We prove thatµ=µ <µ , 2 µ =µ + and “there is a non-reflecting stationary subset ofµ + composed of ordinals of cofinality <μ” imply that there is a μ-complete Souslin tree onµ +.

We investigate various strong notions of rigidity for Souslin trees, separating them under ♢ into a hierarchy. Applying our methods to the automorphism tower problem in group theory, we show under ♢ that there is a group whose automorphism tower is highly malleable by forcing.

In Part I of this series, we presented the microscopic approach to Souslin-tree constructions, and argued that all known ⋄-based constructions of Souslin trees with various additional properties may be rendered as applications of our approach. In this paper, we show that constructions following the same approach may be carried out even in the absence of ⋄. In particular, we obtain a new weak sufficient condition for the existence of Souslin trees at the level of (...) a strongly inaccessible cardinal. We also present a new construction of a Souslin tree with an ascent path, thereby increasing the consistency strength of such a tree's nonexistence from a Mahlo cardinal to a weakly compact cardinal. Section 2 of this paper is targeted at newcomers with minimal background. It offers a comprehensive exposition of the subject of constructing Souslin trees and the challenges involved. (shrink)

We present a variation of the forcing S max as presented in Woodin [4]. Our forcing is a P max -style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree T G which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with T G (...) being this minimal tree. In particular, in the extension this Souslin tree has the property that forcing with it gives a model of Souslin's Hypothesis. (shrink)

We show it is consistent that there is a Souslin tree S such that after forcing with S, S is Kurepa and for all clubs $$C \subset \omega _1$$ C ⊂ ω 1, $$S\upharpoonright C$$ S ↾ C is rigid. This answers the questions in Fuchs (Arch Math Logic 52(1–2):47–66, 2013). Moreover, we show it is consistent with $$\diamondsuit $$ ♢ that for every Souslin tree T there is a dense $$X \subseteq T$$ X ⊆ T which does (...) not contain a copy of T. This is related to a question due to Baumgartner in Baumgartner (Ordered sets (Banff, Alta., 1981), volume 83 of NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., Reidel, Dordrecht-Boston, pp 239–277, 1982). (shrink)

An ℵ1-Souslin tree is a complicated combinatorial object whose existence cannot be decided on the grounds of ZFC alone. But fifteen years after Tennenbaum and Jech independently devised notions of forcing for introducing such a tree, Shelah proved that already the simplest forcing notion—Cohen forcing—adds an ℵ1-Souslin tree. In this article, we identify a rather large class of notions of forcing that, assuming a GCH-type hypothesis, add a λ+-Souslin tree. This class includes Prikry, Magidor, and Radin forcing.

We prove that there are groups in the constructible universe whose automorphism towers are highly malleable by forcing. This is a consequence of the fact that, under a suitable diamond hypothesis, there are sufficiently many highly rigid non-isomorphic Souslin trees whose isomorphism relation can be precisely controlled by forcing.

We investigate the effect after forcing with a coherent Souslin tree on the gap structure of the class of coherent Aronszajn trees ordered by embeddability. We shall show, assuming the relativized version PFA(S) of the proper forcing axiom, that the Souslin tree S forces that the class of Aronszajn trees ordered by the embeddability relation is universal for linear orders of cardinality at most ${\aleph_1}$.

We present a variation of the forcing $\mathbb{S}_{max}$ as presented in Woodin [4]. Our forcing is a $\mathbb{P}_{max}$-style construction where each model condition selects one Souslin tree. In the extension there is a Souslin tree T$_G$ which is the direct limit of the selected Souslin trees in the models of the generic. In some sense, the generic extension is a maximal model of "there exists a minimal Souslin tree," with T$_G$ being this minimal tree. In (...) particular, in the extension this Souslin tree has the property that forcing with it gives a model of Souslin's Hypothesis. (shrink)

We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support iterations in the so-called Case A that does not need a division into forcings that add reals and those who do not.

A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with (...) a review of some generalizations of results from descriptive set theory concerning Baire category which are then used in non-trivial Souslin tree constructions that yield Souslin algebras with a remarkable subalgebra structure. In particular, we use this method to prove that under the diamond principle there is a bi-embeddable though not isomorphic pair of homogeneous Souslin algebras. (shrink)

We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely MA(Γ + ℵ 0 ), and using the results on Souslin forcing we show that MA(Γ + ℵ 0 ) is consistent with the existence of a Souslin tree and with the splitting number s = ℵ 1 . We prove that MA(Γ + ℵ 0 ) proves the additivity of measure. (...) Also we introduce the notion of proper Souslin forcing, and we prove that this property is preserved under countable support iterated forcing. We use these results to show that ZFC + there is an inaccessible cardinal is equiconsistent with ZFC + the Borel conjecture + Σ 1 2 -measurability. (shrink)

Assuming Jensen's principle ◊+ we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and Todorčević.

A highly rigid Souslin tree T is constructed such that forcing with T turns T into a Kurepa tree. Club versions of previously known degrees of rigidity are introduced, as follows: for a rigidity property P, a tree T is said to have property P on clubs if for every club set C (containing 0), the restriction of T to levels in C has property P. The relationships between these rigidity properties for Souslin trees are investigated, and (...) some open questions are stated. (shrink)

We investigate questions involving Aronszajn trees, square principles, and stationary reflection. We first consider two strengthenings of introduced by Brodsky and Rinot for the purpose of constructing κ‐Souslin trees. Answering a question of Rinot, we prove that the weaker of these strengthenings is compatible with stationary reflection at κ but the stronger is not. We then prove that, if μ is a singular cardinal, implies the existence of a special ‐tree with a cf(μ)‐ascent path, thus answering a (...) question of Lücke. (shrink)

We study descriptive set theory in the space ω1 ω 1 by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of Π 1 1 -sets of ω1 ω 1 . We call a family U of trees universal for a class V of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in V (...) can be order-preservingly mapped into a tree in U. It is well known that the class of countable trees with no infinite branches has a universal family of size ℵ 1 . We shall study the smallest cardinality of a universal family for the class of trees of cardinality ≤ℵ 1 with no uncountable branches. We prove that this cardinality can be 1 (under ¬CH) and any regular cardinal κ which satisfies ℵ 2 ≤ κ ≤ 2 ℵ 1 (under CH). This bears immediately on the covering property of the Π 1 1 -subsets of the space ω1 ω 1 . We also study the possible cardinalities of definable subsets of ω1 ω 1 . We show that the statement that every definable subset of ω1 ω 1 has cardinality $ or cardinality 2 ω1 is equiconsistent with ZFC (if n ≥ 3) and with ZFC plus an inaccessible (if n = 2). Finally, we define an analogue of the notion of a Borel set for the space ω1 ω 1 and prove a Souslin-Kleene type theorem for this notion. (shrink)

We study descriptive set theory in the space $^{\omega_1}\omega_1$ by letting trees with no uncountable branches play a similar role as countable ordinals in traditional descriptive set theory. By using such trees, we get, for example, a covering property for the class of $\Pi^1_1$-sets of $^{\omega_1}\omega_1$. We call a family $\mathscr{U}$ of trees universal for a class $\mathscr{V}$ of trees if $\mathscr{U} \subseteq \mathscr{V}$ and every tree in $\mathscr{V}$ can be order-preservingly mapped into a tree in (...) $\mathscr{U}$. It is well known that the class of countable trees with no infinite branches has a universal family of size $\aleph_1$. We shall study the smallest cardinality of a universal family for the class of trees of cardinality $\leq\aleph_1$ with no uncountable branches. We prove that this cardinality can be 1 (under $\neg$CH) and any regular cardinal $\kappa$ which satisfies $\aleph_2 \leq \kappa \leq 2^{\aleph_1}$ (under CH). This bears immediately on the covering property of the $\Pi^1_1$-subsets of the space $^{\omega_1}\omega_1$. We also study the possible cardinalities of definable subsets of $^{\omega_1}\omega_1$. We show that the statement that every definable subset of $^{\omega_1}\omega_1$ has cardinality $<\omega_n$ or cardinality $2^{\omega_1}$ is equiconsistent with ZFC (if $n \geq 3$) and with ZFC plus an inaccessible (if $n = 2$). Finally, we define an analogue of the notion of a Borel set for the space $^{\omega_1}\omega_1$ and prove a Souslin-Kleene type theorem for this notion. (shrink)

Words like well, oh, and you know have long been observed and studied in spontaneous speech. With the proliferation of on-line dialogues, such as instant messaging between friends or back-and-forth postings at websites, there are increasing opportunities to observe them in spontaneous writing. In Experiment 1, the interpretation of discourse markers in on-line debates was compared to proposed functions of those markers identified in other settings. In Experiment 2, the use of discourse markers in spontaneous speech was compared to their (...) uses in spontaneous writing. Results support the conclusion that although they vary in frequency in spoken versus written domains, discourse markers are used similarly across domains, but with particular communicative functions that make them non-interchangeable. (shrink)

The discourse marker use of the word like is considered by many to be superfluously sprinkled into talk, a bad habit best avoided. But a comparison of the use of like in successive tellings of stories demonstrates that like can be anticipated in advance and planned into stories. In this way, like is similar to other words and phrases tellers recycle during story telling. The anticipation of like contrasted with the uses of other discourse markers such as oh, you know, (...) and well, which almost never re-occurred in similar locations across tellings. Um and uh did sometimes re-occur; these uses are contrasted with like. Although discourse markers are generally used on the fly to handle various issues that come up in coordinating talk as it unfolds, like can be used as an integral part of the story -a marked contrast to the prevalent idea that likes are speech tics. (shrink)

To generate empathy in the care of vulnerable older persons requires care providers to reflect critically on their care practices. Ethics education and training must provide them with tools to accomplish such critical reflection. It must also create a pedagogical context in which good care can be taught and cultivated. The care-ethics lab ‘sTimul’ originated in 2008 in Flanders with the stimulation of ethical reflection in care providers and care providers in training as its main goal. Also in 2008, sTimul (...) commenced the organization of empathy sessions as an attempt to achieve this goal by simulation. The empathy session is a practical and fairly straightforward way of working to provoke care providers and care providers in training to engage in ethical reflection. Characteristic of the empathy session in the care-ethics lab is the emphasis on experience as a basis for ethical reflection. (shrink)

We explored the differential impact of auditory information and written contextual information on the recognition of verbal irony in spontaneous speech. Based on relevance theory, we predicted that speakers would provide acoustic disambiguation cues when speaking in situations that lack other sources of information, such as a visual channel. We further predicted that listeners would use this information, in addition to context, when interpreting the utterances. People were presented with spontaneously produced ironic and nonironic utterances from radio talk shows in (...) written or auditory form, with or without written contextual information. When the utterances were read without written contextual information, all utterances were rated as equally ironic. But when they were heard as opposed to read, or when they were presented in irony-biasing contexts, originally ironic utterances were rated as more sarcastic than originally nonironic utterances. This evidence suggests both acoustic and contextual information are used when inferring ironic intent in spontaneous speech, and validates previous manipulations of intonation in studies of irony understanding. (shrink)

Observing others in conversation is a common format for comprehending language, yet little work has been done to understand dialog comprehension. We tested whether overhearers use addressee backchannels as predictive cues for how to integrate information across speaker turns during comprehension of spontaneously produced collaborative narration. In Experiment 1, words that followed specific backchannels were recognized more slowly than words that followed either generic backchannels or pauses. In Experiment 2, we found that when the turn after the backchannel was a (...) continuation of the narrative, specific backchannels prompted the fastest verification of prior information. When the turn after was an elaboration, they prompted the slowest, indicating that overhearers took specific backchannels as cues to integrate preceding talk with subsequent talk. These findings demonstrate that overhearers capitalize on the predictive relationship between backchannels and the development of speakers’ talk, coordinating information across conversational roles. (shrink)

The COVID-19 pandemic has presented a global threat to physical and mental health worldwide. Research has highlighted adverse impacts of COVID-19 on wellbeing but has yet to offer insights as to how wellbeing may be protected. Inspired by developments in wellbeing science and guided by our own theoretical framework, we examined the role of various potentially protective factors in a sample of 138 participants from the United Kingdom. Protective factors included physical activity, tragic optimism, gratitude, social support, and nature connectedness. (...) Initial analysis involved the application of one-sample t-tests, which confirmed that wellbeing in the current sample was significantly lower compared to previous samples. Protective factors were observed to account for up to 50% of variance in wellbeing in a hierarchical linear regression that controlled for a range of sociostructural factors including age, gender, and subjective social status, which impact on wellbeing but lie beyond individual control. Gratitude and tragic optimism emerged as significant contributors to the model. Our results identify key psychological attributes that may be harnessed through various positive psychology strategies to mitigate the adverse impacts of hardship and suffering, consistent with an existential positive psychology of suffering. (shrink)

This article discusses ways to operationalise the concept of culturally appropriate computer-mediated communication, utilising information systems (IS) development methodologies and adopting a postmodern and postcolonial perspective. By way of illustration, it describes progress on the participative development of the Ieramugadu Cultural Information System. This project is designed to develop and evaluate innovative procedures for elicitation, analysis, storage and communication of indigenous cultural heritage information. It is investigating culturally appropriate IS design techniques, multimedia approaches and ways to ensure protection of secret/sacred (...) information. Ethical issues are foregrounded. (shrink)

There has been little previous scholarship regarding the aims, options and impact of ethics education on residential care‐givers. This manuscript details findings from a pragmatic cluster trial evaluating the impact of three different approaches to ethics education. The focus of the article is on one of the interventions, an immersive simulation experience. The simulation experience required residential care‐givers to assume the profile of elderly care‐recipients for a 24‐hr period. The care‐givers were student nurses. The project was reviewed favourably by a (...) university ethics committee, and participants provided informed consent. Data from six postsimulation experience focus groups were analysed thematically and three themes were identified: the experience of vulnerability, dignity in care and the organisation of care. Findings suggest that the immersive simulation experience had a powerful immediate impact as participants described epiphanous insights relating to their care experiences. It is suggested that reflecting on and recording epiphanous events has the potential to sustain ethical care practices. Further research is required to evaluate the impact of different ethics education interventions in different cultural contexts. Exploration is also required regarding the meaning and significance of care epiphanies, those “most delicate and evanescent of moments,” for the sustainability of ethical care. (shrink)

Bilingualism has been identified as a potential cognitive factor linked to delayed onset of dementia as well as boosting executive functions in healthy individuals. However, more recently, this claim has been called into question following several failed replications. It remains unclear whether these contradictory findings reflect how bilingualism is defined between studies, or methodological limitations when measuring the bilingual effect. One key issue is that despite the claims that bilingualism yields general protection to cognitive processes, studies reporting putative bilingual differences (...) are often focused on domain specific experimental paradigms. This study chose a broader approach, by considering the consequences of bilingualism on a wide range of cognitive functions within individuals. We utilised 19 measures of different cognitive functions commonly associated with bilingual effects, to form a “cognitive profile” for 215 non-clinical participants. We recruited Welsh speakers, who as a group of bilinguals were highly homogeneous, as means of isolating the bilingualism criterion. We sought to determine if such analyses would independently classify bilingual/monolingual participant groups based on emergent patterns driven by collected cognitive profiles, such that population differences would emerge. Multiple predictive models were trained to independently recognise the cognitive profiles of bilinguals, older adults and higher education attainment. Despite managing to successfully classify cognitive profiles based on age and education, the model failed to differentiate between bilingual and monolingual cognitive ability at a rate greater than that of chance. Repeated modelling using alternative definitions of bilingualism, and just the older adults, yielded similar results. In all cases then, using our “bottom–up” analytical approach, there was no evidence that bilingualism as a variable indicated differential cognitive performance – as a consequence, we conclude that bilinguals are not cognitively different from their monolingual counterparts, even in older demographics. We suggest that studies that have reported a bilingual advantage could well have confounded other key variables that may be driving reported advantages. We recommend that future research refine the machine learning methods used in this study to further investigate the complex relationship between bilingualism and cognition. (shrink)

We give some restrictions for the search for a model of the club principle with no Souslin trees. We show that ${\diamondsuit(2^\omega, [\omega]^\omega}$ , is almost constant on) together with CH and “all Aronszajn trees are special” is consistent relative to ZFC. This implies the analogous result for a double weakening of the club principle.

Three central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying (...) a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$. (shrink)

An outline is given of the proof that the consistency of a κ⁺-Mahlo cardinal implies that of the statement that I[ω₂] does not include any stationary subsets of Cof(ω₁). An additional discussion of the techniques of this proof includes their use to obtain a model with no ω₂-Aronszajn tree and to add an ω₂-Souslin tree with finite conditions.

We define the property of Π2-compactness of a statement Φ of set theory, meaning roughly that the hard core of the impact of Φ on combinatorics of 1 can be isolated in a canonical model for the statement Φ. We show that the following statements are Π2-compact: “dominating NUMBER = 1,” “cofinality of the meager IDEAL = 1”, “cofinality of the null IDEAL = 1”, “bounding NUMBER = 1”, existence of various types of Souslin trees and variations on (...) uniformity of measure and CATEGORY = 1. Several important new metamathematical patterns among classical statements of set theory are pointed out. (shrink)

In [4], Kunen used iterated ultrapowers to show that ifUis a normalκ-complete nontrivial ultrafilter on a cardinalκthenL[U], the class of sets constructive fromU, has only the ultrafilterU∩L[U] and this ultrafilter depends only onκ. In this paper we extend Kunen's methods to arbitrary sequencesUof ultrafilters and obtain generalizations of these results. In particular we answer Problem 1 of Kunen and Paris [5] which asks whether the number of ultrafilters onκcan be intermediate between 1 and 22κ. If there is a normalκ-complete ultrafilterUonκsuch (...) that {α <κ: α is measurable} ∈Uthen there is an inner model with exactly two normal ultrafilters onκ, and ifκis super-compact then there are inner models havingκ+ +,κ+or any cardinal less than or equal toκnormal ultrafilters.These methods also show that several properties ofLwhich had been shown to hold forL[U] also hold forL[U]: using an idea of Silver we show that inL[U] the generalized continuum hypothesis is true, there is a Souslin tree, and there is awell-ordering of the reals. In addition we generalize a result of Kunen to characterize the countaby complete ultrafilters ofL[U]. (shrink)

l investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed. <κ-directed-closed, or of the form Col (κ. <Λ). These principles come in many variants, depending on the parameters which are allowed. I shall write MPΓ(A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are: • The principles have many consequences, such as <κ-closed-generic $\Sigma _{2}^{1}(H_{\kappa})$ absoluteness, and imply. e.g., that ◇κ holds. I give an application (...) to the automorphism tower problem, showing that there are Souslin trees which are able to realize any equivalence relation, and hence that there are groups whose automorphism tower is highly sensitive to forcing. • The principles can be separated into a hierarchy which is strict, for many κ. • Some of the principles can be combined, in the sense that they can hold at many different κ simultaneously. The possibilities of combining the principles are limited, though: While it is consistent that MP<κ-closed(HκT) holds at all regular κ below any fixed α. the "global" maximality principle, stating that MP<κ-closed(Hκ ∪ {κ}) holds at every regular κ. is inconsistent. In contrast to this, it is equiconsistent with ZFC that the maximality principle for directed-closed forcings without any parameters holds at every regular cardinal. It is also consistent that every local statement with parameters from HκT that's provably <κ-closed-forceably necessary is true, for all regular κ. (shrink)

König, Larson and Yoshinobu initiated the study of principles for guessing generalized clubs, and introduced a construction of a higher Souslin tree from the strong guessing principle.Complementary to the author’s work on the validity of diamond and non-saturation at the successor of singulars, we deal here with a successor of regulars. It is established that even the non-strong guessing principle entails non-saturation, and that, assuming the necessary cardinal arithmetic configuration, entails a diamond-type principle which suffices for the construction of (...) a higher Souslin tree.We also establish the consistency of GCH with the failure of the weakest form of generalized club guessing. This, in particular, settles a question from the original paper. (shrink)

In mathematical practice certain formulas φ are believed to essentially decide all other natural properties of the object x. The purpose of this paper is to exactly quantify such a belief for four formulas φ, namely “x is a Ramsey ultrafilter”, “x is a free Souslin tree”, “x is an extendible strong Lusin set” and “x is a good diamond sequence”.