Andy Clark and Jakob Hohwy. Briefly, PEM maintains that “the brain is an organ that on aver-age and over time continually minimizes the error between the sensory input it predicts on the basis of its model of the world and the actual sensory input” (Hohwy 2014, p. 2). An interesting (...) debate has arisen with regard to which is the more adequate epistemological interpretation of PEM. Indeed, Hohwy maintains that given that PEM supports an inferential view of perception and cognition, PEM has to be considered as conveying an internalist epistemological perspective. Contrary to this view, Clark maintains that it would be incorrect to interpret in such a way the indirectness of the link between the world and our inner model of it, and that PEM may well be combined with an externalist epistemological perspective. The aim of this paper is to assess those two opposite interpretations of PEM. Moreover, it will be suggested that Hohwy’s position may be considerably strengthened by adopting Carlo Cellucci’s view on knowledge (2013). (shrink)

This is the introductory chapter to the anthology: Inner Speech: New Voices, to be published in fall 2018 by OUP. It gives an overview of current debates in philosophy, psychology, and neuroscience concerning inner speech, and situates the chapters of the volume with respect to those debates.

We present a framework for understanding abduction within modal logic and Kripke semantics; worlds of a Kripke frame will represent possible theories, and a change in theory will be understood as a passage from one world to an adjacent possible world. Further, these steps may agree with the accessibility relation or may ‘backtrack’, accordingly as new information refutes or reinforces our present theory. Our formalism can be used to model not only abduction, but also to talk about the inner (...) structure of theories as well as relations between them, allowing us to interpret many ideas from philosophy of science within the well-understood framework of modal logic. (shrink)

What is it to interpret the outputs of an opaque machine learning model? One approach is to develop interpretable machine learning techniques. These techniques aim to show how machine learning models function by providing either model-centric local or global explanations, which can be based on mechanistic interpretations (revealing the inner working mechanisms of models) or non-mechanistic approximations (showing input feature–output data relationships). In this paper, we draw on social philosophy to argue that interpreting machine learning outputs in (...) certain normatively salient domains could require appealing to a third type of explanation that we call “socio-structural” explanation. The relevance of this explanation type is motivated by the fact that machine learning models are not isolated entities but are embedded within and shaped by social structures. Socio-structural explanations aim to illustrate how social structures contribute to and partially explain the outputs of machine learning models. We demonstrate the importance of socio-structural explanations by examining a racially biased healthcare allocation algorithm. Our proposal highlights the need for transparency beyond model interpretability: understanding the outputs of machine learning systems could require a broader analysis that extends beyond the understanding of the machine learning model itself. (shrink)