We introduce notions of strong and eventual strong non-isolation for types in countable, stable theories. For T superstable or small stable we prove a dichotomy theorem: a regular type over a finite domain is either eventually strongly non-isolated or is non-orthogonal to a NENI type . As an application we obtain the upper bound for Lascar’s rank of a superstable theory which is one-based or trivial, and has fewer than 20 non-isomorphic countable models.

We prove that if T is a stable theory with only a finite number of countable models, then T contains a type-definable pseudoplane. We also show that for any stable theory T either T contains a type-definable pseudoplane or T is weakly normal.

We define a generalized notion of rank for stable theories without dense forking chains, and use it to derive that every type is domination-equivalent to a finite product of regular types. We apply this to show that in a small theory admitting finite coding, no realisation of a nonforking extension of some strong type can be algebraic over some realisation of a forking extension.

Semiotic Theory of Learning asks what learning is and what brings it about, challenging the hegemony of psychological and sociological constructions of learning in order to develop a burgeoning literature in semiotics as an educational foundation. Drawing on theoretical research and its application in empirical studies, the book attempts to avoid the problematization of the distinction between theory and practice in semiotics. It covers topics such as signs, significance and semiosis; the ontology of learning; the limits of learning; (...) ecosemiotics; ecology and sexuality. -/- The book is written by five of the key figures in the semiotics field, each committed to the belief that living is a process of interaction through acts of signification with a signifying environment. While the authors are agreed on the value of semiotic frameworks, the book aims not to present an entirely coherent line in every respect, but rather to reflect ongoing scholarship and debates in the area. In light of this, the book offers a range of possible interpretations of major semiotic theorists, unsettling assumptions while offering a fresh, and still developing, series of perspectives on learning from academics grounded in semiotics. -/- Semiotic Theory of Learning is a timely and valuable text that will be of great interest to academics, researchers and postgraduates working in the fields of educational studies, semiotics, psychology, philosophy, applied linguistics and media studies. (shrink)

We use exact saturation to study the complexity of unstable theories, showing that a variant of this notion called pseudo-elementary class (PC)-exact saturation meaningfully reflects combinatorial dividing lines. We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals satisfying mild set-theoretic hypotheses. This had previously (...) been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analog of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories. (shrink)

_Edusemiotics_ addresses an emerging field of inquiry, educational semiotics, as a philosophy of and for education. Using "sign" as a unit of analysis, educational semiotics amalgamates philosophy, educational theory and semiotics. Edusemiotics draws on the intellectual legacy of such philosophers as John Dewey, Charles Sanders Peirce, Gilles Deleuze and others across Anglo-American and continental traditions. This volume investigates the specifics of semiotic knowledge structures and processes, exploring current dilemmas and debates regarding self-identity, learning, transformative and lifelong education, leadership and (...) policy-making, and interrogating an important premise that still haunts contemporary educational philosophy: Cartesian dualism. In defiance of substance dualism and the fragmentation of knowledge that still inform education, the book offers a unifying paradigm for education as edusemiotics and emphasises ethical education in compliance with the semiotic unity between knowledge and action. Chapters contain accessible discussions in the context of educational philosophy and theory, crossing the borders between logic, art, and science together with a provocative theoretical critique. Recently awarded a PESA book award for its contribution to the philosophy of education, _Edusemiotics_ will appeal to an academic readership in education, philosophy and cultural studies, while also being an inspiring resource for students. (shrink)

In this essay, Andrew Stables notes that philosophies such as existentialism, humanism, and environmentalism come in either exploratory or active forms: that is, one can study the nature of existence or the human, or one can ascribe to a way of life in an attempt to improve the world. Among the major influences on active environmentalist thought are humanism, socialism, posthumanism, and post- colonialism. In many cases, however, such ways of thinking can be as damaging or unsuccessful as they may (...) be ameliorative and successful. The search for an environmentalist Way, Stables maintains, must transcend entrenched binaries, oppositions, and ideologies without lapsing into thoroughgoing antirealism or willfully ignoring current realities. Here, he explores the valuable contributions that Daoist and Buddhist texts and practices have to offer in this search. (shrink)

Kueker's conjecture is proved for stable theories, for theories that interpret a linear ordering, and for theories with Skolem functions. The proof of the stable case involves certain results on coordinatization that are of independent interest.

We study groups definable in tame expansions of ω-stable theories. Assuming several tameness conditions, we obtain structural theorems for groups definable and interpretable in these expansions. As our main example, by characterizing independence in the pair, where K is an algebraically closed field and G is a multiplicative subgroup of K× with the Mann property, we show that the pair satisfies the assumptions. In particular, this provides a characterization of definable and interpretable groups in in terms of algebraic groups (...) in K and interpretable groups in G. Furthermore, we compute the Morley rank and the U-rank in and both ranks agree. (shrink)

We construct a type p with preweight ω with respect to itself in a theory with few types. A type with this property must be present in a stable theory with finitely many (but more than one) countable models. The construction is a modification of Hrushovski's important pseudoplane construction.

We prove that every relation-universal model of an ω-stable theory is saturated. We also show there is a large class of ω-stable theories for which every resplendent model is homogeneous.

In this dissertation we study two-cardinal phenomena—both of the admitting cardinals variety and of the Chang’s Conjecture variety—under the assumption that all our models have stable theories. All our results involve two, relatively widely accepted generalizations of the traditional definitions in this area. First, we allow the relevant subsets of our models to be picked out by (perhaps infinitary) partial types; second we consider δ-cardinal problems as well as two-cardinal problems.

In The Song of the Earth, Jonathan Bate promotes ‘ecopoesis’, contrasting it with ‘ecopolitical’ poetry (and by implication, other forms of writing and expression). Like others recently, including Simon James and Michael Bonnett, he appropriates the notion of ‘dwelling’ from Heidegger to add force to this distinction. Bate's argument is effectively that we have more chance of protecting the environment if we engage in ecopoetic activity, involving a sense of immediate response to nature, than if we do not. This has (...) obvious educational implications. If Bate, James and Bonnett are correct, then the educational pursuit of (eco)poetic sensibility will, of itself, contribute to education for a sustainable future by grounding human experience in nature; if their assertions are insupportable, and (eco)poetic sensibility does not afford privileged access to a state of nature, then the assumption cannot be made that the development of such sensibility will contribute to education for sustainability. I shall critique Bate's argument from a pragmatic perspective. (shrink)

We define a hierarchy on the regular types of an ω-stable nonmultidimensional theory, using generalised notions of algebraic and strongly minimal formulae. As an application we show that any resplendent model of an ω-stable finite-dimensional theory is saturated.

We work in the context of ω-stable theories. We obtain a natural, algebraic equivalent of ENI-NDOP and discuss recent joint proofs with Shelah that if an ω-stable theory has either ENI-DOP or is ENI-NDOP and is ENI-deep, then the set of models of T with universe ω is Borel complete.

This paper aims at revealing the various meanings of schools as more than built physical environments from a geographical-phenomenological (or ‘geo-phenomenological’) perspective. This paper consists of five sections: the first explicates the meaning of ‘geo-phenomenology’; the second reveals the meaning of ‘environment’ and a dialectics of strangeness and intimacy through geo-phenomenological analysis; the third examines the meanings of environment as ‘space’ and ‘place’ and the act of naming as the process of constructing meaning between humans and environment; the fourth section (...) attempts to explore the meaning of conceiving school as a particular environment; and the final is the conclusion. (shrink)

Building on Hrushovski's work in [5], we study definable groupoids in stable theories and their relationship with 3-uniqueness and finite internal covers. We introduce the notion of retractability of a definable groupoid (which is slightly stronger than Hrushovski's notion of eliminability), give some criteria for when groupoids are retractable, and show how retractability relates to both 3-uniqueness and the splitness of finite internal covers. One application we give is a new direct method of constructing non-eliminable groupoids from witnesses to (...) the failure of 3-uniqueness. Another application is a proof that any finite internal cover of a stable theory with a centerless liaison groupoid is almost split. (shrink)

Given the tendency for the "lifeworld approach" to be adopted in the domain of environmental theory and education without critical examination of the key concept "lifeworld", this paper attempts to elucidate the ambiguity apparent in Husserl's development of the notion and the implications of this for teaching and learning about nature. The paper consists of three sections. The first section deals with the meaning and limitations of the current lifeworld approach to nature and the implications for environmental and educational (...) thinking. In the second section, the confusion surrounding the concept of lifeworld is traced back to the later Husserl's philosophy. Exploring the meaning of lifeworld in Husserl's philosophy reveals that there may be two lifeworld orientations: one is explicit and objective in its emphasis on the shared and universal; the other is implicit and subjective in its emphasis on the idiosyncratically personal. The final section argues that the implicit and subjective orientation of lifeworld may be more tenable experientially, and as such more conducive to helping environmental and educational thinkers envisage an attentive and responsive approach to teaching and learning about nature. (shrink)

This study examines the relationship between teachers' beliefs and their practices at Key Stage 2 (ages 7-11) in relation to the use of questioning. Data were collected from interviewing and observing Key Stage 2 teachers at four schools in the West of England. A Straussian approach to grounded theory is followed broadly in order to analyse the data. In contrast to the findings of previous studies, which suggested a mismatch between teachers' beliefs and practices in that teachers, in certain (...) respects, do less than they claim, the research revealed that teachers use a variety of skills during their teaching that they may not always be aware of. It is also argued that teachers do not share researchers' language to express the way they teach. (shrink)

We study the class of all strongly constructivizable models having $$\omega $$ -stable theories in a fixed finite rich signature. It is proved that the Tarski–Lindenbaum algebra of this class considered together with a Gödel numbering of the sentences is a Boolean $$\Sigma ^1_1$$ -algebra whose computable ultrafilters form a dense subset in the set of all ultrafilters; moreover, this algebra is universal with respect to the class of all Boolean $$\Sigma ^1_1$$ -algebras. This gives a characterization to the (...) Tarski-Lindenbaum algebra of the class of all strongly constructivizable models with $$\omega $$ -stable theories. (shrink)

In "A proof of Vaught's conjecture for ω-stable theories," the notions of ENI-NDOP and eni-depth have been introduced, which are variants of the notions of NDOP and depth known from Shelah's classification theory. First, we show that for an ω-stable first-order complete theory, ENI-NDOP allows tree decompositions of countable models. Then we discuss the relationship between eni-depth and the complexity of the isomorphism relation for countable models of such a theory in terms of Borel reducibility (...) as introduced by Friedman and Stanley and construct, in particular, a sequence of complete first-order ω-stable theories $(T_\alpha)_{\alpha < \omega_1}$ with increasing and cofinal eni-depth and isomorphism relations which are strictly increasing with respect to Borel reducibility. (shrink)

We show that in a stable first-order theory, the failure of higher dimensional type amalgamation can always be witnessed by algebraic structures that we call n-ary polygroupoids. This generalizes a result of Hrushovski in [16] that failures of 4-amalgamation are witnessed by definable groupoids. The n-ary polygroupoids are definable in a mild expansion of the language.

Recent work on formal theories of truth has revived an approach, due originally to Tarski, on which syntax and truth theories are sharply distinguished—‘disentangled’—from mathematical base theories. In this paper, we defend a novel philosophical constraint on disentangled theories. We argue that these theories must be epistemically stable: they must possess an intrinsic motivation justifying no strictly stronger theory. In a disentangled setting, even if the base and the syntax theory are individually stable, they may be (...) jointly unstable. We contend that this flaw afflicts many proposals discussed in the literature; we defend a new, stable disentangled theory, double second-order arithmetic. (shrink)

For a NIP theory T, a sufficiently saturated model ${\mathfrak C}$ of T, and an invariant (over some small subset of ${\mathfrak C}$ ) global type p, we prove that there exists a finest relatively type-definable over a small set of parameters from ${\mathfrak C}$ equivalence relation on the set of realizations of p which has stable quotient. This is a counterpart for equivalence relations of the main result of [2] on the existence of maximal stable quotients (...) of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in [3]. (shrink)

We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U þ -rank 1 must arise from definable stable sets.

In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.

Stable adaptive strategy of Homo sapiens (SESH) is a superposition of three different adaptive data arrays: biological, socio-cultural and technological modules, based on three independent processes of generation and replication of an adaptive information – genetic, socio-cultural and symbolic transmissions (inheritance). Third component SESH focused equally to the adaptive transformation of the environment and carrier of SESH. With the advent of High Hume technology, risk has reached the existential significance level. The existential level of technical risk is, by definition, (...) an evolutionary risk as possible leads to the genesis of disappearance of humanity as a species. The emergence of bioethics has to consider as a form of modern (transdisciplinary) scientific concept and sociocultural adaptation for regulate human identity in the global-evolutionary transformation and performs the function of self-preservation. (shrink)

We consider various concepts associated with the revision theory of truth of Gupta and Belnap. We categorize the notions definable using their theory of circular definitions as those notions universally definable over the next stable set. We give a simplified account of varied revision sequences-as a generalised algorithmic theory of truth. This enables something of a unification with the Kripkean theory of truth using supervaluation schemes.

Kojman, M. and S. Shelah, The universality spectrum of stable unsuperstable theories, Annals of Pure and Applied Logic 58 57–72. It is shown that if T is stable unsuperstable, and 1 [brvbar]T[brvbar], T stable and κ<κ then there is a universal tree of height κ + 1 in cardinality λ.

Part IIb presents some of the most important theorems for stable equilibrium states that can be deduced from the four postulates of the unified theory presented in Part I. It is shown for the first time that the canonical and grand canonical distributions are the only distributions that are stable. Moreover, it is shown that reversible adiabatic processes exist which cannot be described by the dynamical equation of quantum mechanics. A number of conditions are discussed that must (...) be satisfied by the general equation of motion which is yet to be discovered. (shrink)

We show that every reduct of a stable. CM-trivial theory of finite U-rank is CM-trivial.

Let M be an arbitrary structure. Then we say that an M -formula φ defines a stable set inM if every formula φ ∧ α is stable. We prove: If G is an M -definable group and every definable stable subset of G has U -rank at most n , then G has a maximal connected stable normal subgroup H such that G /H is purely unstable. The assumptions hold for example if M is interpretable in (...) an o-minimal structure.More generally, an M -definable set X is weakly stable if the M -induced structure on X is stable. We observe that, by results of Shelah, every weakly stable set in theories with NIP is stable. (shrink)