It is known Hrushovski and Pillay that a group G definable in the field \ of p-adic numbers is definably locally isomorphic to the group \\) of p-adic points of a algebraic group H over \. We observe here that if H is commutative then G is commutative-by-finite. This shows in particular that any one-dimensional group G definable in \ is commutative-by-finite. This result extends to groups definable in p-adically closed fields. We prove our results in the more general (...) context of geometric structures. (shrink)

As clinicians, this case1 raises both personal and professional challenges. A key issue is who carries legal parental responsibility for the difficult decisions that may be required around life-sustaining care in baby T. Medicolegally, we understand that the surrogate mother holds legal parental responsibility for baby T until this can be transferred to the intended parents.2 But this process can take many months to complete, after the birth of baby. As M is now critically ill and unable to engage in (...) any discussion around the care of her baby, who becomes the legal guardian of the baby, for complex decision making that involves either reorientation of care away from intensive care with the inevitable consequence of death in this extremely premature baby, or continuing this life-sustaining treatment with a high likelihood for major neonatal morbidity and longer term disability? Does it pass onto one of the IPs as the genetic father, or does this responsibility fall on the neonatal consultant managing the baby in intensive care? And if an antenatal surrogacy agreement had existed, is it legally binding for inclusion of the IPs perspectives in the decision-making, especially if our medical decision making turns out to be at discord with the beliefs and wishes of the IPs for baby T? Box 1 ### Current surrogacy proceedings in the UK.2 6–8 The Law Commission UK5 is currently reviewing the decision on whether legal parenthood to intended parents can be effected immediately at the time of birth of the baby. Parental orders. (shrink)

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof.Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure (...) and the latter the archetypal stable structure. In this sense we try here to situate our work ono-minimal structures [PS] in a general topological context. Note, however, that thep-adic numbers, and structures definable therein, will also fit into our analysis.In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions. (shrink)

The COVID-19 challenge is unprecedented. It has caused enormous trauma, disrupted economies, social life, mass transportation, work and employment, supply chains, leisure, sport, international relations, academic programmes; literally everything. Churches and religious communities have not been spared; they have been severely affected and, in all likelihood, permanently transformed by the pandemic. The pre-COVID-19 world is gone, replaced by a ‘new normal’. The new landscape calls for both resilience and adaptation, embracing new ways of doing things and of being church. Churches (...) have to adapt; they have to ask themselves questions about the implications for being church in this ‘new normal’ context. This article aims to explore the impact of the coronavirus on the mission and theology of the church. (shrink)

Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.

We give a commentary on Newelski's suggestion or conjecture [8] that topological dynamics, in the sense of Ellis [3], applied to the action of a definable group $G(M)$ on its “external type space” $S_{G,\textit{ext}}(M)$, can explain, account for, or give rise to, the quotient $G/G^{00}$, at least for suitable groups in NIP theories. We give a positive answer for measure-stable (or $fsg$) groups in NIP theories. As part of our analysis we show the existence of “externally definable” generics of $G(M)$ (...) for measure-stable groups. We also point out that for $G$ definably amenable (in a NIP theory) $G/G^{00}$ can be recovered, via the Ellis theory, from a natural Ellis semigroup structure on the space of global $f$-generic types. (shrink)

We point out that the theory of difference fields with algebraically closed fixed field has no model companion.

We give some sufficient conditions for a predicate P in a complete theory T to be "stably embedded". Let P be P with its "induced θ-definable structure". The conditions are that P (or rather its theory) is "rosy", P has NIP in T and that P is stably 1-embedded in T. This generalizes a recent result of Hasson and Onshuus [6] which deals with the case where P is o-minimal in T. Our proofs make use of the theory of strict (...) nonforking and weight in NIP theories ([3], [10]). (shrink)

We prove that any field definable in (Q p, +, ·) is definably isomorphic to a finite extension ofQ p.

We study type-definable subgroups of small index in definable groups, and the structure on the quotient, in first order structures. We raise some conjectures in the case where the ambient structure is o-minimal. The gist is that in this o-minimal case, any definable group G should have a smallest type-definable subgroup of bounded index, and that the quotient, when equipped with the logic topology, should be a compact Lie group of the "right" dimension. I give positive answers to the conjectures (...) in the special cases when G is 1-dimensional, and when G is definably simple. (shrink)

We introduce and study the notions of a PAC-substructure of a stable structure, and a bounded substructure of an arbitrary substructure, generalizing [10]. We give precise definitions and equivalences, saying what it means for properties such as PAC to be first order, study some examples (such as differentially closed fields) in detail, relate the material to generic automorphisms, and generalize a "descent theorem" for pseudo-algebraically closed fields to the stable context. We also point out that the elementary invariants of pseudo-algebraically (...) closed fields from [6] are also valid for pseudo-differentially closed fields. (shrink)

The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, and (...) the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (shrink)

The Reformed tradition, emerging in the 16th-century Reformation, consists of a variety of sources that often lead to complex and differing views about beliefs, doctrines and ethics. However, this tradition and theology have always stressed the significance of social justice and diakonia as important aspects of faith and ministry, even though its great sense of diversity has often nuanced and stressed different levels of understanding and engagement of social justice. This article aims to show that social justice and diakonia are (...) integral to Reformed tradition and practice. Using mainly the methodologies of literature review and contextuality (the author’s context), this article establishes that social justice is grounded in the history, theology, spirituality, confessions and polity of the Reformed faith. The latter aspects are also contained in the notion that to be Reformed is to be ecumenical. In this sense, Reformed tradition is concerned about the whole world and all creation. Contribution: The diverse and complex nature of Reformed tradition and theology often creates a sense of confusion on how Reformed Christians understand social justice and diakonia. This article offers a significant contribution to establishing that social justice is an integral aspect of Reformed tradition. By firmly grounding social justice in the history, theology, spirituality, confessions and polity of the Reformed faith, the author makes a significant contribution to a debate that has pervaded Reformed churches over many centuries. (shrink)

We continue the study of simple theories begun in [3] and [5]. We first find the right analogue of definability of types. We then develop the theory of generic types and stabilizers for groups definable in simple theories. The general ideology is that the role of formulas (or definability) in stable theories is replaced by partial types (or ∞-definability) in simple theories.

In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank (...) context than [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid. (shrink)

Les corps algébriquement clos, réels clos et pseudo-finis n'ont, pour chaque entier n, qu'un nombre fini d'extensions de degré n; nous montrons qu'ils partagent cette propriété avec tous les corps qui, comme eux, satisfont une propriété très rudimentaire de préservation de la dimension, de nature modèle-théorique. Ce résultat est atteint en montrant qu'une certaine action du groupe GLn d'un tel corps n'a qu'un nombre fini d'orbites. /// La korpoj algebre fermataj, reale fermataj kaj pseudofinataj ne havas, pri ciu integro n, (...) pli ol finitan nombron de sternejoj kun grado n; ni montras ke ili kunpartigas tiu ci propecon kun ciu tiel korpoj kiuj kontentigas kiel ili unu tre simplan modelteorian propecon de konserveco de la amplekso. Pri atingi la resultaton, ni montras ke kelka ago de la grupo GLn de tiel korpo havas sole finitan nombron de orbitoj. (shrink)

This article traces the historical impact of the church in transforming, developing and changing society. It looks at how the church in selected periods in history, mainly in the reformation era, worked towards the transformation of society and communities. Extracting from these the author attempts to show how this can be applied to the church in the South African context. Further, the author explores the changing dynamics and characteristics of being church today and establishes that it is not so much (...) about what the church believes, but in what it does that matters most in this day and age. The author, with the latter thought in mind, expresses that community transformation has to be an integral part of the ministry of the church today. (shrink)

CM-triviality of a stable theory is a notion introduced by Hrushovski [1]. The importance of this property is first that it holds of Hrushovski's new non 1-based strongly minimal sets, and second that it is still quite a restrictive property, and forbids the existence of definable fields or simple groups (see [2]). In [5], Frank Wagner posed some questions aboutCM-triviality, asking in particular whether a structure of finite rank, which is “coordinatized” byCM-trivial types of rank 1, is itselfCM-trivial. (Actually Wagner (...) worked in a slightly more general context, adapting the definitions to a certain “local” framework, in which algebraic closure is replaced byP-closure, forPsome family of types. We will, however, remain in the standard context, and will just remark here that it is routine to translate our results into Wagner's framework, as well as to generalise to the superstable theory/regular type context.) In any case we answer Wagner's question positively. Also in an attempt to put forward some concrete conjectures about the possible geometries of strongly minimal sets (or stable theories) we tentatively suggest a hierarchy of geometric properties of forking, the first two levels of which correspond to 1-basedness andCM-triviality respectively. We do not know whether this is a strict hierarchy (or even whether these are the “right” notions), but we conjecture that it is, and moreover that a counterexample to Cherlin's conjecture can be found at level three in the hierarchy. (shrink)

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.

The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as well (...) as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint, arXiv:1201.5221v1]. (shrink)

We consider the theory P of pairs F (...) is generically free. (shrink)

Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L * be L together with a unary predicate P. Let T * be the L * -theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an |M| + -saturated elementary extension of N (and (...) M is the interpretation of P). Using the definability of types result, we show that T * is complete and we give a simple set of axioms for T * . We also show that for every L * -formula φ(x) there is an L-formula ψ(x) such that $T^\ast \models (\forall \mathbf{x})(P(\mathbf{x}) \rightarrow (\phi(\mathbf{x}) \mapsto \psi (\mathbf{x}))$ . This yields the following result: Let M be a Dedekind complete model of T. Let φ(x, y) be an L-formula where l(y) = k. Let $\mathbf{X} = \{X \subset M^k$ : for some a in an elementary extension N of M, X = φ (a,y N ∩ M k }. Then there is a formula ψ(y, z) of L such that X = {ψ (y, b) M : b in M}. (shrink)