We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups ${\mathbb {Z}, Z(p^{\infty})}$ , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. (...) For example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism. (shrink)

In this article, we formally define and investigate the computational complexity of the definability problem for open first-order formulas with equality. Given a logic $\boldsymbol{\mathcal{L}}$, the $\boldsymbol{\mathcal{L}}$-definability problem for finite structures takes as an input a finite structure $\boldsymbol{A}$ and a target relation $T$ over the domain of $\boldsymbol{A}$ and determines whether there is a formula of $\boldsymbol{\mathcal{L}}$ whose interpretation in $\boldsymbol{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas is (...) coNP-complete. We also investigate the parametric complexity of the problem and prove that if the size and the arity of the target relation $T$ are taken as parameters, then open definability is $\textrm{coW}[1]$-complete for every vocabulary $\tau $ with at least one, at least binary, relation. (shrink)

Contemporary architecture is increasingly grounded in science and mathematics. Architectural discourse has shifted radically from the sometimes disorienting Derridean deconstruction, to engaging scientific terms such as fractals, chaos, complexity, nonlinearity, and evolving systems. That's where the architectural action is -- at least for cutting-edge architects and thinkers -- and every practicing architect and student needs to become conversant with these terms and know what they mean. Unfortunately, the vast majority of architecture faculty are unprepared to explain them to students, (...) not having had a scientific education themselves. Here is an architecture book by an architect/scientist, just in time to help architects in the new millennium. Alexander discusses many of the scientific terms arising in cutting-edge architecture, and explains them to those who don't have scientific training or advanced mathematical knowledge. We find discussions of the evolution of forms; the importance of process in design; iteration; genetic algorithms; sequences of transformations; different levels of scale (i.e. fractals); etc. They are explained here by an architect who is also a scientist, because he wants to change the way architects think and build. Alexander is not merely popularizing other scientists' results and making them accessible to architects: he is in fact presenting new and original scientific work that ties many of these concepts together in a way that will be useful to architects. (shrink)

The article presents proofs of the context freeness of a family of typelogical grammars, namely all grammars that are based on a uni- ormultimodal logic of pure residuation, possibly enriched with thestructural rules of Permutation and Expansion for binary modes.

We define the Scott complexity of a countable structure to be the least complexity of a Scott sentence for that structure. This is a finer notion of complexity than Scott rank: it distinguishes between whether the simplest Scott sentence is $\Sigma _{\alpha }$, $\Pi _{\alpha }$, or $\mathrm {d-}\Sigma _{\alpha }$. We give a complete classification of the possible Scott complexities, including an example of a structure whose simplest Scott sentence is $\Sigma _{\lambda + (...) 1}$ for $\lambda $ a limit ordinal. This answers a question left open by A. Miller.We also construct examples of computable structures of high Scott rank with Scott complexities $\Sigma _{\omega _1^{CK}+1}$ and $\mathrm {d-}\Sigma _{\omega _1^{CK}+1}$. There are three other possible Scott complexities for a computable structure of high Scott rank: $\Pi _{\omega _1^{CK}}$, $\Pi _{\omega _1^{CK}+1}$, $\Sigma _{\omega _1^{CK}+1}$. Examples of these were already known. Our examples are computable structures of Scott rank $\omega _1^{CK}+1$ which, after naming finitely many constants, have Scott rank $\omega _1^{CK}$. The existence of such structures was an open question. (shrink)

We study partitions of Fraïssé limits of classes of finite relational structures where the partitions are encoded by infinite binary strings which are random in the sense of Kolmogorov-Chaitin.

Computable model theory, also called effective or recursive model theory, studies algorithmic properties of mathematical structures, their relations, and isomorphisms. These properties can be described syntactically or semantically. One of the major tasks of computable model theory is to obtain, whenever possible, computability-theoretic versions of various classical model-theoretic notions and results. For example, in the 1950's, Fröhlich and Shepherdson realized that the concept of a computable function can make van der Waerden's intuitive notion of an explicit field precise. This led (...) to the notion of a computable structure. In 1960, Rabin proved that every computable field has a computable algebraic closure. However, not every classical result “effectivizes”. Unlike Vaught's theorem that no complete theory has exactly two nonisomorphic countable models, Millar's and Kudaibergenov's result establishes that there is a complete decidable theory that has exactly two nonisomorphic countable models with computable elementary diagrams. In the 1970's, Metakides and Nerode [58], [59] and Remmel [71], [72], [73] used more advanced methods of computability theory to investigate algorithmic properties of fields, vector spaces, and other mathematical structures. (shrink)

We say that a first order formula Φ distinguishes a structure M over a vocabulary L from another structure M' over the same vocabulary if Φ is true on M but false on M'. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M'. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure M'. We prove that every n-element (...) structure M is identifiable by a formula with quantifier rank less than (1- 1/2k)n+k²-k+4 and at most one quantifier alternation, where k is the maximum relation arity of M. Moreover, if the automorphism group of M contains no transposition of two elements, the same result holds for definability rather than identification. The Bernays-Schönfinkel class consists of prenex formulas in which the existential quantifiers all precede the universal quantifiers. We prove that every n-element structure M is identifiable by a formula in the Bernays-Schönfinkel class with less than (1-1/2k²+2)n+k quantifiers. If in this class of identifying formulas we restrict the number of universal quantifiers to k, then less than n-√ n+k²+k quantifiers suffice to identify M and, as long as we keep the number of universal quantifiers bounded by a constant, at total n-O(√ n) quantifiers are necessary. (shrink)

Every countable structure has a sentence of the infinitary logic $\mathcal {L}_{\omega _1 \omega }$ which characterizes that structure up to isomorphism among countable structures. Such a sentence is called a Scott sentence, and can be thought of as a description of the structure. The least complexity of a Scott sentence for a structure can be thought of as a measurement of the complexity of describing the structure. We begin with an introduction to (...) the area, with short and simple proofs where possible, followed by a survey of recent advances. (shrink)

This article explores the influence of textual structures on the acquisition of knowledge in popularization discourses related to biopesticides. Following a terminological insight into the linguistic and cognitive complexities of the notion, we proceed to a semantic analysis of press articles in major Anglo-Saxon newspapers, focusing on the explanation strategies used by the media to simplify their presentation. We show that in the mediation process, biopesticides are systematically described as being environmentally friendly, and opposed to chemical pesticides, consistently shown to (...) be detrimental to the environment. We hypothesize that this simplistic binary framing is exploited by journalists as it is easy to understand in a context of agroecological transition and commonly shared distrust for chemical pesticides. In that respect, it proves to be adequate for purposes of explanation and simplification. However, by outlining this discursive process based on duality, we aim to draw attention to the potential misconceptions of plant protection issues in general, and biopesticides in particular. We ultimately demonstrate that when adopting other perspectives, this dual framing proves to be inadequate for the construction of knowledge by any non-expert press reader. (shrink)

The Continuous Parameter Estimation Model is applied to develop individual genomic-level heritabilities for the latent hierarchical structure and developmental dynamics of Life History (LH) strategy LH strategies relate to the allocations of bioenergetic resources into different domains of fitness. LH has moderate to high population-level heritability in humans, both at the level of the high-order Super-K Factor and the lower-order factors, the K-Factor, Covitality Factor, and General Factor of Personality (GFP). Several important questions remain unexplored. We developed measures of (...) genome-level heritabilities employing an American sample of 316 monozygotic (MZ) and 274 dizygotic (DZ) twin dyads and a Swedish sample of 863 MZ and 475 DZ twin dyads. This novel heritability index measures individual genetic transmissibility, therefore opening new avenues for analyzing complex interactions among heritable traits inaccessible to standard structural equations methods. For these samples: (1) moderate to high heritability of factor loadings of Super-K on its lower-order factors is demonstrated, evidencing biological preparedness, genetic accommodation, and the gene-culture coevolution of biased epigenetic rules of development; (2) moderate to high heritability of the magnitudes of the effect of the higher-order factors upon their loadings on their constituent factors, evidencing genetic constraints upon phenotypic plasticity; and (3) that heritability of the LH factors, of factor loadings, and of the magnitudes of the correlations among factors are weaker among those with slower LH speeds, demonstrating that inter-individual variation in transmissibility is a function of individual socioecological selection pressures. (shrink)

This text addresses the problems of complex systems in understanding natural phenomena and the behaviour of systems related to human activity, from a science and humanities perspective. It discusses molecular behaviour and structures, and offers examples of ecological and environmental modelling.

Levin and Schnorr (independently) introduced the monotone complexity, Km(α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α ↾ n) ≤ Km(β ↾ n) + c for all n. The monotone degree of α is the set of all β such that α ≤Km β and (...) β ≤Km α. We show the monotone degrees contain an antichain of size 2N0, a countable dense linear ordering (of degrees of cardinality 2N0), and a minimal pair. Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K -degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K -degrees was constructed by Csima and Montalbán. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We define a real to be (Km, K)-trivial if for some constant c, Km(α ↾ n) ≤ K(n)+c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km, K)-trivial. Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering. (shrink)

Globalization is a ubiquitousyet highly elusive term. The debate on the cont and meaning of globalization is still waged largely in binary terms; for example, globalization is understood either as increasing standardization or as increasing difference. This article argues that the effects of globalization are best understood in terms of the following three sets of simultaneous contradictions: convergence and divergence, inclusion and exclusion, and centralization and decentralization. These contradictions can be fruitfully “unpacked” and examined through critical stakeholder analysis (CSA). (...) This technique focuses on broader structural commonalties and differences among key stakeholder groups, factors that are not incorporated in mainstream stakeholder theory. The associated normative agenda of CSA promotes a common understanding of key issues within a focal organizational field, the negotiated settlement of zero-sum situations that optimize collective stakeholder interests within that field, and the containment and minimization of negative externalities within as well as between fields. (shrink)

We show that for random bit strings, Up, with probability, image, the first order quantifier depth D) needed to distinguish non-isomorphic structures is Θ, with high probability. Further, we show that, with high probability, for random ordered graphs, G≤,p with edge probability image, D)=Θ, contrasting with the results for random graphs, Gp, given by Kim et al. [J.H. Kim, O. Pikhurko, J. Spencer, O. Verbitsky, How complex are random graphs in first order logic? Random Structures and Algorithms 26 119–145] of (...) D)=log1/pn+O. (shrink)

We show that for random bit strings, Up, with probability, image, the first order quantifier depth D) needed to distinguish non-isomorphic structures is Θ, with high probability. Further, we show that, with high probability, for random ordered graphs, G≤,p with edge probability image, D)=Θ, contrasting with the results for random graphs, Gp, given by Kim et al. [J.H. Kim, O. Pikhurko, J. Spencer, O. Verbitsky, How complex are random graphs in first order logic? Random Structures and Algorithms 26 119–145] of (...) D)=log1/pn+O. (shrink)

It has been suggested that biological theories differ from physical theories because the subject matter of biology differs from the subject matter of physics especially in the fact that living bodies are more complex than nonliving bodies. It is shown that the interactional complexity of living bodies can only be expressed by invoking biological theories. The claim that living bodies are complex is, therefore, ultimately a claim about the nature of scientific theories rather than a claim about the nature (...) of the subject matter of biology resting upon a presystematic judgement. (shrink)

The SRL of King is a sound, complete and decidable logic designed specifically to support formalisms for the HPSG of Pollard and Sag. The SRL notion of modellability in a signature is particularly important for HPSG, and the present paper modifies an elegant method due to Blackburn and Spaan in order to prove that – modellability in each computable signature is 1 0 – modellability in some finite signature is 1 0 -hard, and – modellability in some finite signature is (...) decidable. (shrink)

This thesis provides a comparative analysis of the different ways Hegel and Sartre understand that consciousness can be alienated. Because understanding the various ways Hegel and Sartre hold that consciousness can be alienated is not possible without first understanding what each thinker understands by consciousness, I first identify and outline the different ways Hegel and Sartre conceptualise consciousness’s ontological structure before identifying the various ways each thinker understands that consciousness can be alienated. The general argument developed shows that while (...) Hegel and Sartre agree that alienation is a constitutive aspect of consciousness’s existence and are, therefore, allies in the battle against it, Sartre’s analysis of consciousness’s ontological structure is unable to provide the same depth of analysis as Hegel’s. Put differently, I believe it is Hegel’s analysis of consciousness’s ontological structure that provides an analysis of alienation that is more nuanced, subtle, complex, and multi-dimensional than the account Sartre’s provides. To support my argument, I first explore Sartre’s understanding of consciousness’s ontological structure. This discloses that, because Sartre defines consciousness as ontologically nothing, he holds that consciousness is defined in strict ontological opposition to objectivity. Consciousness’s ontological nothingness leads Sartre to hold that consciousness is free to choose its mode of being. This leads me to identify what Sartre holds to be constitutive of authentic and inauthentic modes of being. But while Sartre distinguishes between the ontological structure of consciousness and its experiences, I argue that Hegel: 1) does not introduce a distinction between consciousness’s ontological structure and its mode of being, but holds that consciousness’s self-understanding and ontological structure develop through its experiences; and 2) holds that consciousness is not ontologically opposed to objectivity, but is a spiritual synthesis of subjectivity and objectivity. I show that because Hegel holds that consciousness’s intentional object is an aspect of its ontological structure, rather than something simply opposed to itself, and because he recognises that consciousness must learn what it is ontologically by experiencing numerous different relations with its object, he is able to show that while the subject/object binary opposition of Sartre’s analysis of consciousness’s ontological structure describes one potential ontological relation consciousness can have to its object, it is not the only one. Indeed, Hegel’s analysis of consciousness’s ontological structure insists that consciousness will only truly understand its ontological structure if it learns to not think of itself in terms of the subject/object dichotomy and, instead, realises that it is a spiritual synthesis of subjectivity and objectivity. To show how this fundamental difference manifests itself throughout their thought, I identify and compare what each thinker’s understanding of consciousness’s ontological structure means in terms of consciousness’s relation to the world, alienation, authenticity, ethics, self-transformation, and social relations. (shrink)

Real causal systems are complicated. Despite this, causal learning research has traditionally emphasized how causal relations can be induced on the basis of idealized events, i.e. those that have been mapped to binary variables and abstracted from time. For example, participants may be asked to assess the efficacy of a headache-relief pill on the basis of multiple patients who take the pill (or not) and find their headache relieved (or not). In contrast, the current study examines learning via interactions (...) with continuous dynamic systems, systems that include continuous variables that interact over time (and that can be continuously observed in real time by the learner). To explore such systems, we develop a new framework that represents a causal system as a network of stationary Gauss--Markov (``\OU'') processes and show how such \textit{OU networks} can express complex dynamic phenomena such as feedback loops and oscillations. To assess adult's abilities to learn such systems, we conducted an experiment in which participants were asked to identify the causal relationships of a number of OU networks, potentially carrying out multiple, temporally-extended interventions. We compared their judgments to a normative model for learning OU networks as well as a range of alternative and heuristic learning models from the literature. We found that, although participants exhibited substantial learning of such systems, they committed certain systematic errors. These successes and failures were best accounted for by a model that describes people as focusing on pairs of variables, rather than evaluating the evidence with respect to the full space of possible structural models. We argue that our approach provides both a principled framework for exploring the space of dynamic learning environments as well as new algorithmic insights into how people interact successfully with a continuous causal world. (shrink)

This paper studies algorithmic learning theory applied to algebraic structures. In previous papers, we have defined our framework, where a learner, given a family of structures, receives larger and larger pieces of an arbitrary copy of a structure in the family and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. The learning is successful if there is a learner that eventually stabilizes to a correct conjecture. Here, we analyze the (...) number of mind changes that are needed to learn a given family K. We give a descriptive set-theoretic interpretation of such mind change complexity. We also study how bounding the Turing degree of learners affects the mind change complexity of a given family of algebraic structures. (shrink)

Background:Moral courage is defined as courage to act according to one’s own ethical values and principles even at the risk of negative consequences for the individual. In a complex nursing practice, ethical considerations are integral. Moral courage is needed throughout nurses’ career.Aim:To analyse graduating nursing students’ moral courage and the factors associated with it in six European countries.Research design:A cross-sectional design, using a structured questionnaire, as part of a larger international ProCompNurse study. In the questionnaire, moral courage was assessed with (...) a single question (visual analogue scale 0–100), the questionnaire also covered several background variables.Participants and research context:The sample comprised graduating nursing students (n = 1796) from all participating countries. To get a comprehensive view about graduating nursing students’ moral courage, the views of nurse managers (n = 538) and patients (n = 1327) from the same units in which the graduating nursing students practised were also explored, with parallel questionnaires.Ethical considerations:Ethical approvals and research permissions were obtained according to national standards in every country and all participants gave their informed consent.Results:The mean of graduating nursing students’ self-assessed moral courage was 77.8 (standard deviation 17.0; on a 0–100 scale), with statistically significant differences between countries. Higher moral courage was associated with many factors, especially the level of professional competence. The managers assessed the graduating nursing students’ moral courage lower (66.5; standard deviation 18.4) and the patients slightly higher (80.6; standard deviation 19.4) than the graduating nursing students themselves.Discussion and conclusions:In all countries, the graduating nursing students’ moral courage was assessed as rather high, with differences between countries and populations. These differences and associations between moral courage and ethics education require further research. (shrink)

Current formal dialectical models postulate normative rules that enable discussants to conduct dialogical interactions without committing fallacies. Though the rules for conducting a dialogue are supposed to apply to interactions between actual arguers, they are without exception theoretically motivated. This creates a gap between model and reality, because dialogue participants typically leave important content-related elements implicit. Therefore, analysts cannot readily relate normative rules to actual debates in ways that will be empirically confirmable. This paper details a new, data-driven method for (...) describing discussants’ actual reply structures, wherein corpus studies serve to acknowledge the complexity of natural argumentation. Rather than refer exclusively to propositional content as an indicator of arguing pro/contra a given claim, the proposed approach to dialogue structure tracks the sequence of dialogical moves itself. This arguably improves the applicability of theoretical dialectical models to empirical data, and thus advances the study of dialogue systems. (shrink)

In previous work, we have combined computable structure theory and algorithmic learning theory to study which families of algebraic structures are learnable in the limit (up to isomorphism). In this paper, we measure the computational power that is needed to learn finite families of structures. In particular, we prove that, if a family of structures is both finite and learnable, then any oracle which computes the Halting set is able to achieve such a learning. On the other hand, we (...) construct a pair of structures which is learnable but no computable learner can learn it. (shrink)

Limiting identification of r.e. indexes for r.e. languages and limiting identification of programs for computable functions have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of "intrinsic" complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, (...) Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification. The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages. Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open. (shrink)

We investigate the computational complexity the class of Γ-categorical computable structures. We show that hyperarithmetic categoricity is Π11-complete, while computable categoricity is Π04-hard.

Limiting identification of r.e. indexes for r.e. languages (from a presentation of elements of the language) and limiting identification of programs for computable functions (from a graph of the function) have served as models for investigating the boundaries of learnability. Recently, a new approach to the study of "intrinsic" complexity of identification in the limit has been proposed. This approach, instead of dealing with the resource requirements of the learning algorithm, uses the notion of reducibility from recursion theory to (...) compare and to capture the intuitive difficulty of learning various classes of concepts. Freivalds, Kinber, and Smith have studied this approach for function identification and Jain and Sharma have studied it for language identification. The present paper explores the structure of these reducibilities in the context of language identification. It is shown that there is an infinite hierarchy of language classes that represent learning problems of increasing difficulty. It is also shown that the language classes in this hierarchy are incomparable, under the reductions introduced, to the collection of pattern languages. Richness of the structure of intrinsic complexity is demonstrated by proving that any finite, acyclic, directed graph can be embedded in the reducibility structure. However, it is also established that this structure is not dense. The question of embedding any infinite, acyclic, directed graph is open. (shrink)

We consider the complexity (in terms of the arithmetical hierarchy) of the various quantifier levels of the diagram of a computably presented metric structure. As the truth value of a sentence of continuous logic may be any real in [0, 1], we introduce two kinds of diagrams at each level: the closed diagram, which encapsulates weak inequalities of the form $$\phi ^\mathcal {M}\le r$$, and the open diagram, which encapsulates strict inequalities of the form $$\phi ^\mathcal {M}< r$$. (...) We show that the closed and open $$\Sigma _N$$ diagrams are $$\Pi ^0_{N+1}$$ and $$\Sigma ^0_N$$ respectively, and that the closed and open $$\Pi _N$$ diagrams are $$\Pi ^0_N$$ and $$\Sigma ^0_{N + 1}$$ respectively. We then introduce effective infinitary formulas of continuous logic and extend our results to the hyperarithmetical hierarchy. Finally, we demonstrate that our results are optimal. (shrink)

Ethics is a foundational competency in healthcare inherent in everyday nursing practice. Therefore, the promotion of qualified nurses’ and nursing students’ moral competence is essential to ensure ethically high-quality and sustainable healthcare. The aim of this integrative literature review is to identify the factors contributing to the promotion of qualified nurses’ and nursing students’ moral competence. The review has been registered in PROSPERO (CRD42023386947) and reported according to the PRISMA guideline. Focusing on qualified nurses’ and nursing students’ moral competence, a (...) literature search was undertaken in January 2023 in six scientific databases: CINAHL, Cochrane Library, PsycINFO, PubMed Medline, Scopus and Web of Science. Empirical studies written in English without time limitation were eligible for inclusion. A total of 29 full texts were retrieved and included out of 5233 citations. Quality appraisal was employed using Joanna Briggs Institute checklists and the Mixed Method Appraisal Tool. Data were analysed using inductive content analysis. Research about the factors contributing to the promotion of qualified nurses’ and nursing students’ moral competence is limited and mainly explored using descriptive research designs. The contributing factors were identified as comprising two main categories: (1) human factors, consisting of four categories: individual, social, managerial and professional factors, and ten sub-categories; and (2) structural factors, consisting of four categories: educational, environmental, organisational and societal factors, and eight sub-categories. This review provides knowledge about the factors contributing to the promotion of qualified nurses’ and nursing students’ moral competence for the use of researchers, nurse educators, managers, organisations and policymakers. More research about the contributing factors is needed using complex intervention, implementation and multiple methods designs to ensure ethically sustainable healthcare. (shrink)

We calculate the possible Scott ranks of countable models of Peano arithmetic. We show that no non-standard model can have Scott rank less than $\omega $ and that non-standard models of true arithmetic must have Scott rank greater than $\omega $. Other than that there are no restrictions. By giving a reduction via $\Delta ^{\mathrm {in}}_{1}$ bi-interpretability from the class of linear orderings to the canonical structural $\omega $ -jump of models of an arbitrary completion T of $\mathrm {PA}$ we (...) show that every countable ordinal $\alpha>\omega $ is realized as the Scott rank of a model of T. (shrink)