In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma /\Delta $ for each definable convex subgroup $\Delta $, and sorts for the quotient groups $\Gamma /(\Delta + \ell \Gamma )$ where $\Delta $ is a definable convex subgroup and $\ell (...) \in \mathbb {N}_{\geq 2}$. We refer to these sorts as the quotient sorts. For the dp-minimal case we obtain a complete elimination of imaginaries if we also add constants to distinguish the cosets of $\ell \Gamma $ in $\Gamma $, where $\ell \in \mathbb {N}_{\geq 2}$. (shrink)

Let TP be the theory obtained by adding a generic predicate to an o-minimal theory T. We prove that if T admits elimination of imaginaries, then TP also admits elimination of imaginaries.

We give a complete characterization of Boolean algebras admitting weak elimination of imaginaries in terms of elementary invariants.

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 study the degree of elimination of imaginaries needed for the three main applications: to have canonical bases for types over models, to define strong types as types over algebraically closed sets and to have a Galois correspondence between definably closed sets B such that A ⊆ B ⊆ acl and closed subgroups of the Galois group Aut/A). We also characterize when the topology of the Galois group is the quotient topology.

A nontrivial ring with unit eliminates imaginaries just in case its complete theory has the following property: every definable m-ary equivalence relation E may be defined by a formula f = f, where f is an m-ary definable function. We show that for certain natural expansions of the field of p-adic numbers, elimination of imaginaries fails or is independent of ZPC. Similar results hold for certain fields of formal power series.

Given an infinite Boolean algebra B, we find a natural class of equation image-definable equivalence relations equation image such that every imaginary element from Beq is interdefinable with an element from a sort determined by some equivalence relation from equation image. It follows that B together with the family of sorts determined by equation image admits elimination of imaginaries in a suitable multisorted language. The paper generalizes author's earlier results concerning definable equivalence relations and weak elimination of (...) imaginaries for Boolean algebras, obtained in 10. (shrink)

We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.

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.

In response to both scientific and individualistic pressures, the law is increasingly giving in to the temptation of eugenics and transhumanism, encouraging the elimination of ‘undesirable’ embryos and genetic manipulation that will eventually lead to the modification of humanity.

We show that the ordered field of real numbers with restricted $\mathbb{R}_{\mathscr{H}}$-definable analytic functions admits quantifier elimination if we add a function symbol $^{-1}$ for the function $x\mapsto \frac{1}{x}$ (with $0^{-1}=0$ by convention), where $\mathbb{R}_{\mathscr{H}}$ is the real field augmented by the functions in the family $\mathscr{H}$ of restricted parts (real and imaginary) of holomorphic functions which satisfies certain conditions. Further, with another condition on $\mathscr{H}$ we show that the structure ($\mathbb{R}_{\mathscr{H}}$, constants) is strongly model complete.

The objective of this article is to characterise elimination of finite generalised imaginaries as defined in [9] in terms of group cohomology. As an application, I consider series of Zariski geometries constructed [10, 23, 24] by Hrushovski and Zilber and indicate how their nondefinability in algebraically closed fields is connected to eliminability of certain generalised imaginaries.

Elimination of imaginaries for 1-variable definable equivalence relations is proved for a theory of algebraically closed valued fields with new sorts for the disc spaces. The proof is constructive, and is based upon a new framework for proving elimination of imaginaries, in terms of prototypes which form a canonical family of formulas for defining each set that is definable with parameters. The proof also depends upon the formal development of the tree-like structure of valued fields, in (...) terms of valued trees, and a decomposition of valued trees which is used in the coding of certain sets of discs. (shrink)

Multiple modernities has emerged as the post-Huntingtonian paradigm in the study of secularism and religion, and the concepts ‘imaginary’ or ‘ verstehen’ are the most common candidates guiding research aiming to articulate this multiplicity. This article revisits Shmuel Eisenstadt’s original ‘Multiple Modernities’ thesis, Charles Taylor’s concept ‘imaginary’ and Max Weber’s ‘ verstehen’, and offers concise examples on how they are put into practice in the current literature on secularism and religion. I argue that the original Eisenstadt thesis is built upon (...) interactions of modernities, and the ‘imaginary’ and ‘ verstehen’ analytics eliminate from sight non-isomorphic relations between ideas and actions, despite the ample presence of both interactions and non-isomorphic relations in the politics of secularism and modernity. Turning multiple modernities into an exercise in typologies of non-interacting modernities articulated in isomorphic relations between ideas and actions produces new kinds of post-Huntingtonian culturalism. I finally sketch a comparative politics of new meanings as the counter-hypothesis to which ‘imaginary’, ‘ verstehen’ and non-interacting typology analytics of modernities have to respond. (shrink)

If K is a field of finite Morley rank, then for any parameter set $A \subseteq K^{eq}$ the prime model over A is equal to the model-theoretic algebraic closure of A. A field of finite Morley rank eliminates imaginaries. Simlar results hold for minimal groups of finite Morley rank with infinite acl( $\emptyset$ ).

In this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ω m + 1.

These photographs, essays, drawings, and original texts document the rich agit-art legacy of the Situationist International, a group of European artists and writers who emerged from such avant-garde movements as COBRA, Lettrisme, and the Imaginary Bauhaus and from the breakup of surrealism to launch a strategy of art as cultural critique.

The following strong form of density of definable types is introduced for theoriesTadmitting a fibered dimension functiond: given a modelMofTand a definable setX⊆Mn, there is a definable typepinX, definable over a code forXand of the samed-dimension asX. Both o-minimal theories and the theory of closed ordered differential fields are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

A Steiner triple system (STS) is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite STS has a Fraïssé limit M_F. Here, we show that the theory T of M_F is the model completion of the theory of STSs. We also prove that T is not small and it has quantifier (...) class='Hi'>elimination, TP2, NSOP1, elimination of hyperimaginaries and weak elimination of imaginaries. (shrink)

In Dellunde et al. 997–1015), we determined the complete theory Te of modules of separably closed fields of characteristic p and imperfection degree e, eω{∞}. Here, for 0≠eω, we describe the closed set of the Ziegler spectrum corresponding to Te. Further, we establish a correspondence between certain submodules and n-types and we investigate several notions of dimensions and their relationships with the Lascar rank. Finally, we show that Te has uniform p.p. elimination of imaginaries and deduce uniform weak (...) elimination of imaginaries. (shrink)

Let T be a theory. If T eliminates \, it need not follow that \ eliminates \, as shown by the example of the p-adics. We give a criterion to determine whether \ eliminates \. Specifically, we show that \ eliminates \ if and only if \ is eliminated on all interpretable sets of “unary imaginaries.” This criterion can be applied in cases where a full description of \ is unknown. As an application, we show that \ eliminates \ (...) when T is a C-minimal expansion of ACVF. (shrink)

Abstract‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid. We give a full characterization of the ‐definable stably embedded sets of T, showing in particular that, up to the relation of having the same definable closure, there is only one non‐trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries but not elimination of finite imaginaries.

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in, we establish a relative Galois correspondence for relatively definable subgroups of (...) the group of differential order automorphisms. (shrink)

This chapter argues that the general philosophy of science should learn metaphilosophical lessons from the case of metaphysical underdetermination, as it occurs in non-relativistic quantum mechanics. Section presents the traditional discussion of metaphysical underdetermination regarding the individuality and non-individuality of quantum particles. Section discusses three reactions to it found in the literature: eliminativism about individuality; conservatism about individuality; eliminativism about objects. Section wraps it all up with metametaphysical considerations regarding the epistemology of metaphysics of science.

Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to (...) the predicate. For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate. (shrink)

Following the keen interests in citizenship education across the fields of education, this study delves into the ways in which we conceptualize good citizenship. To do so, I focus on two theoretical concepts (i.e., imaginary and cultural mythology) and the provincial level of education policy(ies) and the K-12 curriculum contexts in Canada. Based on my theoretical ground and critical discourse analysis of the un/official documents for Alberta education, I indicate diversity as one crucial element of a cultural mythology: it, as (...) a sign in a second-order semiological system, serves to disseminate a monolithic and depoliticized vision of good citizenship based within a particular imaginary of the idealized Canada. This study thus elucidates not only our inherited cultural biases about the popular meanings of good citizenship, but also their invisible onto-epistemological presuppositions perpetuating the unchanged institutionalized structures of privilege and their exploitations of poor and minority peoples (e.g., elimination of Indigenous peoples and their land sovereignty). Hence, this study offers critical insights to address unequal relations of power in our prevalent notion(s) of citizenship we as educators presume to teach. This work, by doing so, follows the decolonial imperative to rupture the social inequality and discrimination we all strive to resist. (shrink)

According to the logical positivists, signs (words and pictures) of imaginary beings have no referent (Goodman). The semiotic theory behind this assumption is dualistic and Cartesian: signs vs. nonsigns as well as the mental vs. the material world are in fundamental opposition. Peirce’s semiotics is based on the premise of the sign as a mediator between such opposites: signs do not refer to referents, they represent objects to a mind, but the object of a sign can be existent or nonexistent, (...) a feeling or an idea, something merely imaginary or even something false. The paper examines the nature of imaginary, nonexistent, and nonfigurative objects, such as unicorns, phoenixes, or nonrepresentative paintings, and shows why verbal or visual representations of imaginary beings and even nonfigurative paintings, which seem to represent nothing, are fully developed signs with a variety of objects determining their representation. (shrink)

Although it was traditionally thought that self-reference is a crucial ingredient of semantic paradoxes, Yablo (1993, 2004) showed that this was not so by displaying an infinite series of sentences none of which is self-referential but which, taken together, are paradoxical. Yablo's paradox consists of a countable series of linearly ordered sentences s(0), s(1), s(2),... , where each s(i) says: For each k > i, s(k) is false (or equivalently: For no k > i is s(k) true). We generalize Yablo's (...) results along two dimensions. First, we study the behavior of generalized Yablo-series in which each sentence s(i) has the form: For Q k > i, s(k) is true, where Q is a generalized quantifier (e.g., no, every, infinitely many, etc). We show that under broad conditions all the sentences in the series must have the same truth value, and we derive a characterization of those values of Q for which the series is paradoxical. Second, we show that in the Strong Kleene trivalent logic Yablo's results are a special case of a more general fact: under certain conditions, any semantic phenomenon that involves self-reference can be emulated without self-reference. Various translation procedures that eliminate self-reference from a non-quantificational language are defined and characterized. An Appendix sketches an extension to quantificational languages, as well as a new argument that Yablo's paradox and the translations we offer do not involve self-reference. (shrink)

A uniform construction for sequent calculi for finite-valued first-order logics with distribution quantifiers is exhibited. Completeness, cut-elimination and midsequent theorems are established. As an application, an analog of Herbrand’s theorem for the four-valued knowledge-representation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.

Building on past constructive criticism, the present study provides further methodological development focused on the elimination of bias that may occur during first-person observation. First, various sources of errors that may accompany introspection are distinguished based on previous critical literature. Four main errors are classified, namely attentional, attributional, conceptual, and expressional error. Furthermore, methodological recommendations for the possible elimination of these errors have been determined based on the analysis and focused excerpting of introspective scientific literature. The following groups (...) of methodological recommendations were determined: 1) a better focusing of the subject’s attention to their mental processes, 2) providing suitable stimuli, and 3) the sharing of introspective experience between subjects. Furthermore, the potential of adjustments in introspective research designs for eliminating attentional, attributional, conceptual, and expressional error is discussed. (shrink)

We prove weak elimination of imaginary elements for Boolean orderings with finitely many atoms. As a consequence we obtain equivalence of the two notions of o-minimality for Boolean ordered structures, introduced by C. Toffalori. We investigate atoms in Boolean algebras induced by algebraically closed subsets of Boolean ordered structures. We prove uniqueness of prime models in strongly o-minimal theories of Boolean ordered structures.

In this paper a new method, elimination of Skolem functions for monotone formulas, is developed which makes it possible to determine precisely the arithmetical strength of instances of various non-constructive function existence principles. This is achieved by reducing the use of such instances in a given proof to instances of certain arithmetical principles. Our framework are systems ${\cal T}^{\omega} :={\rm G}_n{\rm A}^{\omega} +{\rm AC}$ -qf $+\Delta$ , where (G $_n$ A $^{\omega})_{n \in {\Bbb N}}$ is a hierarchy of (weak) (...) subsystems of arithmetic in all finite types (introduced in [14]), AC-qf is the schema of quantifier-free choice in all types and $\Delta$ is a set of certain analytical principles which e.g. includes the binary König's lemma. We apply this method to show that the arithmetical closures of single instances of $\Pi^0_1$ -comprehension and -choice contribute to the growth of extractable bounds from proofs relative to ${\cal T}^\omega$ only by a primitive recursive functional in the sense of Kleene. In subsequent papers these results are widely generalized and the method is used to determine the arithmetical content of single sequences of instances of the Bolzano-Weierstraß principle for bounded sequences in ${\Bbb R}^d$ , the Ascoli-lemma and others. (shrink)

We are in places. Some places beckon us, some are to be avoided, and some are banal. However, this emplacement urges reflection. In this essay I consider the role of place in environmental experiences, beginning with analysis of the concepts of place and space that motivate the development of four environmental imaginaries (extractive, wilderness, managed, and reciprocal). Ultimately, through a discussion of fences, I aim to show how place-meanings are materially inscribed on the landscape while evidencing the value of (...) place-based analysis to understanding the ways these landscapes are shaped by and serve to shape experience. (shrink)

This paper provides a procedure which, from any Boolean system of sentences, outputs another Boolean system called the ‘_m_-cycle unwinding’ of the original Boolean system for any positive integer _m_. We prove that for all \(m>1\), this procedure eliminates the direct self-reference in that the _m_-cycle unwinding of any Boolean system must be indirectly self-referential. More importantly, this procedure can preserve the primary periods of Boolean paradoxes: whenever _m_ is relatively prime to all primary periods of a Boolean paradox, this (...) paradox and its _m_-cycle unwinding have the same primary periods. In this way, we can produce an indirectly self-referential Boolean paradox with the same periodic characteristics as a known Boolean paradox. (shrink)

We consider two-sorted theories of vector spaces and prove a criterion for the assertion that such a theory allows elimination of quantifiers over vector variables.

Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

In [2], Singer proved that the theory of ordered differential fields has a model completion, i.e, the theory of closed ordered differential fields, CODF. As a result, CODF admits elimination of quantifiers. In this paper we give an algorithm to eliminate the quantifiers of CODF-formulas.

In the sequent systems for exponential-free linear logic and BCK logic a procedure of elimination of maximum cuts, cuts which correspond to maximum segments from natural deduction derivations, will be presented.

This paper extends theorems of Belegradek about poly-regular groups of finite rank to certain poly-regular groups of infinite rank. A model-theoretic property aiding these investigations is the elimination of unbounded quantifiers, and the paper establishes both a general model-theoretic test for this property and results about bounded quantifiers in the special context of ordered Abelian groups.