We construct a model in which for all 1 ≤ n < ω, there is no stationary subset of ${\aleph_{n+1} \cap {\rm cof}(\aleph_n)}$ which carries a partial square.

We introduce the weak partial square principles View the MathML source and View the MathML source, which combine the ideas of a weak square sequence and a partial square sequence. We construct models in which weak partial square principles fail. The main result of the paper is that □λ,κ does not imply View the MathML source.

We present a characterization of weakly compact cardinals in terms of generalized stationarity. We apply this characterization to construct a model with no partial square sequences.

We present a technique for destroying stationary subsets of Pκκ+ using partial square sequences. We combine this method with Gitik’s poset for changing the cofinality of a cardinal without adding bounded sets to prove a variety of consistency results concerning saturated ideals and the set S.

We present a construction of a global square sequence in extender models with λ-indexing.

We investigate the relationship between weak square principles and simultaneous reflection of stationary sets.

We extend the construction of a global square sequence in extender models from Zeman [8] to a construction of coherent non-threadable sequences and give a characterization of stationary reflection at inaccessibles similar to Jensen’s characterization in L.

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof carries a partial square.

The paper offers a logical characterisation of multi-step actions in the context of deontic notions of obligation, permission and prohibition. Deontic notions for sequentially composed actions are founded on deontic notions for one-step actions. The present work includes a formal study of situations where execution of a multi-step action has been unsuccessful and provides normative analysis of such actions.

This is a continuation of our paper where we show that Rado's Conjecture can trivialize ‐sequences in some cases when ϑ is not necessarily a successor cardinal.

Over the years, two models of freedom have emerged as competitors: the alternative-possibilities model and the actual-sequence model. This paper is a partial defense of the actual-sequence model. My defense relies on two strategies. The first strategy consists in de-emphasizing the role of examples in arguing for a model of freedom. Imagine that, as some people think, Frankfurt-style cases fail to undermine the alternative-possibilities model. What follows from this? Not much, I argue. In particular, I note that (...) the counterparts of Frankfurt-style cases also fail to undermine the actual-sequence model. My second strategy of defense consists in revitalizing the original motivation for the actual-sequence model, by revamping it, isolating it from claims that do not fully capture the same idea, and arguing that it can be developed in a successful way. (shrink)

Significant research has been devoted to detecting people in images and videos. In this paper we describe a human de- tection method that augments widely used edge-based fea- tures with texture and color information, providing us with a much richer descriptor set. This augmentation results in an extremely high-dimensional feature space (more than 170,000 dimensions). In such high-dimensional spaces, classical machine learning algorithms such as SVMs are nearly intractable with respect to training. Furthermore, the number of training samples is much (...) smaller than the dimensionality of the feature space, by at least an order of magnitude. Finally, the extraction of features from a densely sampled grid structure leads to a high degree of multicollinearity. To circumvent these data characteristics, we employ Partial Least Squares (PLS) analysis, an effi- cient dimensionality reduction technique, one which pre- serves significant discriminative information, to project the data onto a much lower dimensional subspace (20 dimen- sions, reduced from the original 170,000). Our human de- tection system, employing PLS analysis over the enriched descriptor set, is shown to outperform state-of-the-art tech- niques on three varied datasets including the popular INRIA pedestrian dataset, the low-resolution gray-scale Daim- lerChrysler pedestrian dataset, and the ETHZ pedestrian dataset consisting of full-length videos of crowded scenes. (shrink)

The paper studies a domain theoretical notion of primitive recursion over partial sequences in the context of Scott domains. Based on a non-monotone coding of partial sequences, this notion supports a rich concept of parallelism in the sense of Plotkin. The complexity of these functions is analysed by a hierarchy of classes ${\cal E}^{\bot}_n$ similar to the Grzegorczyk classes. The functions considered are characterised by a function algebra ${\cal R}^{\bot}$ generated by continuity preserving operations starting from computable initial (...) functions. Its layers ${\cal R}^{\bot}_n$ are related to those above by showing $\forall n \ge 2.{\cal E}^{\bot}_{n+1} ={\cal R}^{\bot}_n$ , thus generalising results of Schwichtenberg/Müller and Niggl. (shrink)

The COVID-19 pandemic cropping up at the end of 2019 started to pose a threat to millions of people’s health and life after a few weeks. Nevertheless, the COVID-19 pandemic gave rise to social and economic problems that have changed the progress steps of individuals and the whole nation. In this study, the work conditions for employees from Taiwan, Malaysia, and the Chinese mainland are explored and compared, and the relationship between support mechanisms and innovation behaviors is evaluated with a (...) view of the social cognitive career theory. This study adopts the cross-sectional survey and purposive sampling to collect questionnaires. A total of 623 copies of a questionnaire from Taiwanese, 440 copies from Malaysians, and 513 copies from mainlanders were collected in this study to compare the three groups in developing employees’ IBs. Smart-partial least squares for partial least squares structural equation modeling was applied in the structural model to conduct a verification of the hypotheses and comparative analysis in this study. According to the findings, compared with employees from the Chinese mainland, the Taiwanese and Malaysian samples show more significant paths regarding employee employability, IB, prior knowledge, perceived organizational support, self-efficacy, and job performance. Our results will offer more insights and advice concerning theories of human resource. (shrink)

PurposeDrawing upon the stimulus-organism-response model, this paper aims to investigate the effects of stores’ environmental components on Chinese consumers’ emotions and intentions to purchase luxury brands.Design/Methodology/ApproachData were collected from Chinese consumers who have purchased luxury brands from retail stores. Partial least squares-structural equation modeling and fuzzy-set qualitative comparative analysis were employed for data analysis.FindingsThe results of PLS-SEM indicated that three dimensions of the store environment directly and significantly influenced Chinese consumers’ emotions. However, fsQCA revealed greater heterogeneity among respondents by (...) highlighting stores’ environmental components and Chinese consumers’ emotions.Originality/ValueThis study makes significant contributions to the marketing literature by examining the S-O-R model alongside fsQCA approach to conditionally predict outcomes in a luxury brand context. The present study may be one of the first to examine the effects of stores’ environmental components on Chinese consumers’ emotions and purchase intentions, applying both symmetrical test and asymmetrical test approaches to determine significant components of the store environment as well as component combinations that predict Chinese consumers’ emotions and purchase intentions. (shrink)

Viale introduced covering matrices in his proof that SCH follows from PFA. In the course of the proof and subsequent work with Sharon, he isolated two reflection principles, CP and S, which, under certain circumstances, are satisfied by all covering matrices of a certain shape. Using square sequences, we construct covering matrices for which CP and S fail. This leads naturally to an investigation of square principles intermediate between □κ and □ for a regular cardinal κ. We provide (...) a detailed picture of the implications between these square principles. (shrink)

Various semantics for studying the square of opposition and the hexagon of opposition have been proposed recently. We interpret sentences by imprecise (set-valued) probability assessments on a finite sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square and of the hexagon in terms of acceptability. Then, we show how to construct probabilistic versions of the square and of the hexagon of opposition by forming (...) suitable tripartitions of the set of all coherent assessments. Finally, as an application, we present new versions of the square and of the hexagon involving generalized quantifiers. (shrink)

Various semantics for studying the square of opposition have been proposed recently. So far, only [14] studied a probabilistic version of the square where the sentences were interpreted by (negated) defaults. We extend this work by interpreting sentences by imprecise (set-valued) probability assessments on a sequence of conditional events. We introduce the acceptability of a sentence within coherence-based probability theory. We analyze the relations of the square in terms of acceptability and show how to construct probabilistic (...) versions of the square of opposition by forming suitable tripartitions. Finally, as an application, we present a new square involving generalized quantifiers. (shrink)

If is a random sequence, then the sequence is clearly not random; however, seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 215–229] and K. Tadaki [A generalisation of Chaitin’s halting probability Ω and halting self-similar sets, Hokkaido Math. J. 31 219–253] have studied the degree of randomness of sequences or reals by measuring (...) their “degree of compression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ1-dimension in terms of strong Martin-Löf ε-tests , and we show that ε-randomness for ε is different than the classical 1-randomness. (shrink)

We show that MRP + MA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . This generalizes a result by Weiß who showed that PFA implies that ITP(λ, ω 2 ) holds for all cardinal λ ≥ ω 2 . Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA (...) with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω 2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω 1 ) for all λ ≥ ω 2. (shrink)

In their comment on Sandberg, Timmermans, Overgaard, and Cleeremans , Dienes and Seth argue that increased sensitivity of the Perceptual Awareness Scale is a consequence of the scale being less exclusive rather than more exhaustive. According to Dienes and Seth, this is because PAS may measure some conscious content, though not necessarily relevant conscious content, “If one saw a square but was only aware of seeing a flash of something, then one has not consciously seen a square.” In (...) this reply, we claim that there is a difference between conscious visual experience, which may be partial, and the resulting conscious content, which is conceptual. Whereas PAS measures the first, confidence judgments and post-decision wagering measure the second. (shrink)

In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with (...) the classical tree-embeddability, but in this article we are interested in more general situations. We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega}}$$\end{document}. Somewhat surprisingly, changing a little bit the well-partially ordered addresses, the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑΩΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega^\Omega}}$$\end{document}. Our results contribute to the research program on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings. (shrink)

We give the proof of a theorem of Jensen and Zeman on the existence of a global □ sequence in the Core Model below a measurable cardinal κ of Mitchell order ) equal to κ++, and use it to prove the following theorem on mutual stationarity at n.Let ω1 denote the first uncountable cardinal of V and set to be the class of ordinals of cofinality ω1.TheoremIf every sequence n m. In particular, there is such a model in (...) which for all sufficiently large m<ω, the class of measurables λ with oM≥ωm is, in V, stationary below m+2. (shrink)

This paper gives a new, proof-theoretic explanation of partial-order reasoning about time in a nonmonotonic theory of action. The explanation relies on the technique of lifting ground proof systems to compute results using variables and unification. The ground theory uses argumentation in modal logic for sound and complete reasoning about specifications whose semantics follows Gelfond and Lifschitz’s language. The proof theory of modal logic A represents inertia by rules that can be instantiated by sequences of time steps or events. (...) Lifting such rules introduces string variables and associates each proof with a set of string equations; these equations are equivalent to a set of partial-order tree-constraints that can be solved efficiently. The defeasible occlusion of inertia likewise imposes partial-order constraints in the lifted system. By deriving an auxiliary partial order representation of action from the underlying logic, not the input formulas or proofs found, this paper strengthens the connection between practical planners and formal theories of action. Moreover, the general correctness of the theory of action justifies partial-order representations not only for forward reasoning from a completely specified start state, but also for explanatory reasoning and for reasoning by cases. (shrink)

This thesis proposes and presents two new models for belief representation and belief revision. The first model is the B-structures model which relies on a notion of partial language splitting and tolerates some amount of inconsistency while retaining classical logic. The model preserves an agent's ability to answer queries in a coherent way using Belnap's four-valued logic. Axioms analogous to the AGM axioms hold for this new model. The distinction between implicit and explicit beliefs is represented and psychologically plausible, (...) computationally tractable procedures for query answering and belief base revision are obtained. ;The second model presents a method for relevance sensitive non-monotonic inference from belief sequences which incorporates insights pertaining to prioritized inference and relevance sensitive, inconsistency tolerant belief revision. Our model uses a finite, logically open sequence of propositional formulas as a representation for beliefs and defines a notion of inference from maxiconsistent subsets of formulas guided by two orderings: a temporal sequencing and an ordering based on relevance relations between the conclusion and formulas in the sequence. The relevance relations are ternary as opposed to standard binary axiomatizations. The inference operation thus defined easily handles iterated revision by maintaining a revision history, blocks the derivation of inconsistent answers from a possibly inconsistent sequence and maintains the distinction between explicit and implicit beliefs. In doing so, it provides a finitely representable formalism and a plausible model of reasoning for automated agents. (shrink)

Given any subset A of ω1 there is a proper partial order which forces that the predicate xA and the predicate xω1A can be expressed by -provably incompatible Σ3 formulas over the structure Hω2,,NSω1. Also, if there is an inaccessible cardinal, then there is a proper partial order which forces the existence of a well-order of Hω2 definable over Hω2,,NSω1 by a provably antisymmetric Σ3 formula with two free variables. The proofs of these results involve a technique for (...) manipulating the guessing properties of club-sequences defined on stationary subsets of ω1 at will in such a way that the Σ3 theory of Hω2,,NSω1 with countable ordinals as parameters is forced to code a prescribed subset of ω1. On the other hand, using theorems due to Woodin it can be shown that, in the presence of sufficiently strong large cardinals, the above results are close to optimal from the point of view of the Levy hierarchy. (shrink)

We prove that, in any fine structural extender model with Jensen’s λ-indexing, there is a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square(\kappa^{+})}$$\end{document} -sequence if and only if there is a pair of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \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} which, in turn, is equivalent to the existence (...) of a family of size \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} of stationary subsets of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\kappa^{+} \cap {\rm {cof}}( < \kappa)}$$\end{document} without common reflection point of cofinality \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}. By a result of Burke/jensen, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square_\kappa}$$\end{document} fails whenever \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} is a subcompact cardinal. Our result shows that in extender models, it is still possible to construct a canonical \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\square(\kappa^{+})}$$\end{document} -sequence where \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} is the first subcompact. (shrink)

It is shown that the class of Kolmogorov-Loveland stochastic sequences is not closed under selecting subsequences by monotonic computable selection rules. This result gives a strong negative answer to the question whether the Kolmogorov-Loveland stochastic sequences are closed under selecting sequences by Kolmogorov-Loveland selection rules, i.e., by not necessarily monotonic, partial computable selection rules. The following previously known results are obtained as corollaries. The Mises-Wald-Church stochastic sequences are not closed under computable permutations, hence in particular they form a strict (...) superclass of the class of Kolmogorov-Loveland stochastic sequences. The Kolmogorov-Loveland selection rules are not closed under composition. (shrink)

Examines Marx's account in Capital of a machine burned at the stake in a public square in 1685 as an emblem of modern attitudes to technology.

Though the current research stream has provided some risk factors for envy at the workplace, little is still known about the drivers and consequences of envy. Based on Vecchio’s theory, this study investigates the ripple effect of the span of supervision on envy. Moreover, it sheds light on the moderating role of meaningful work in their relationship. The data comprising sample size 439 were collected from confrères of four fast food companies listed on the Stock Exchange of Pakistan. Partial (...) Least Square Structural Equation Modeling technique was implemented through SmartPLS 3.3.2 to analyze the measurement and structural relationships. The results demonstrate that a narrow span of supervision will increase work engagement, and reduce instigated incivility via decreasing envy and resource depletion in sequence. Moreover, meaningful work would help regulate the inimical stream of dénouement of envy. Theoretical and practical implications, along with the limitations and future directions, have also been discussed. (shrink)

Recent studies have indicated that it is possible to protect individuals from HIV infection using passive infusion of monoclonal antibodies. However, in order for monoclonal antibodies to confer robust protection, the antibodies must be capable of neutralizing many possible strains of the virus. This is particularly challenging in the context of a highly diverse pathogen like HIV. It is therefore of great interest to leverage existing observational data sources to discover antibodies that are able to neutralize HIV viruses via residues (...) where existing antibodies show modest protection. Such information feeds directly into the clinical trial pipeline for monoclonal antibody therapies by providing information on whether and to what extent combinations of antibodies can generate superior protection and strategies for analyzing past clinical trials to identify in vivo evidence of antibody resistance. These observational data include genetic features of many diverse HIV genetic sequences, as well as in vitro measures of antibody resistance. The statistical learning problem we are interested in is developing statistical methodology that can be used to analyze these data to identify important genetic features that are significantly associated with antibody resistance. This is a challenging problem owing to the high-dimensional and strongly correlated nature of the genetic sequence data. To overcome these challenges, we propose an outcome-adaptive, collaborative targeted minimum loss-based estimation approach using random forests. We demonstrate via simulation that the approach enjoys important statistical benefits over existing approaches in terms of bias, mean squared error, and type I error. We apply the approach to the Compile, Analyze, and Tally Nab Panels database to identify AA positions that are potentially causally related to resistance to neutralization by several different antibodies. (shrink)

Let f be a computable function from finite sequences of 0ʼs and 1ʼs to real numbers. We prove that strong f-randomness implies strong f-randomness relative to a PA-degree. We also prove: if X is strongly f-random and Turing reducible to Y where Y is Martin-Löf random relative to Z, then X is strongly f-random relative to Z. In addition, we prove analogous propagation results for other notions of partial randomness, including non-K-triviality and autocomplexity. We prove that f-randomness relative to (...) a PA-degree implies strong f-randomness, hence f-randomness does not imply f-randomness relative to a PA-degree. (shrink)

With an analysis of the structure and the sequence of analogy, the paper is mainly a critique to the partial reducibility thesis: a thesis sustaining that analogy, besides a strictly analogical step, is in the remaining part reducible to balancing. Thus, the paper points out some problems raised by the partial reducibility thesis, such as the contingency of reducibility or the fact that a proper analogy is done under the cover of a balancing. The main point is, (...) however, the claim that analogy and balancing have opposite normative conditions, being this premise the reason to a structural explanation for the unacceptability of the reducibility enterprise. (shrink)

The major aim of this paper is to present some reflections about the political domain and the common good that may be helpful in answering the following issue: How can religions contribute to the common good? The problematic background of this paper can be summarized by the so-called Dilemma of Böckenförde, which presents the difficulties secular states have in creating social capital, and by the Habermasian notion of a “post-secular society,” an expression used by the German philosopher to summarize the (...) curious situations of Western secularized states, where religions continue to play important public roles. I will first discuss the notion of “neutralization” with the support of Carl Schmitt. Then I will present Chantal Mouffe’s doctrine of “agonistic pluralism” and her partial legitimization of the presence of religions in the political domain. Finally, I will criticize Mouffe’s approach with the help of Alasdair MacIntyre’s phenomenology of social practices in order to stress the importance of public religions in contemporary liberal societies. (shrink)

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our (...) main computability concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

We analyse the connection between the computability and continuity of functions in the case of homomorphisms between topological algebraic structures. Inspired by the Pour-El and Richards equivalence theorem between computability and boundedness for closed linear operators on Banach spaces, we study the rather general situation of partial homomorphisms between metric partial universal algebras. First, we develop a set of basic notions and results that reveal some of the delicate algebraic, topological and effective properties of partial algebras. Our (...) main computability concepts are based on numerations and include those of effective metric partial algebras and effective partial homomorphisms. We prove a general equivalence theorem that includes a version of the Pour-El and Richards Theorem, and has other applications. Finally, the Pour-El and Richards axioms for computable sequence structures on Banach spaces are generalised to computable partial sequence structures on metric algebras, and we prove their equivalence with our computability model based on numerations. (shrink)

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216 89–117] and Steel [PFA implies AD L, J. Symbolic Logic70 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a (...) singular strong limit cardinal or PFA. (shrink)

I analyze the hierarchies of the bounded and the weak bounded forcing axioms, with a focus on their versions for the class of subcomplete forcings, in terms of implications and consistency strengths. For the weak hierarchy, I provide level-by-level equiconsistencies with an appropriate hierarchy of partially remarkable cardinals. I also show that the subcomplete forcing axiom implies Larson’s ordinal reflection principle atω2, and that its effect on the failure of weak squares is very similar to that of Martin’s Maximum.