This article is a logical continuation of the Henri Lombardi and Franz-Viktor Kuhlmann article [9]. We address some classical points of the theory of valued fields with an elementary and constructive point of view. We deal with Krull valuations, and not simply discrete valuations. First of all, we show how to construct the Henselization of a valued field; we restrict to fields in which one has at one's disposal algorithmic tools to test the nullity or the (...) valuation ring membership. It is therefore a work that differs as much in spirit as in field of application from that of Mines, Richman and Bridges , who address the framework of Heyting fields and discrete valuation. We show then in a constructive way a batch of classical results in Henselian fields, notably factorization criteria and Krasner's Lemma. We conclude by a construction of the inertia field of a valued field. (shrink)

When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map [Formula: see text], which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for t. Driven by this observation, we study the (...) theory [Formula: see text] of valued fields equipped with a linear form [Formula: see text] which restricts to the residue map on the valuation ring. We prove that [Formula: see text] does not admit a model companion. In addition, we show that [Formula: see text] is undecidable whenever k is an infinite field. As a consequence, we get that [Formula: see text] is undecidable, where [Formula: see text] maps f to its complex residue at 0. (shrink)

We prove that every ultraproduct of p-adics is inp-minimal. More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that the total residue map [math], which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for (...) t. Driven by this observation, we study the theory [math] of valued fields equipped with a linear form [math] which restricts to the residue map on the valuation ring. We prove that [math] does not admit a model companion. In addition, we show that [math] is undecidable whenever k is an infinite field. As a consequence, we get that [math] is undecidable, where [math] maps f to its complex residue at 0. (shrink)

In this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$, algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal (...) related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}})$ is inp-minimal, and that the ring of Witt vectors $W$ over $\mathbb {F}_p^{\text {alg}}$ is not strong. This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R})$ are definable. Similarly, all types over the quotient field of $W$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye.These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups.prepared by Pierre Touchard.E-mail: [email protected]: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9. (shrink)

We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility (...) relation in the value group which is sufficient to find such an embedding. By checking that algebraically closed valued fields and p-adically closed valued fields of infinite transcendence degree have the Mal’cev property, we show that they have computable dimension \. (shrink)

In order to maximize human performance, defence forces continue to explore, develop, and apply human performance enhancement (HPE) methods, ranging from pharmaceuticals to (bio)technological enhancement. This raises ethical, legal, and societal concerns and requires organizing a careful reflection and deliberation process, with relevant stakeholders. We discuss a range of ethical, legal, and societal aspects (ELSA), which people involved in the development and deployment of HPE can use for such reflection and deliberation. A realistic military scenario with proposed HPE application can (...) serve as a starting point for such an iterative and participatory process. Stakeholders can discuss this application, modify its features, and design appropriate processes around it – for instance, procedures for informed consent. We propose that organizing aspects into these three categories – ethical, legal, and societal – can help involve appropriate interlocutors at different moments: legal aspects with people in strategy or management roles, from the start of a project; ethical aspects with people in operations and medical roles, during development; and societal aspects with people in communication and personnel roles, during deployment. Notably, we developed and discussed this framework and the three aspects in close collaboration with personnel from the military. (shrink)

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)

Debates over vaccine mandates raise intense emotions, as reflected in the current controversy over whether to mandate the vaccine against human papilloma virus (HPV), the virus that can cause cervical cancer. Public health ethics so far has failed to facilitate meaningful dialogue between the opposing sides. When stripped of its emotional charge, the debate can be framed as a contest between competing ethical values. This framework can be conceptualized graphically as a conflict between autonomy on the one hand, which militates (...) against government intrusion, and beneficence, utilitarianism, justice, and nonmaleficence on the other, which may lend support to intervention. When applied to the HPV vaccine, this framework would support a mandate based on utilitarianism, if certain conditions are met and if herd immunity is a realistic objective. (shrink)

The first two steps of the construction of motivic integration in the fundamental work of Hrushovski and Kazhdan [8] have been presented in Yin [12]. In this paper we present the final third step. As in Yin [12], we limit our attention to the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated substructure S in the language . A canonical description of the kernel of the homomorphism is obtained.

We extend the characterization of extremal valued fields given in [2] to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finitep-degree, the images of all additive (...) polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in [8] for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1. (shrink)

We prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.

We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric' sorts which suffice to code all imaginaries in the corresponding algebraic setting.

We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K 2 to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.

I argue that there are some situations in which it is praiseworthy to be motivated only by moral rightness de dicto, even if this results in wrongdoing. I consider a set of cases that are challenging for views that dispute this, prioritising concern for what is morally important in moral evaluation. In these cases, the agent is not concerned about what is morally important, does the wrong thing, but nevertheless seems praiseworthy rather than blameworthy. I argue that the views under (...) discussion cannot accommodate this, and should be amended to recognise that it is often praiseworthy to be motivated to do what is right. (shrink)

I offer a line of argument that aims at the conclusion that the notion of radically different and incommensurable systems of value is incoherent, which would mean that the presumption of some significant common ground of valuation is rationally required in value inquiry.

We define a notion of residue field domination for valued fields which generalizes stable domination in algebraically closed valued fields. We prove that a real closed valued field is dominated by the sorts internal to the residue field, over the value group, both in the pure field and in the geometric sorts. These results characterize forking and þ-forking in real closed valued fields (and also algebraically closed valued fields). We lay some (...) groundwork for extending these results to a power-bounded T-convex theory. (shrink)

In this note we show that groups with definable generics in a separably closed valued field K of finite imperfection degree can be embedded into groups definable in the algebraic closure of K.

The paper shows elimination of imaginaries for real closed valued fields to suitable sorts. We also show that this result is in some sense optimal. The paper includes a quantifier elimination theorem for real closed valued fields in a language with sorts for the field, value group and residue field.

Motivated by the Ax–Kochen/Ershov principle, a large number of questions about Henselian valued fields have been shown to reduce to analogous questions about the value group and residue field. In this article, we investigate the burden of Henselian valued fields in the three-sorted Denef–Pas language. If T is a theory of Henselian valued fields admitting relative quantifier elimination (in any characteristic), we show that the burden of T is equal to the sum of the (...) burdens of its value group and residue field. As a consequence, T is NTP2 if and only if its residue field and value group are; the same is true for the statements “T is strong” and “T has finite burden.”. (shrink)

The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a (...) point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields. (shrink)

I must begin by apologizing for taking a somewhat well-worn subject for my theme. My reason is that I have not yet found a recent treatment of it which is altogether to my satisfaction. Most of them seem to me too often to approach the subject from a point of view which, in a way, expects too much from the study of Plato or any other ancient author, and consequently either makes exaggerated claims for it or fails to do justice (...) to what we really can learn. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Commentary on “Sanity and Irresponsibility”Lloyd Fields (bio)AbstractI make two criticisms of Wilson’s proposal to dispense with a loaded axiological criterion of sanity. First, Edwards’s axiological criterion of sanity, which Wilson accepts, involves the requirement of impartiality, which at least excludes some standards of right and wrong. Second, value pluralism applies only to morally acceptable forms of life and thus presupposes a standard of right and wrong. I conclude (...) by noting that mental illnesses are not sufficiently characterised by extreme and persistent deviance from practical rationality, as conceived by Edwards and Wilson.KeywordsSanity, responsibility, mental illness, value pluralismWilson’s paper is set in an important tradition of which Susan Wolf is a notable exemplar. He tackles the crucially important issue of how practical rationality is related to mental health and of how certain forms of irrationality are connected with mental illness. This issue must be taken into account in order to avoid infecting the criteria of sanity with “idiosyncratic or culturally biased norms.” Like Wolf (1989, 137–151), Wilson holds that sanity is a necessary condition of responsibility. But Wilson rejects the second of Wolf’s two criteria of sanity on the grounds that it invokes a definite normative theory of right and wrong. He proposes instead that whether an agent is sane should be decided on the basis of whether s/he displays practical rationality, as defined by Edwards’s list of seven items. Following Edwards, Wilson implicitly equates “sanity” with “mental health” and “insanity” with “mental illness.” By rejecting an axiological criterion of sanity that depends on a normative theory of right and wrong, Wilson is able to ensure that individuals like JoJo are not judged to be insane and hence not morally responsible for their actions. Wilson is right to stress the importance of our not applying notions such as “mental illness” or “insanity” too widely, especially in such a way as to allow blameworthy persons to escape blame and punishment.But are Wilson’s own criteria of sanity free from dependence on a normative theory, or a normative standard, of right and wrong? First, let us note that Wilson states Edwards’s seventh condition of practical rationality as follows: “holding values that are or would be adopted under the conditions of (negative) freedom.” Edwards’s own formulation of this condition is the following: “having values which have been (or would be) adopted under conditions of freedom, enlightenment and impartiality” (1982, 315). Wilson explains “freedom” as the absence of coercion. It is uncertain what “enlightenment” means in this context. Impartiality would seem to include being impartial between oneself and others and between one’s own tribe, nation or race and that of others. Although this requirement is perhaps too general to be regarded as a sufficient standard [End Page 303] of right and wrong, it is at least part of such normative theories as utilitarianism and Kant’s moral theory, while it is inconsistent with ethical egoism, i.e., the view that an action is right or wrong depending on whether it serves or frustrates the agent’s overall best interests. One could dispute Wilson’s claim, therefore, that Edwards’s seventh item is not a “loaded” axiological criterion.Second, Wilson maintains that we can avoid assuming a loaded axiological criterion for sanity by adopting the position of value pluralism. Wolf’s axiological criterion for sanity is that the agent should be capable of knowing the difference between right and wrong. Wilson holds that this criterion, as presented, invokes a normative theory of right and wrong. The criterion would not do so if it were applied in accordance with value pluralism, which, Wilson says, “rejects the notion that there is one monolithic theory of right and wrong without denying that an actor may behave virtuously or viciously.” Wilson appears to assume that an “unloaded” version of Wolf’s axiological criterion would be more or less equivalent to Edwards’s seventh condition of practical rationality.Value pluralism maintains that there are incompatible forms of life that display distinct virtues. “Whichever form of life one is pursuing there are virtues which elude one because they are available only to people pursuing alternative and... (shrink)

In this dissertation I offer a theory of intrinsic value based on contextualist principles drawn from the value theories of John Dewey and Alfred North Whitehead. The point of departure for the argument is the contextualist view that the qualitative patters representing in experience objects of states of affairs to which we attribute values provide necessary, but not sufficient, conditions to elicit particular valuations, and ground the evaluative judgments we make. The sufficient conditions for valuation include a broader context of (...) memories, beliefs, and peripheral qualia that combine in an integral experiential whole whose value is not reducible to that of its parts. It is first argued that value is, as G.E. Moore suggested, a simple, indefinable property. This property is emergent from experiential qualitative patterns in the sense that value is conditioned by the patterns, but is not itself a condition of them. An analysis of experiential qualitative pattern is then offered, which indicates how some parts of these patterns are focused on in evaluative judgment as crucial for the reproduction of particular values through practical action, whereas the influence of other parts is largely ignored, thus forming a relatively constant experiential context. Finally, a theory of evaluative judgment is offered based on contextualist principles. It is suggested that a contextualist theory can answer the challenge of value relativism by showing how evaluative judgment is 'position relative' rather than 'author relative'. That is, two subjects many come to different evaluative judgments on account of their differing positions or situations, while both may agree that the other's judgment is sound given his or her situation. (shrink)

We construct Hrushovski–Kazhdan style motivic integration in certain expansions of ACVF. Such an expansion is typically obtained by adding a full section or a cross-section from the RV-sort into the VF-sort and some extra structure in the RV-sort. The construction of integration, that is, the inverse of the lifting map , is rather straightforward. What is a bit surprising is that the kernel of is still generated by one element, exactly as in the case of integration in ACVF. The overall (...) construction is more or less parallel to the main construction of Hrushovski and Kazhdan , as presented in Yin and Yin . As an application, we show uniform rationality of Igusa zeta functions for non-archimedean local fields with unbounded ramification degrees. (shrink)

Let (K, v) be a henselian valued field of characteristic 0. Then K admits a definable partition on each piece of which the leading term of a polynomial in one variable can be computed as a definable function of the leading term of a linear map. The main step in obtaining this partition is an answer to the question, given a polynomial f(x) ∈ K[x], what is v(f(x))? Two applications are given: first, a constructive quantifier elimination relative to the (...) leading terms, suggesting a relative decision procedure; second, a presentation of every definable subset of K as the pullback of a definable set in the leading terms subjected to a linear translation. (shrink)

In [12]. T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language L₂ which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a "positivstellensatz" theorem for the differential analogue of the theory (...) of real-series closed fields in the valued D-field setting. (shrink)

I argue that level-incoherence is epistemically valuable in a specific set of epistemic environments: those in which it is easy to acquire justified false beliefs about normative requirements of epistemic rationality. I argue that in these environments level-incoherence is the rationally dominant strategy. Nevertheless, level-incoherent combinations exhibit a distinctive tension, and this tension has been thought by many to indicate that level-incoherence is always irrational. Although this idea has proved resilient, I argue that it is incorrect. I evaluate three candidate (...) explanations for the distinctive tension exhibited by level-incoherent combinations, only one of which is the traditional view (which I call the ‘Prohibition View’) that epistemic level-incoherence is prohibited by epistemic rationality. I argue instead for the ‘Inquiry View’, according to which level-incoherence is not rationally criticisable but is a reason to undertake further inquiry. (shrink)

The term ‘social capital’ is a way of defining the intangible resources of community, shared values and trust upon which we draw in daily life. It has achieved considerable international currency across the social sciences through the very different work of Pierre Bourdieu in France and James Coleman and Robert Putnam in the United States, and has been widely taken up within politics and sociology as an explanation for the decline in social cohesion and community values in western societies. It (...) has also been adopted by policy makers, particularly in international governmental bodies such as the World Bank. (shrink)

Is there really such a thing as moral progress? Do we get any better as time goes on? It is a question which must often exercise the minds of those who reflect on moral questions at all. And it is a frequent topic of discussion, both in private conversations and in the written contributions of a good many of our popular philosophers. Of some of these contributions one may safely say that their chief value is as a warning against the (...) dangers of hasty generalization on a subject such as this. And, taking advantage of such a warning, I shall make no attempt to give a final or dogmatic answer to this question. All I can try to do, within the limits of a single lecture, is to disentangle some of the most important considerations that seem to be in people's minds in discussing this question, and to estimate their importance as arguments on one side or the other. (shrink)

ABSTRACT Section 1 discusses properties and propositions, and some of the motivation for an account in which property instantiation and propositional truth behave ‘naively’. Section 2 generalizes a standard Kripke construction for naive properties and propositions, in a language with modal operators but no conditionals. Whereas Kripke uses a 3-valued value space, the generalized account allows for a broad array of value spaces, including the unit interval [0,1]. This is put to use in Section 3, where I add to (...) the language a conditional suitable for restricting quantification. The shift from a value space based on the ‘mini-space’ {0,, 1} to one based on the ‘mini-space’ [0,1] leads to more satisfactory results than I was able to achieve in previous work: a vast variety of paradoxical sentences can now be treated very simply. In Section 4 I make a further addition to the language, a conditional modeled on the ordinary English conditional, paying particular attention to how it interacts with the restricted quantifier conditional. This is all done in the [0,1] framework, and two alternatives are considered for how the ordinary conditional is to be handled; one of them results from adding a tweak to a construction by Ross Brady. Section 5 discusses a further alternative, a standard relevance conditional (for the ordinary conditional, perhaps for use with a different quantifier-restricting conditional), but argues that it is not promising. Section 6 discusses the identity conditions of properties and propositions (again in the setting of a value space based on [0,1]); the issue of achieving naivety for coarse-grained properties is seen to be more complicated than some brief remarks in Field [2010] suggested, but a way to get a fair degree of coarse-grainedness is shown. (shrink)

We present two of the three major steps in the construction of motivic integration, that is, a homomorphism between Grothendieck semigroups that are associated with a first-order theory of algebraically closed valued fields, in the fundamental work of Hrushovski and Kazhdan [8]. We limit our attention to a simple major subclass of V-minimal theories of the form ACV FS, that is, the theory of algebraically closed valued fields of pure characteristic 0 expanded by a -generated substructure (...) S in the language . The main advantage of this subclass is the presence of syntax. It enables us to simplify the arguments with many different technical details while following the major steps of the Hrushovski–Kazhdan theory. (shrink)

A relational-self theory claims that one’s self is constituted by one’s relationships. The type of ethics that is said to arise from this concept of self is often called an ethics of care, whereby the focus of ethical deliberation is on preserving and nurturing those relationships. Some environmental philosophers advocating a relational-self theory tend to assume that the particular relationships that constitute the self will prioritize the natural world. I question this assumption by introducing the problem of artifact relationships. It (...) is unclear whether a relational-self theory recognizes relationships with the artificial world as beingmeaningful in any moral sense, and whether such relationships, if they can exist, should be accorded equal value to relationships with the natural world. The problem of artifact relationships becomes particularly apparent when the relational-self theory is linked to place-based ethics. If our ethics are to develop from our relations to place, and our place is largely an artificial world, is there not a danger that our ethical deliberations will tend to neglect the natural world? I adapt Holmes Rolston’s concept of “storied residence” to show how the inclusion of the artificial world will lead to different questions regarding one’s resident environment, and perhaps a different emphasis on what is valued. My aim in raising these questions is to challenge the optimism that writers such as Karen Warren and Jim Cheney have shown in supporting relational-self theories and place-based ethics. I conclude that the challenge to develop a relational-self/place-based ethic does not appear to have been met within Western environmental philosophy, which has perpetuated a silence on the matter of our embedment in the artificial world. (shrink)

Religious therapeutics is the term I use to designate relations between health and spirituality, and medicine and religion. Dimensions of religious therapeutics include religious meanings that inform medical theory, religious means of healing, health as part of religious life, and religion as a remedy for human suffering. Classical Yoga is analyzed to establish an initial matrix of religious therapeutics with 5 branches: philosophical foundations, soteriology, value theory, physical practice, and cultivation of consciousness. Through comparative criticism of classical Yoga, the study (...) presents a heuristic of religious therapeutics: a model for interpreting relations among healing and liberative functions in world religions. ;Body and health are of instrumental but not ultimate value in classical Yoga: the body is used to transcend itself for attainment of Yoga's soteriological goal, realization of self as pure consciousness. Yoga's Samkhya-based metaphysics contains an unreconciled dualism, and while practice of Yoga is paradigmatic of mind/body holism, Yoga prescribes realization of a spiritual self, independent of material and psychological nature. The study rehabilitates the body in respect of the compatibility of embodiedness with religiousness. ;Other Indian and world traditions suggest dimensions of religious therapeutics both resonant with classical Yoga and lacking from it. India's Ayurvedic medicine represents the sixth branch of religious therapeutics: medical therapeutics. I distill from Ayurvedic and Western sources a set of determinants of health: biological, medical/psychological, cultural, and metaphysical. Significant determinants of health are wholeness, self-identity, and freedom; these are incorporated in discussion of the complementary functions of medicine and religion, grounding the claim that in classical Yoga, liberation is healing in an ultimate sense. ;Tantric yogas utilize material nature for human spiritual progress, and unlike classical Yoga, esteem nature, body/mind, the feminine, and relationality. Tantra provides another branch of religious therapeutics: aesthetic therapeutics. The study anticipates elements of health/medicine in western religions, Buddhism, and Lakota religious philosophy. Sacred speech and song are explored to demonstrate comparative inquiry into religious therapeutics; some Native American Indian and Hindu applications of sacred language are considered. Finally, the model of religious therapeutics is supplemented with community, embracing ecological, social, and religious relationality and communication. (shrink)

The Expanse does not provide an easy answer to the vexing question on making a decision when competing, but considering conflicts of values on the show can help us reason about tough choices in real life. Sometimes, scientific progress conflicts with the prudential value of self‐preservation. This chapter explains three ways of understanding value conflicts: as situations in which every option is forbidden, situations in which every option is permissible, or situations in which some options are obligatory and some options (...) are forbidden. The idea that all these different kinds of value can be traded with a common currency is tempting. But contra Keats, beauty is not truth; and it certainly is not goodness, as Tolstoy says. Epistemic, moral, prudential, and aesthetic values are very different. (shrink)

We give some general criteria for the stable embeddedness of a definable set. We use these criteria to establish the stable embeddedness in algebraically closed valued fields of two definable sets: The set of balls of a given radius r < 1 contained in the valuation ring and the set of balls of a given multiplicative radius r < 1. We also show that in an algebraically closed valued field a 0-definable set is stably embedded if and (...) only if its algebraic closure is stably embedded. (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 (...) class='Hi'>valued fields. (shrink)

If properties are to play a useful role in semantics, it is hard to avoid assuming the naïve theory of properties: for any predicate Θ(x), there is a property such that an object o has it if and only if Θ(o). Yet this appears to lead to various paradoxes. I show that no paradoxes arise as long as the logic is weakened appropriately; the main difficulty is finding a semantics that can handle a conditional obeying reasonable laws without engendering paradox. (...) I employ a semantics which is infinite-valued, with the values only partially ordered. Can the solution be adapted to naïve set theory? Probably not, but limiting naïve comprehension in set theory is perfectly satisfactory, whereas this is not so in a property theory used for semantics. (shrink)

We present a canonical form for definable subsets of algebraically closed valued fields by means of decompositions into sets of a simple form, and do the same for definable subsets of real closed valued fields. Both cases involve discs, forming "Swiss cheeses" in the algebraically closed case, and cuts in the real closed case. As a step in the development, we give a proof for the fact that in "most" valued fields F, if f(x),g(x) (...) ∈ F[ x] and v is the valuation map, then the set {x : v(f(x)) ≤ v(g(x))} is a Boolean combination of discs; in fact, it is a finite union of Swiss cheeses. The development also depends on the introduction of "valued trees", which we define formally. (shrink)

In, Marker and Steinhorn characterized models of an o‐minimal theory such that all types over M realized in N are definable. In this article we characterize pairs of algebraically closed valued fields satisfying the same property. In o‐minimal theories, a pair of models for which all 1‐types over M realized in N are definable has already the desired property. Although it is true that if M is an algebraically closed valued field such that all 1‐types over M (...) are definable then all types over M are definable, we build a counterexample for the relative statement, i.e., we show for any that there is a pair of algebraically closed valued fields such that all n‐types over M realized in N are definable but there is an ‐type over M realized in N which is not definable. (shrink)

The theory of algebraically closed non-Archimedean valued fields is proved to eliminate quantifiers in an analytic language similar to the one used by Cluckers, Lipshitz, and Robinson. The proof makes use of a uniform parameterized normalization theorem which is also proved in this paper. This theorem also has other consequences in the geometry of definable sets. The method of proving quantifier elimination in this paper for an analytic language does not require the algebraic quantifier elimination theorem of Weispfenning, (...) unlike the customary method of proof used in similar earlier analytic quantifier elimination theorems. (shrink)

We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist (...) a uniform definition by an existential formula and neither by a universal formula for the valuation rings of all the finite extensions of a given Henselian valued field. We also show that there is no existential formula of the ring language defining Zp inside Qp uniformly for all p. For any fixed finite extension of Qp, we give an existential formula and a universal formula in the ring language which define the valuation ring. (shrink)