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 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.

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].

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

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)

We prove a number of results relating the concepts of Keisler measures, generic stability, randomizations and NIP formulas. Among other things, we do the following: (1) We introduce the notion of a Keisler–Morley measure, which plays the role of a Morley sequence for a Keisler measure. We prove that if [Formula: see text] is fim over [Formula: see text], then for any Keisler–Morley measure [Formula: see text] in [Formula: see text] over [Formula: see text] and any formula [Formula: see text], (...) [Formula: see text]. We also show that any measure satisfying this conclusion must be fam. (2) We study the map, defined by Ben Yaacov, taking a definable measure [Formula: see text] to a type [Formula: see text] in the randomization. We prove that this map commutes with Morley products, and that if [Formula: see text] is fim then [Formula: see text] is generically stable. (3) We characterize when generically stable types are closed under Morley products by means of a variation of ict-patterns. Moreover, we show that NTP2 theories satisfy this property. (4) We prove that if a local measure admits a suitably tame global extension, then it has finite packing numbers with respect to any definable family. We also characterize NIP formulas via the existence of tame extensions for local measures. (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)

In this paper, we prove that definable ring topologies on NIP fields are closely connected to NIP integral domains. More precisely, we show that up to elementary equivalence, any NIP topological field arises from an NIP integral domain. As an application, we prove several results about definable ring topologies on NIP fields, including the following. Let [Formula: see text] be an NIP field or expansion of a field. Let [Formula: see text] be a definable ring topology on [Formula: see text]. (...) Then [Formula: see text] is a field topology, and [Formula: see text] is locally bounded. If [Formula: see text] has characteristic [Formula: see text] or finite dp-rank, then [Formula: see text] is “generalized [Formula: see text]-henselian” in the sense of Dittman, Walsberg and Ye, meaning that the implicit function theorem holds for polynomials. If [Formula: see text] has finite dp-rank, then [Formula: see text] must be a topology of “finite breadth” (a [Formula: see text]-topology). Using these techniques, we give some reformulations of the conjecture that NIP local rings are henselian. (shrink)

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)

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.

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)

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 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.

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)

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 give several new equivalences of NIP for formulas and new proofs of known results using Talagrand (Ann Probab 15:837–870, 1987) and Haydon et al. (in: Functional Analysis Proceedings, The University of Texas at Austin 1987–1989, Lecture Notes in Mathematics, Springer, New York, 1991). We emphasize that Keisler measures are more complicated than types (even in the NIP context), in an analytic sense. Among other things, we show that for a first order theory T and a formula $$\phi (x,y)$$, the (...) following are equivalent. (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)

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.

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)

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)

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...

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 an important (yet unpublished) research note, Ben Yaacov describes how to turn global Keisler measures into types over a monster model of the randomization. These transfer methods allow one to turn questions involving measures into those involving types (in continuous logic). Assuming that T is NIP, we show that the Morley product commutes with the transfer map for finitely satisfiable measures. We characterize when the Morley product commutes with the restriction map for pairs of global finitely satisfiable types in (...) the randomization. We end by making some brief observations about the Ellis semigroup in this context. (shrink)

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)

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).

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)

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.

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.

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)

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)

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 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 study expansions of an algebraically closed field K or a real closed field R with a linearly independent subgroup G of the multiplicative group of the field or the unit circle group \\), satisfying a density/codensity condition. Since the set G is neither algebraically closed nor algebraically independent, the expansion can be viewed as “intermediate” between the two other types of dense/codense expansions of geometric theories: lovely pairs and H-structures. We show that in both the algebraically closed field and (...) real closed field cases, the resulting theory is near model complete and the expansion preserves many nice model theoretic conditions related to the complexity of definable sets such as stability and NIP. (shrink)

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)

We study randomizations of definable groups. Whenever the underlying theory is stable or NIP and the group is definably amenable, we show its randomization is definably connected.

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.

We explore several model-theoretic aspects of D-sets, which were studied in detail by Adeleke and Neumann. We characterize ultrahomogeneity in the class of colored D-sets and classify unbounded order-indiscernible sequences in such structures. We use these results to provide a characterization of distal colored D-sets and prove that all colored D-sets are monadically NIP.

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 give explicit formulas witnessing IP, IP $_{\!n}$, or TP2 in fields with Artin–Schreier extensions. We use them to control p-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP $_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP $_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.

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)