Generalising work of Berenstein, Dolich and Onshuus (Preprint 145 on MODNET Preprint server, 2008) and Günaydın and Hieronymi (Preprint 146 on MODNET Preprint server, 2010), we give sufficient conditions for a theory TP to inherit N I P from T, where TP is an expansion of the theory T by a unary predicate P. We apply our result to theories, studied by Belegradek and Zilber (J. Lond. Math. Soc. 78:563–579, 2008), of the real field with a subgroup of the unit (...) circle. (shrink)

A field K in a ring language $\mathcal {L}$ is finitely undecidable if $\mbox {Cons}(T)$ is undecidable for every nonempty finite $T \subseteq {\mathtt{Th}}(K; \mathcal {L})$. We extend a construction of Ziegler and (among other results) use a first-order classification of Anscombe and Jahnke to prove every NIP henselian nontrivially valued field is finitely undecidable. We conclude (assuming the NIP Fields Conjecture) that every NIP field is finitely undecidable. This work is drawn from the author’s PhD thesis [48, Chapter 3].

We show that any theory of tame henselian valued fields is NIP if and only if the theory of its residue field and the theory of its value group are NIP. Moreover, we show that if is a henselian valued field of residue characteristic \=p\) such that if \, depending on the characteristic of K either the degree of imperfection or the index of the pth powers is finite, then is NIP iff Kv is NIP and v is roughly separably (...) tame. (shrink)

The derivation on the differential-valued field Tlogof logarithmic transseries induces on its value group${{\rm{\Gamma }}_{{\rm{log}}}}$a certain mapψ. The structure${\rm{\Gamma }} = \left$is a divisible asymptotic couple. In [7] we began a study of the first-order theory of$\left$where, among other things, we proved that the theory$T_{{\rm{log}}} = Th\left$has a universal axiomatization, is model complete and admits elimination of quantifiers in a natural first-order language. In that paper we posed the question whetherTloghas NIP. In this paper, we answer that question in the (...) affirmative:Tlogdoes have NIP. Our method of proof relies on a complete survey of the 1-types ofTlog, which, in the presence of QE, is equivalent to a characterization of all simple extensions${\rm{\Gamma }}\left\langle \alpha \right\rangle$of${\rm{\Gamma }}$. We also show thatTlogdoes not have the Steinitz exchange property and we weigh in on the relationship between models ofTlogand the so-calledprecontraction groupsof [9]. (shrink)

Speaking of dental floss contaminated with bacteria, I may separate the dental floss that is sterile from the dental floss that isn’t sterile. The definite description “the dental floss that isn’t sterile” contracts its reference to just the dental floss near bacteria, although it, the dental floss whole, isn’t sterile. To accommodate the definite descriptions that contract their reference, received definitions for ⌜the Φ⌝ are amended from to read as in : ⌜the Φ⌝ refers to that which any Φ is (...) part of and is the least such. ⌜the Φ⌝ refers to that which any Φ overlaps and is the least such. If definite description is to be based on a purely logical notion of plural and mass predication, it is further amended. Like overt demonstratives—⌜this/these Φ⌝, ⌜that/those Φ⌝—any definite description in natural language is also perspectival, scanning everywhere the description Φ is satisfied:⌜the Φ⌝ refers to the least Φ that overlap Φ anywhere there is Φ. (shrink)

The following conjecture is due to Shelah–Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non‐trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

We study one way in which stable phenomena can exist in an NIP theory. We start by defining a notion of ‘pure instability’ that we call ‘distality’ in which no such phenomenon occurs. O-minimal theories and the p-adics for example are distal. Next, we try to understand what happens when distality fails. Given a type p over a sufficiently saturated model, we extract, in some sense, the stable part of p and define a notion of stable independence which is implied (...) by non-forking and has bounded weight. (shrink)

In recent years, two schools of thought have emerged with regard to the Cambrian “explosion”. One argues that it was very quick, with phyla tumbling into existence in a virtual geological instant. The other view has a more relaxed temporal perspective. It looks to slow aeons of cryptic metazoan history, which led to a final breakthrough in the Cambrian, not in evolution but of fossilization potential. Yet both views have serious difficulties. Now, in a recent issue of Biological Reviews, Graham (...) Budd and Sören Jensen(1) argue for a third way. In an intriguing blend of functional morphology, the fossil record and cladistic thinking, they suggest that the assembly of metazoan bodyplans took place in a surprisingly straightforward manner. BioEssays 22:1053–1056, 2000. © 2000 John Wiley & Sons, Inc. (shrink)

‘Disease interception’ describes the treatment of a disease in its clinically inapparent phase and is increasingly used in medical literature. However, no precise definition, much less an ethical evaluation, has been developed yet. This article starts with a definition of ‘disease interception’ by distinguishing it from other preventions. It then analyses the ethical and social implications of the concept in light of the four principles of medical ethics by Beauchamp and Childress. The term ‘disease interception’ refers to a form of (...) secondary prevention applied in a short interception window intended to prevent a preclinical disease from developing further. We propose the definition ‘early and targeted secondary prevention by treatment’. The ethical evaluation of the concept shows that while it promises to be beneficial, it raises a number of ethical and social challenges regarding patient autonomy and justice. In order to ensure decision-making that respects patient autonomy, commercially motivated metaphors such as ‘disease interception’ should make way for precise definitions. Future research should not only focus on how to detect clinically inapparent diseases but also on the ethical question, when this is justifiable and what consequences it has for the individual and society as a whole. (shrink)

We generalize the Unstable Formula Theorem characterization of stable theories from Shelah [11], that a theory T is stable just in case any infinite indiscernible sequence in a model of T is an indiscernible set. We use a generalized form of indiscernibles from [11], in our notation, a sequence of parameters from an L-structure M, , indexed by an L′-structure I is L′-generalized indiscernible inM if qftpL′=qftpL′ implies tpL=tpL for all same-length, finite ¯,j from I. Let Tg be the theory (...) of linearly ordered graphs in the language with signature Lg={<,R}. Let Kg be the class of all finite models of Tg. We show that a theory T has NIP if and only if any Lg-generalized indiscernible in a model of T indexed by an Lg-structure with age equal to Kg is an indiscernible sequence. (shrink)

During late embryonic and early postnatal development, synaptic connections are extensively modified so that some functional connections are weakened and eliminated from a neural circuit while others are strengthened and maintained. The mechanisms that underlie synapse elimination are beginning to be understood from studies of the neuromuscular junction. A recent paper(1) provides some intriguing insights into the role proteases may play in the developmental disassembly of neuromuscular synapses.

We study invariant types in NIP theories. Amongst other things: we prove a definable version of the [Formula: see text]-theorem in theories of small or medium directionality; we construct a canonical retraction from the space of [Formula: see text]-invariant types to that of [Formula: see text]-finitely satisfiable types; we show some amalgamation results for invariant types and list a number of open questions.

Let [Formula: see text] be an NIP field and let [Formula: see text] be a Henselian valuation on [Formula: see text]. We ask whether [Formula: see text] is NIP as a valued field. By a result of Shelah, we know that if [Formula: see text] is externally definable, then [Formula: see text] is NIP. Using the definability of the canonical [Formula: see text]-Henselian valuation, we show that whenever the residue field of [Formula: see text] is not separably closed, then [Formula: (...) see text] is externally definable. In the case of separably closed residue field, we show that [Formula: see text] is NIP as a pure valued field. (shrink)

We initiate an account of Shelah’s notion of “strong dependence” in terms of generically stable measures, proving a measure analogue of the fact that a stable theory $T$ is “strongly dependent” if and only if all types have almost finite weight.

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 study and characterize stability, the negation of the independence property (NIP) and the negation of the strict order property (NSOP) in terms of topological and measure theoretical properties of classes of functions. We study a measure theoretic property, Talagrand's stability, and explain the relationship between this property and the NIP in continuous logic. Using a result of Bourgain, Fremlin, and Talagrand, we prove almost definability and Baire 1 definability of coheirs assuming the NIP. We show that a formula has (...) the strict order property if and only if there is a convergent sequence of continuous functions on the space of φ‐types such that its limit is not continuous. We deduce from this a theorem of Shelah and point out the correspondence between this theorem and the Eberlein‐Šmulian theorem. (shrink)

We define the notion has the NIP (not the independence property) in A, where A is a subset of a model, and give some equivalences by translating results from function theory. We also discuss the number of coheirs when A is not necessarily countable, and revisit the notion “ has the NOP (not the order property) in a model M”.

We study generically stable measures in the local, NIP context. We show that in this setting, a measure is generically stable if and only if it admits a natural finite approximation.

We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian, then [Formula: see text] and [Formula: see text] are comparable. As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting (...) a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian. (shrink)

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)

A theory [Formula: see text] is said to have exact saturation at a singular cardinal [Formula: see text] if it has a [Formula: see text]-saturated model which is not [Formula: see text]-saturated. We show, under some set-theoretic assumptions, that any simple theory has exact saturation. Also, an NIP theory has exact saturation if and only if it is not distal. This gives a new characterization of distality.

We investigate some dynamical features of the actions of automorphisms in the context of model theory. We interpret a few notions such as compact systems, entropy and symbolic representations from the theory of dynamical systems in the realm of model theory. In this direction, we settle a number of characterizations of NIP theories in terms of dynamics of automorphisms and invariant measures. For example, it is shown that the property of NIP corresponds to the compactness property of some associated systems (...) and also to the zero entropy property of automorphisms. These results give a correspondence between some notions of tameness in model theory and ergodic theory. Moreover, we study the concept of symbolic representation and consider it in some well known mathematical objects such as the circle group, Bohr sets, Sturmian sequences, the structure \\), and random graphs with a model theoretic point of view in mind. We establish certain characterizations for stability theoretic dividing lines, such as independence property, order property and strict order property in terms of associated symbolic representations. At the end, we propose some applications of symbolic representations and these characterizations by giving a proof for a classical theorem by Shelah and also introducing some invariants associated to the types and elements of models. (shrink)

In light of a gap found by Krupiński, we give a new proof of associativity for the Morley product of invariant measures in NIP theories.

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets. This settles a conjecture of La...

Combining two results from machine learning theory we prove that a formula is NIP if and only if it satisfies uniform definability of types over finite sets. This settles a conjecture of La...

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 algebraic implications of the non-independence property and variants thereof on infinite fields, motivated by the conjecture that all such fields which are neither real closed nor separably closed admit a henselian valuation. Our results mainly focus on Hahn fields and build up on Will Johnson’s “The canonical topology on dp-minimal fields” :1850007, 2018).

Let G be a group definable in a monster model $\germ{C}$ of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U þ (G) < ∞. We prove that if G acts definably on a definable set of U þ -rank 1, then, under some general assumption about this action, there is an infinite field interpretable in $\germ{C}$ . We conclude that if G is not solvable-by-finite and it acts faithfully and definably on (...) a definable set of U þ -rank 1, then there is an infinite field interpretable in $\germ{C}$ . As an immediate consequence, we get that if G has a definable subgroup H such that U þ (G) = U þ (H) + 1 and $G/\bigcap _{g\in G}H^{g}$ is not solvable-by-finite, then an infinite field interpretable in $\germ{C}$ also exists. (shrink)

We introduce and study (weakly) semi-equational theories, generalizing equationality in stable theories (in the sense of Srour) to the NIP context. In particular, we establish a connection to distality via one-sided strong honest definitions; demonstrate that certain trees are semi-equational, while algebraically closed valued fields are not weakly semi-equational; and obtain a general criterion for weak semi-equationality of an expansion of a distal structure by a new predicate.

We prove that NIP valued fields of positive characteristic are henselian, and we begin to generalize the known results on dp-minimal fields to dp-finite fields. On any unstable dp-finite field K, we define a type-definable group of “infinitesimals,” corresponding to a canonical group topology on (K, +). We reduce the classification of positive characteristic dp-finite fields to the construction of non-trivial Aut(K/A)-invariant valuation rings.

We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...) fork over $\emptyset$ , precisely as in stable groups, answering positively an earlier problem posed by the first two authors. (shrink)

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type (...) and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types. (shrink)

We refine results of Gannon [6, Theorem 4.7] and Simon [22, Lemma 2.8] on convergence of Morley sequences. We then introduce the notion of eventual $NIP$, as a property of a model, and prove a variant of [15, Corollary 2.2]. Finally, we give new characterizations of generically stable types (for countable theories) and reinforce the main result of Pillay [17] on the model-theoretic meaning of Grothendieck’s double limit theorem.

In NIP theories, generically stable Keisler measures can be characterized in several ways. We analyze these various forms of “generic stability” in arbitrary theories. Among other things, we show that the standard definition of generic stability for types coincides with the notion of a frequency interpretation measure. We also give combinatorial examples of types in NSOP theories that are finitely approximated but not generically stable, as well as ϕ-types in simple theories that are definable and finitely satisfiable in a small (...) model, but not finitely approximated. Our proofs demonstrate interesting connections to classical results from Ramsey theory for finite graphs and hypergraphs. (shrink)

We give a simplified proof of elimination of imaginaries in ACVF, based on ideas of Hrushovski. This proof manages to avoid many of the technical issues which arose in the original proof by Haskell, Hrushovski, and Macpherson.

We prove that in $\operatorname {NTP}_{\operatorname {2}}$ theories the dp-rank of a type can be witnessed by indiscernible sequences of tuples satisfying that type. If the type has dp-rank infinity, then this can be witnessed by singletons.

The Non-Identity Problem (NIP) has been recognized as a hindrance in justifying compensation for historical injustice. Since NIP applies to individuals, an attractive way of trying to remove the obstacle is by shifting the focus from the allegedly harmed individuals to the harmed group. However, critical examination of this move shows that (a) there are groups—most conspicuously African Americans—who were _created_ by the unjust wrongs for which compensation is now claimed and hence fall under the same category as any wrongful (...) life case on the individual level, and (b) even when the group already existed when the wrongs were perpetrated, with the long passage of time it changed its identity in the narrative sense, which means that its contemporary members are attached to it as it is defined today but not as it was when the wrong was done. The paper suggests, therefore, that the Group Non-Identity Problem (GNIP) is the parallel impediment to that of NIP in doing historical justice to social groups which were maltreated in the past. (shrink)

Orthogonality in model theory captures the idea of absence of non-trivial interactions between definable sets. We introduce a somewhat opposite notion of cohesiveness, capturing the idea of interaction among all parts of a given definable set. A cohesive set is indecomposable, in the sense that if it is internal to the product of two orthogonal sets, then it is internal to one of the two. We prove that a definable group in an o-minimal structure is a product of cohesive orthogonal (...) subsets. If the group has dimension one, or it is definably simple, then it is itself cohesive. As an application, we show that an abelian group definable in the disjoint union of finitely many o-minimal structures is a quotient, by a discrete normal subgroup, of a direct product of locally definable groups in the single structures. (shrink)

Philosophers concerned with procreative ethics have long been puzzled by Parfit’s Non-Identity Problem (NIP). Various solutions have been proposed, but I argue that we have not solved the problem on its own narrow person-affecting terms, i.e., in terms of the identified individuals affected by procreative decisions and acts, especially future children. Thus, the core problem remains unsolved. This is a nagging concern for all who hold the common intuition that actions that harm no one are permissible. I argue against Harmon’s (...) and Woodward’s direct, narrow person-affecting solutions, and in favor of a new solution to the NIP. My solution, or, rather, dissolution, is based on the argument that merely possible people, i.e., hypothetical people who could possibly, but will not actually, exist, are morally irrelevant. I show that the NIP only arises when we concern ourselves with merely possible people. Once we are careful to restrict our concerns to only those that do or will exist, the NIP is dissolved. (shrink)

We show that a complete first-order theory T is distal provided it has a model M such that the theory of the Shelah expansion of M is distal.

We develop “local NIP group theory” in the context of pseudofinite groups. In particular, given a sufficiently saturated pseudofinite structure G expanding a group, and left invariant NIP formula δ...

When in doubt, for the embryo. New arguments on the moral status of human embryos. - In the first part of our essay we distinguish the philosophical from the legal and political level of the embryo debate and describe our indirect justification strategy. It consists in renouncing a determination of the dignity-giving φ-properties and instead starting from premises that are undoubted by all discussion partners. In the second part we reconstruct and criticize the species, continuum, identity and potentiality arguments. The (...) species argument only has a certain plausibility, if at all, as a critical argument. From the continuum argument and identity argument we take over the idea of numerical identity (NI) and link it with the core idea of the potentiality argument (P). So we come to the NIP argument: -/- (NIP) -/- (1) Every living human body that is the bearer of (or has) potential φ-properties has dignity. (2) Every viable human embryo is a living human body that is the bearer of (or has) potential φ properties. Therefore, (3) Every viable human embryo has dignity. -/- Reversibly comatose people and newborns are protected because they have the potential to have actual personal characteristics in the future; our thesis is that embryos that are capable of development also have the same potential in moral terms. The basic idea of numerical identity, with which we support the second premise, is that every human being, from embryonic existence to adulthood, forms a physical unity. In a detailed part, we deal with the crown princess, gametes, parthenogenesis, somatic cell, pronuclear stage, biological heteronome early embryo, multiple, fusion, Siamese twin, hydatidiform mole and finally trophoblast problems. In a third part, the indirect argument is supplemented by a metatheoretical cautionary argument. It states that in situations where there is doubt as to whether a being falls within the scope of a moral norm, but there are sufficiently strong reasons for this subsumption, it must be assumed that this is the case if the contrary assumption and the positive effects it may have are in no acceptable proportion to the moral harm that would result if that subsumption were not made. The main result of our considerations is therefore: When in doubt, for the embryo. (shrink)