The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove (...) from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ. (shrink)

We extend the construction of Mitchell and Steel to produce iterable fine structure models which may contain Woodin limits of Woodin cardinals, and more. The precise level reached is that of a cardinal which is both a Woodin cardinal and a limit of cardinals strong past it.

We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than $\aleph_{1}$, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important (...) since a proof using finite supports is more amenable to generalizations to cardinals greater than $\aleph_{1}$. (shrink)

Halley et al. (2023) highlight important equity challenges emerging out of existing health policy’s reliance on self- and family advocacy. As advocacy capacity varies dramatically across groups, pu...

Jullien's indecomposability theorem states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis. We identify the strength of the theorem relative to standard reverse mathematics markers. We show that it lies strictly between weak [Formula: see text] choice and [Formula: see text] comprehension.

A $\Sigma _{1}^{2}$ truth for λ is a pair 〈Q, ψ〉 so that Q ⊆ Hλ, ψ is a first order formula with one free variable, and there exists B ⊆ Hλ+ such that (Hλ+; ε, B) $(H_{\lambda +};\in ,B)\vDash \psi [Q]$ . A cardinal λ is $\Sigma _{1}^{2}$ indescribable just in case that for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $\overline{\lambda}<\lambda $ so that $\overline{\lambda}$ is a cardinal and $\langle Q\cap H_{\overline{\lambda}},\psi \rangle $ is a (...) $\Sigma _{1}^{2}$ truth for $\overline{\lambda}$ . More generally, an interval of cardinals [κ, λ] with κ ≤ λ is $\Sigma _{1}^{2}$ indescribable if for every $\Sigma _{1}^{2}$ truth 〈Q, ψ〉 for λ, there exists $??\leq \overline{\lambda}<\kappa,??\subseteq H_{\overline{\lambda}}$ , and π: $H_{\overline{\lambda}}\rightarrow H_{\lambda}$ so that $??$ is a cardinal, $\langle ??,\psi \rangle $ is a $\Sigma _{1}^{2}$ truth for $??$ , and π is elementary from $(H_{\overline{\lambda}};\in,??,??)$ with $\pi \,|\,??={\rm id}$ . We prove that the restriction of the proper forcing axiom to c-linked posets requires a $\Sigma _{1}^{2}$ indescribable cardinal in L, and that the restriction of the proper forcing axiom to c⁺-linked posets, in a proper forcing extension of a fine structural model, requires a $\Sigma _{1}^{2}$ indescribable 1-gap [κ, κ⁺]. These results show that the respective forward directions obtained in " Hierarchies of Forcing Axioms I" by Neeman and Schimmerling are optimal. (shrink)

Microdevelopment is the process of change in abilities, knowledge and understanding during short time-spans. This book presents a new process-orientated view of development and learning based on recent innovations in psychology research. Instead of characterising abilities at different ages, researchers investigate processes of development and learning that evolve through time and explain what enables progress in them. Four themes are highlighted: variability, mechanisms that create transitions to higher levels of knowledge, interrelations between changes in the short-term scale of microdevelopment and (...) the crucial effect of context. Learning and development are analysed in and out of school, in the individual's activities and through social interaction, in relation to simple and complex problems and in everyday behaviour and novel tasks. With contributions from the foremost researchers in the field Microdevelopment will be essential reading for all interested in cognitive and developmental science. (shrink)

We present equiconsistency results at the level of subcompact cardinals. Assuming SBHδ, a special case of the Strategic Branches Hypothesis, we prove that if δ is a Woodin cardinal and both □ and □δ fail, then δ is subcompact in a class inner model. If in addition □ fails, we prove that δ is Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact in a class inner model. These results are optimal, and lead to equiconsistencies. As a corollary (...) we also see that assuming the existence of a Woodin cardinal δ so that SBHδ holds, the Proper Forcing Axiom implies the existence of a class inner model with a Π12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^2}$$\end{document} subcompact cardinal. Our methods generalize to higher levels of the large cardinal hierarchy, that involve long extenders, and large cardinal axioms up to δ is δ+ supercompact for all n < ω. We state some results at this level, and indicate how they are proved. (shrink)

Three studies examined the effect of primed psychological distance on level of perceptual construal, using Navon’s paradigm of composite letters (global letters that are made of local letters). Relative to a control group, thinking of the more distant future (Study 1), about more distant spatial locations (Study 2), and about more distant social relations (Study 3) facilitated perception of global letters relative to local letters. Proximal times, spatial locations, and social relations had the opposite effect. The results are discussed within (...) the framework of Construal Level Theory of psychological distance (Liberman & Trope, 2008; Trope & Liberman, 2003). (shrink)

Listening effort has been known to characterize speech recognition in noise regardless of hearing sensitivity and age. Whereas the behavioral measure of dual-task paradigm effectively manifests the cognitive cost that listeners exert when processing speech in background noise, there is no consensus as to a clinical procedure that might best express LE. In order to assess the cognitive load underlying speech recognition in noise and promote counselling for coping strategies, a feasible clinical paradigm is warranted. The ecological validity of such (...) a paradigm might best be demonstrated in middle-aged adults, exhibiting intact hearing sensitivity on one hand, however, experiencing difficulties in degraded listening conditions, unaware of the implicated cognitive cost of speech recognition in noise. To this end, we constructed a dual-task paradigm that consists of a primary task of sentences-in-noise recognition and a secondary task of simple visual colored-shape matching. Research objective was to develop a clinical paradigm for the assessment of LE in middle-aged adults. Participants were 17 middle-aged adults and 23 young adults. All participants had normal hearing according to age. Speech stimuli consisted of the Hebrew Matrix sentences in noise test. SRTn was obtained for 80% correct identification. Visual stimuli were colored geometric shapes. Outcome measures were obtained initially for each task separately, to establish performance ability, and then obtained simultaneously. Reaction time and accuracy in the secondary task were the defined metrics for LE. Results: LE was indicated for both groups, however, was more pronounced in the middle-aged, manifested in the visual accuracy and reaction time metrics. Both groups maintained the 80% correct recognition-in-noise in the dual-task, however, the middle-aged group necessitated a better SNR of 1.4dB than the normal hearing group. Moreover, the middle-aged group was taxed in a greater prolongation of reaction time, in order to uphold the correct recognition. Conclusion: a dual-task paradigm consisting of sentences-in-noise primary task combined with a simple secondary task successfully showed different manifestations of LE in middle-aged adults compared to young adults, thus approximating the use of such a paradigm in a clinical setting. (shrink)

Answering a question of Sakai :29–45, 2013), we show that the existence of an ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _1$$\end{document}-Erdős cardinal suffices to obtain the consistency of Chang’s Conjecture with □ω1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\omega _1, 2}$$\end{document}. By a result of Donder, volume 872 of lecture notes in mathematics. Springer, Berlin, pp 55–97, 1981) this is best possible. We also give an answer to another question of Sakai relating to (...) the incompatibility of □λ,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\square _{\lambda, 2}$$\end{document} and ↠\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \twoheadrightarrow $$\end{document} for uncountable κ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa $$\end{document}. (shrink)

Name der Zeitschrift: Language and Cognition - An Interdisciplinary Journal of Language and Cognitive Science Jahrgang: 5 Heft: 4 Seiten: 409-436.

Jullien's indecomposability theorem (INDEC) states that if a scattered countable linear order is indecomposable, then it is either indecomposable to the left, or indecomposable to the right. The theorem was shown by Montalbán to be a theorem of hyperarithmetic analysis, and then, in the base system RCA₀ plus ${\mathrm{\Sigma }}_{1}^{1}\text{\hspace{0.17em}}$ induction, it was shown by Neeman to have strength strictly between weak ${\mathrm{\Sigma }}_{1}^{1}$ choice and ${\mathrm{\Delta }}_{1}^{1}$ comprehension. We prove in this paper that ${\mathrm{\Sigma }}_{1}^{1}$ induction is needed (...) for the reversal of INDEC, that is for the proof that INDEC implies weak ${\mathrm{\Sigma }}_{1}^{1}$ choice. This is in contrast with the typical situation in reverse mathematics, where reversals can usually be refined to use only ${\mathrm{\Sigma }}_{1}^{0}$ induction. (shrink)

In this paper I shall present a method for proving determinacy from large cardinals which, in many cases, seems to yield optimal results. One of the main applications extends theorems of Martin, Steel and Woodin about determinacy within the projective hierarchy. The method can also be used to give a new proof of Woodin's theorem about determinacy in L.The reason we look for optimal determinacy proofs is not only vanity. Such proofs serve to tighten the connection between large cardinals and (...) descriptive set theory, letting us bring our knowledge of one subject to bear on the other, and thus increasing our understanding of both. A classic example of this is the Harrington-Martin proof that -determinacy implies -determinacy. This is an example of a transfer theorem, which assumes a certain determinacy hypothesis and proves a stronger one. While the statement of the theorem makes no mention of large cardinals, its proof goes through 0#, first proving that-determinacy ⇒ 0# exists,and then that0# exists ⇒ -determinacyMore recent examples of the connection between large cardinals and descriptive set theory include Steel's proof thatADL ⇒ HODL ⊨ GCH,see [9], and several results of Woodin about models of AD+, a strengthening of the axiom of determinacy AD which Woodin has introduced. These proofs not only use large cardinals, but also reveal a deep, structural connection between descriptive set theoretic notions and notions related to large cardinals. (shrink)

We present two applications of forcing with finite sequences of models as side conditions, adding objects of size \. The first involves adding a \ sequence and variants of such sequences. The second involves adding partial weak specializing functions for trees of height \.

We prove new upper bound theorems on the consistency strengths of SPFA (θ), SPFA(θ-linked) and SPFA(θ⁺-cc). Our results are in terms of (θ, Γ)-subcompactness, which is a new large cardinal notion that combines the ideas behind subcompactness and Γ-indescribability. Our upper bound for SPFA(c-linked) has a corresponding lower bound, which is due to Neeman and appears in his follow-up to this paper. As a corollary, SPFA(c-linked) and PFA(c-linked) are each equiconsistent with the existence of a $\Sigma _{1}^{2}$ -indescribable cardinal. (...) Our upper bound for SPFA(c-c.c.) is a $\Sigma _{2}^{2}$ -indescribable cardinal, which is consistent with V = L. Our upper bound for SPFA(c⁺-linked) is a cardinal κ that is $(\kappa ^{+},\Sigma _{1}^{2})$ -subcompact, which is strictly weaker than κ⁺-supercompact. The axiom MM(c) is a consequence of SPFA(c⁺-linked) by a slight refinement of a theorem of Shelah. Our upper bound for SPFA(c⁺⁺-c.c.) is a cardinal κ that is $(\kappa ^{+},\Sigma _{2}^{2})$ -subcompact, which is also strictly weaker than κ⁺-supercompact. (shrink)

We construct coverings which unravel given Π11 sets. This in turn is used to prove, from optimal large cardinal assumptions, the determinacy of games with payoff and the determinacy of games with payoff in the σ algebra generated by Π11 sets.

The article discusses some of the major issues which need to be examined in a gendered reading of citizenship. However, its basic claim is that a comparative study of citizenship should consider the issue of women's citizenship not only by contrast to that of men, but also in relation to women's affiliation to dominant or subordinate groups, their ethnicity, origin and urban or rural residence. It should also take into consideration global and transnational positionings of these citizenships. The article challenges (...) the gender-blind and Westocentric character of many of the most hegemonic theorizations of citizenship, focusing in particular on the questions of membership in ‘the community’, group rights and social difference and the ways binaries of public/private and active/passive have been constructed to differentiate between different kinds of citizenships. The article argues that in order to be able to analyse adequately people's citizenship, especially in this era of ethnicization on the one hand and globalization on the other hand, and with the rapid pace at which relationships between states and their civil societies are changing, citizenship should best be analysed as a multi-tiered construct which applies, at the same time to people's membership in sub-, cross- and supra-national collectivities as well as in states. (shrink)

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 show that every sufficiently iterable countable mouse has a unique iteration strategy whose associated iteration maps are lexicographically minimal. This enables us to extend the results of [3] on the good behavior of the standard parameter from tame mice to arbitrary mice.

We produce counterexamples to the unique and cofinal branches hypotheses, assuming (slightly less than) the existence of a cardinal which is strong past a Woodin cardinal.

Let o(κ) denote the Mitchell order of κ. We show how to reduce long games which run to the first ordinal admissible in the play, to iteration games on models with a cardinal κ so that (1) κ is a limit of Woodin cardinals: and (2) o(κ) = κ⁺⁺. We use the reduction to derive several optimal determinacy results on games which run to the first admissible in the play.

We define a class of finite state automata acting on transfinite sequences, and use these automata to prove that no singular cardinal can be defined by a monadic second order formula over the ordinals.

PurposeThis paper discusses the need for health information specialists who can promote patient empowerment by tailoring the information patients receive as they cope with illness. The objectives of this study are to distinguish the various stages of coping with illness, examine the informational needs of patients during these stages, explore how web information contributes to patient empowerment, and describe the potential role of the health information specialist.Design/methodology/approachIn order to meet the study's objectives, a qualitative research method was used in which (...) 110 in‐depth interviews were conducted with patients who told their story of coping throughout the course of illness. By distinguishing and understanding the different stages of coping with illness, we came to distinguish the distinct informational needs during the coping process, and understand how web information contributes to patient empowerment.FindingsThe process of coping with illness has four stages. In each stage, the patient's informational needs differ, as does his ability to absorb and process information. Health systems do not provide information to match the coping stage of the patient. Patients turn to professional and unprofessional internet sites in their search for medical knowledge. The current study, integrated with other studies that show patients' difficulties using the internet, emphasizes the importance of the information specialist in the patient empowerment process.Originality/valueThe study will aid policy makers in the process of empowering patients by demonstrating patients' dynamic informational needs when coping with illness. This study proposes a role for the information specialists that will enable them to expand into the health domain. This role is the health information specialist, a professional who will, among other things, learn how health systems work, identify types of web information sources, assess medical site quality, recognize patients' coping stages, and adapt information to individual characteristics. (shrink)

We present two ways in which the model L(R) is canonical assuming the existence of large cardinals. We show that the theory of this model, with ordinal parameters, cannot be changed by small forcing; we show further that a set of ordinals in V cannot be added to L(R) by small forcing. The large cardinal needed corresponds to the consistency strength of AD L (R); roughly ω Woodin cardinals.

We show that Mitchell order on downward closed extenders below rank-to-rank type is wellfounded.

We prove determinacy for open length ω1 games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω1, with payoff conditions involving the entire run, the club filter on ω1, and a sequence of ω1 disjoint stationary subsets of ω1. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption.

The paper examines the effects of intersecting social divisions on constructions of multi‐layered citizenships and the politics of belonging in contemporary Britain. It starts with conceptual clarifications of the notions of citizenship, belonging and intersectionality and then turns to examine contemporary politics of belonging in contemporary Britain, focusing on the current debate on the ‘death of multiculturalism’ and on ‘social cohesion’. It illustrates how the use of civic and democratic values as signifiers of belonging can end up as exclusionary, rather (...) than inclusionary in that discourse. (shrink)