We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are "boundedly algebraically compact" in the language $(+,-,\cdot,\wedge,\vee,\leq)$ , and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with (...) any first-order language. The proofs can be translated into "naive set theory" in a uniform way. (shrink)

Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex V-definable subgroup U of 〈Mⁿ, +〉 and a lattice L of rank n. As two consequences, we (...) derive Pillay's conjecture for a saturated M as above and we show that the o-minimal fundamental group of G is isomorphic to L. (shrink)

We consider R-torsionfree modules over group rings RG, where R is a Dedekind domain and G is a finite group. In the first part of the paper [4] we compared the theory T of all R-torsionfree RG-modules and the theory T0 of RG-lattices , and we realized that they are almost always different. Now we compare their behaviour with respect to decidability, when RG-lattices are of finite, or wild representation type.

We investigate the variety corresponding to a logic, which is the combination of ukasiewicz Logic and Product Logic, and in which Gödel Logic is interpretable. We present an alternative axiomatization of such variety. We also investigate the variety, called the variety of algebras, corresponding to the logic obtained from by the adding of a constant and of a defining axiom for one half. We also connect algebras with structures, called f-semifields, arising from the theory of lattice-ordered rings, (...) and prove that every algebra can be regarded as a structure whose domain is the interval [0, 1] of an f-semifield, and whose operations are the truncations of the operations of to [0, 1]. We prove that such a structure is uniquely determined by up to isomorphism, and we establish an equivalence between the category of algebras and that of f-semifields. (shrink)

We show undecidability for lattices over a group ring ${\vec Z} \, G$ where $G$ has a cyclic subgroup of order $p^3$ for some odd prime $p$ . Then we discuss the decision problem for ${\vec Z} \, G$ -lattices where $G$ is a cyclic group of order 8, and we point out that a positive answer implies – in some sense – the solution of the “wild $\Leftrightarrow$ undecidable” conjecture.

The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order |$p$| and |$p$|-rings, |$p$| prime and lift this result to the axled Łukasiewicz–Moisil algebra |$L \cong B_s \times P$| and the ring |$\prod ^s F_2 \times \prod ^l F_p$|⁠, where |$B_s$| is a Boolean algebra of order |$2^s$|⁠, |$P$| a |$p$|-valued Post (...) algebra of order |$p^l$| and |$F_p$| is the prime field of order |$p$|⁠. (shrink)

We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain (...) various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang’s MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element. (shrink)

Special groups [M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, Memoirs Amer. Math. Soc., vol. 689, Amer. Math. Soc., Providence, RI, 2000] are a first-order axiomatization of the theory of quadratic forms. In Section 2 we investigate reduced special groups which are a lattice under their natural representation partial order ; we show that this lattice property is preserved under most of the standard constructions on RSGs; in particular finite RSGs and (...) RSGs of finite chain length are lattice ordered. We prove that the lattice property fails for the RSGs of function fields of real algebraic varieties over a uniquely ordered field dense in its real closure, unless their stability index is 1 . We show that Open Problem 1 has a positive answer for the RSG of the field Q . In the final section we explore the meaning of Open Problem 1 for formally real fields, in terms of their orders and real valuations; we introduce the notion of “parameter-rank” of a positive-primitive first-order formula of the language for special groups. (shrink)

This article turns its attention to the accounts that Foucault and Derrida made following their encounters with archives, and it relates these accounts to the files of the former East German secret police. Derrida and Foucault located differing qualities of authority in the archives that they consulted, yet they are shown here to converge around a problem of non-integrity in the structuration of the archive as supposed guarantor of epistemological sovereignty. A terminology of sovereign integrity dominates the Stasi's files, so (...) that they sit in stark contrast with the literary and cinematic texts that grapple with the Stasi's legacy — texts that are beset with images of inconsistency and perforation. When read in dialogue with the poststructuralist accounts of the archive, these spy files and the cultural works that emerged after their opening enable new reflection on the ethics of visiting archives, as an act of doing justice that nonetheless risks collapsing the fragments of complex pasts into the narrative wholes of the political present. (shrink)

Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.

We study the variety Var() of lattice-ordered monoids generated by the natural numbers. In particular, we show that it contains all 2-generated positively ordered lattice-ordered monoids satisfying appropriate distributive laws. Moreover, we establish that the cancellative totally ordered members of Var() are submonoids of ultrapowers of and can be embedded into ordered fields. In addition, the structure of ultrapowers relevant to the finitely generated case is analyzed. Finally, we provide a complete isomorphy invariant (...) in the two-generated case. (shrink)

Part I of this paper is developed in the tradition of Stone-type dualities, where we present a new topological representation for general lattices (influenced by and abstracting over both Goldblatt's [17] and Urquhart's [46]), identifying them as the lattices of stable compact-opens of their dual Stone spaces (stability refering to a closure operator on subsets). The representation is functorial and is extended to a full duality.In part II, we consider lattice-ordered algebras (lattices with additional operators), extending the Jónsson (...) and Tarski representation results [30] for Boolean algebras with Operators. Our work can be seen as developing, and indeed completing, Dunn's project of gaggle theory [13, 14]. We consider general lattices (rather than Boolean algebras), with a broad class of operators, which we dubb normal, and which includes the Jónsson-Tarski additive operators. Representation of l-algebras is extended to full duality. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Eclecticism and the Technologies of Discernment in Pietist PedagogyKelly J. WhitmerWhile the Franckesche Stiftungen (the Francke Foundations) of Halle/Saale are perhaps best known today as the institutional centre of German Pietism, throughout much of the eighteenth century they were widely regarded as a pedagogically innovative Schulstadt (or city of schools). The founder of this Schulstadt, August Hermann Francke (1663–1727), was many things to many people: Pietist, radical Lutheran, theologian, (...) pedagogue, professor of Greek and Oriental languages, preacher, member of the Berlin Academy of Sciences and proponent of a Protestant mission. Like most participants in what was initially a grassroots movement oriented around the personality of Philipp Jakob Spener (1635–1705), a minister active in Frankfurt am Main around 1670 who sought to breathe new life into the Lutheran church,2 Francke rarely referred to himself as a Pietist. From his post at the University of Halle (founded in 1694), this Spenerian prote´gé set out to create a new "universal seminar" that would make the assimilation of many diverse points of view, identities, and competing strands of knowledge its [End Page 545] defining mandate. "My purpose," Francke wrote in a letter to a Pastor Ring from Grünebartau (October 29, 1712), "is to develop a universal institution, useful for bringing into a good and holy order all that is important for the Christianity of all orders of society and relates, in some way, to the healing power of the soul."3 His Schulstadt would "bring into order" a myriad of relationships around a single point and relied, quite deliberately I propose, on an eclectic methodology to do this.In recent years, the relationship of Francke's Schulstadt to the implementation of Brandenburg-Prussia's first compulsory schooling system and teacher training institutes has been better documented than in the past.4 Scholars of education have paid closer attention to the methodological innovations of Pietist pedagogy, and its points of convergence with the ideas of John Locke (1632–1704), Jean-Jacques Rousseau (1712–1778) and Johann Heinrich Pestalozzi (1746–1827) have become subjects of mounting interest.5 Francke's significance for the history of education has been assessed mainly in terms of his ability to realize the pedagogically inclined reform projects of seventeenth-century figures, such as Johann Amos Comenius (1592–1670).6 We know that the thousands of children who participated in the life of the Schulstadt very often left Halle to become actively involved as teachers and ministers throughout the German states. Several played pivotal roles in the development of the late eighteenth-century educational reform movement known as Philanthropismus and its associated practices and institutions.7 [End Page 546]Perhaps the most striking feature of Francke's Schulstadt was its curriculum, which featured innovative training regimens in applied mathematics for children from diverse social backgrounds.8 The scale and rigor of these programs were to be found nowhere else at the time and can be linked to the overtly pedagogical mandate of the Berlin Academy of Sciences (f. 1701). These regimens deliberately juxtaposed artisanal and learned knowledge and stressed the importance of the eye as a conciliatory medium.9 Despite having acquired largely anti-empiricist reputations over the years due to their lingering interest in alchemy and their roles in securing the banishment of the "rationalist" mathematician and philosopher, Christian Wolff (1679–1754), from their community in 1723, careful consideration of what Halle Pietists were teaching in their schools suggests a different view. These historical actors reconciled categories of knowledge production and transmission often framed as oppositional—despite what several scholars have recently shown to be their clear interconnectedness.10I argue here for the predominance of eclecticism in the Schulstadt, including a pronounced emphasis on the "middling" mathematical and mechanical sciences as tools of conciliation and self-improvement. I seek to illustrate how eclecticism was much more than a Philosophiebegriff that was well received by some Pietists, as Walter Sparn has argued, but that it is better understood as a set of conciliatory practices, traces of which resonate in Pietist lesson plans.11 Teachers in Francke's schools integrated experimental philosophy into an educational program that was introducing children to the exercises of... (shrink)

We propose a notion of -minimality for partially ordered structures. Then we study -minimal partially ordered structures such that is a Boolean algebra. We prove that they admit prime models over arbitrary subsets and we characterize -categoricity in their setting. Finally, we classify -minimal Boolean algebras as well as -minimal measure spaces.

We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and (...) its equivalent algebraic semantics BAL is definitionally equivalent to the variety of abelian lattice ordered groups, that is, the categories of the algebras in BAL and of ℓ–groups are isomorphic (see [10], Ch.4, 4). We also prove the deduction theorem for Bal and we study different kinds of semantic consequence associated to Bal. Finally, we prove the co-NP-completeness of the tautology problem of Bal. (shrink)

It is a well-known fact that MV-algebras, the algebraic counterpart of Łukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓ-groups in a straightforward manner. In this paper, we consider several logics differing from Łukasiewicz logics in that they contain further connectives: the PŁ-, PŁ'-, PŁ'△-, and ŁΠ-logics. For all their algebraic (...) counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras - the partial algebras - the representing rings. (shrink)

So many times have we heard it told and even recounted it ourselves, that the tale of Descartes’ metaphysical adventure is something we can slip our philosophical feet into without feeling the slightest pinch. The story, or perhaps, only its plot, is this: Descartes, in order to discover whether anything is certain, attempted to doubt everything; though he succeeded in casting at least a shadow of doubt on vast areas of belief, happily one item, though only one, emerged from the (...) inquiry in a respectable condition, indubitable. The remainder of the tale relates how he proceeded to deduce truth after truth from that single original certainty. (shrink)

Whenever a structure with a particularly interesting computability-theoretic property is found, it is natural to ask whether similar examples can be found within well-known classes of algebraic structures, such as groups, rings, lattices, and so forth. One way to give positive answers to this question is to adapt the original proof to the new setting. However, this can be an unnecessary duplication of effort, and lacks generality. Another method is to code the original structure into a structure in the (...) given class in a way that is effective enough to preserve the property in which we are interested. In this paper, we show how to transfer a number of computability-theoretic properties from directed graphs to structures in the following classes: symmetric, irreflexive graphs; partial orderings; lattices; rings ; integral domains of arbitrary characteristic; commutative semigroups; and 2-step nilpotent groups. This allows us to show that several theorems about degree spectra of relations on computable structures, nonpreservation of computable categoricity, and degree spectra of structures remain true when we restrict our attention to structures in any of the classes on this list. The codings we present are general enough to be viewed as establishing that the theories mentioned above are computably complete in the sense that, for a wide range of computability-theoretic nonstructure type properties, if there are any examples of structures with such properties then there are such examples that are models of each of these theories. (shrink)

We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail Abelian (...) ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. (shrink)

Kolmogorov introduced an informal calculus of problems in an attempt to provide a classical semantics for intuitionistic logic. This was later formalised by Medvedev and Muchnik as what has come to be called the Medvedev and Muchnik lattices. However, they only formalised this for propositional logic, while Kolmogorov also discussed the universal quantifier. We extend the work of Medvedev to first-order logic, using the notion of a first-order hyperdoctrine from categorical logic, to a structure which we will call the hyperdoctrine (...) of mass problems. We study the intermediate logic that the hyperdoctrine of mass problems gives us, and we study the theories of subintervals of the hyperdoctrine of mass problems in an attempt to obtain an analogue of Skvortsova’s result that there is a factor of the Medvedev lattice characterising intuitionistic propositional logic. Finally, we consider Heyting arithmetic in the hyperdoctrine of mass problems and prove an analogue of Tennenbaum’s theorem on computable models of arithmetic. (shrink)

We show the non-arithmeticity of 1st order theories of lattices of Σ n sentences modulo provable equivalence in a formal theory, of diagonalizable algebras of a wider class of arithmetic theories than has been previously known, and of the lattice of degrees of interpretability over PA. The first two results are applications of Nies’ theorem on the non-arithmeticity of the 1st order theory of the lattice of r.e. ideals on any effectively dense r.e. Boolean algebra. The theorem on (...) degrees of interpretability relies on an adaptation of techniques leading to Nies’ theorem. (shrink)

The theory of concept lattices is approached from the point of view of fuzzy logic. The notions of partial order, lattice order, and formal concept are generalized for fuzzy setting. Presented is a theorem characterizing the hierarchical structure of formal fuzzy concepts arising in a given formal fuzzy context. Also, as an application of the present approach, Dedekind–MacNeille completion of a partial fuzzy order is described. The approach and results provide foundations for formal concept analysis of vague data—the propositions (...) “object x has attribute y”, which form the input data to formal concept analysis, are now allowed to have also intermediate truth values, meeting reality better. (shrink)

In a new integration, we show that the visual-spatial structuring of time converges with auditory-spatial left–right judgments for time-related words. In Experiment 1, participants placed past and future-related words respectively to the left and right of the midpoint on a horizontal line, reproducing earlier findings. In Experiment 2, neutral and time-related words were presented over headphones. Participants were asked to indicate whether words were louder on the left or right channel. On critical experimental trials, words were presented equally loud binaurally. (...) As predicted, participants judged future words to be louder on the right channel more often than past-related words. Furthermore, there was a significant cross-modal overlap between the visual-spatial ordering and the auditory judgments , which were continuously related. These findings provide support for the assumption that space and time have certain invariant properties that share a common structure across modalities. (shrink)

It is shown that the Boolean center of complemented elements in a bounded integral residuated lattice characterizes direct decompositions. Generalizing both Boolean products and poset sums of residuated lattices, the concepts of poset product, Priestley product and Esakia product of algebras are defined and used to prove decomposition theorems for various ordered algebras. In particular, we show that FLw-algebras decompose as a poset product over any finite set of join irreducible strongly central elements, and that bounded n-potent GBL-algebras (...) are represented as Esakia products of simple n-potent MV-algebras. (shrink)

The films The Bling Ring, Spring Breakers, and Drive, were all dismissed for their depthlessness. This article argues that we need to explore the depths and variety of their engagement with surface in order to fully appreciate what these films are trying to say. The article proposes that these films in fact employ three different “strategies” of surface engagement, in and through their aesthetics; The Bling Ring relies on a sense of “skimming”, Spring Breakers engages ideas of “drifting”, while Drive (...) promotes a sense of “gliding” or “coasting”. Analysis of these strategies of surface aesthetics reveals that the films make dialectic categories of depth and surface, sign and meaning, form and content, indistinguishable, and it is precisely in doing so that they offer complex critique on the crimes they display, and the state of our current hyper-mediated and networked world. These films are not only about the stories they tell but abo... (shrink)

Freely falling point-like objects converge toward the center of the Earth. Hence the gravitational field of the Earth is inhomogeneous, and possesses a tidal component. The free fall of an extended quantum mechanical object such as a hydrogen atom prepared in a high principal-quantum-number state, i.e. a circular Rydberg atom, is predicted to fall more slowly than a classical point-like object, when both objects are dropped from the same height above the Earth’s surface. This indicates that, apart from transitions between (...) quantum states, the atom exhibits a kind of quantum mechanical incompressibility during free fall in inhomogeneous, tidal gravitational fields like those of the Earth.A superconducting ring-like system with a persistent current circulating around it behaves like the circular Rydberg atom during free fall. Like the electronic wavefunction of the freely falling atom, the Cooper-pair wavefunction is quantum mechanically incompressible. The ions in the lattice of the superconductor, however, are not incompressible, since they do not possess a globally coherent quantum phase. The resulting difference during free fall in the response of the nonlocalizable Cooper pairs of electrons and the localizable ions to inhomogeneous gravitational fields is predicted to lead to a charge separation effect, which in turn leads to a large Coulomb force that opposes the convergence caused by the tidal gravitational force on the superconducting system.A “Cavendish-like” experiment is proposed for observing the charge separation effect induced by inhomogeneous gravitational fields in a superconducting circuit. The charge separation effect is determined to be limited by a pair-breaking process that occurs when low frequency gravitational perturbations are present. (shrink)

Many properties of compacta have “textbook” definitions which are phrased in lattice-theoretic terms that, ostensibly, apply only to the full closed-set lattice of a space. We provide a simple criterion for identifying such definitions that may be paraphrased in terms that apply to all lattice bases of the space, thereby making model-theoretic tools available to study the defined properties. In this note we are primarily interested in properties of continua related to unicoherence; i.e., properties that speak to (...) the existence of “holes” in a continuum and in certain of its subcontinua. (shrink)

In this article, we present results from a literature review of intrinsic, instrumental, and relational values of nature conducted for the Intergovernmental Science-Policy Platform on Biodiversity and Ecosystem Services, as part of the Methodological Assessment of the Diverse Values and Valuations of Nature. We identify the most frequently recurring meanings in the heterogeneous use of different value types and their association with worldviews and other key concepts. From frequent uses, we determine a core meaning for each value type, which is (...) sufficiently inclusive to serve as an umbrella over different understandings in the literature and specific enough to help highlight its difference from the other types of values. Finally, we discuss convergences, overlapping areas, and fuzzy boundaries between different value types to facilitate dialogue, reduce misunderstandings, and improve the methods for valuation of nature's contributions to people, including ecosystem services, to inform policy and direct future research. © 2023 The Author(s). Published by Oxford University Press on behalf of the American Institute of Biological Sciences. (shrink)

We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields.

We study the first-order theory of polynomial rings over a GCD domain and of the ring of formal entire functions over a non-Archimedean field in the language $\{ 1, +, \bot \}$. We show that these structures interpret the first-order theory of the semi-ring of natural numbers. Moreover, this interpretation depends only on the characteristic of the original ring, and thus we obtain uniform undecidability results for these polynomial and entire functions rings of a fixed characteristic. This work (...) enhances results of Raphael Robinson on essential undecidability of some polynomial or formal power series rings in languages that contain no symbols related to the polynomial or power series ring structure itself. (shrink)

A first order representation (f.o.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions "one f.o.r. is at least as expressive as another relative to a class of spaces" and "one class of spaces is definable in another relative to an f.o.r.", and prove some general statements. Following this we compare some well-known classes of spaces and (...) first order representations. A principal result is that if X and Y are two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positive-universal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting. (shrink)

In any variety of bounded integral residuated lattice-ordered commutative monoids the class of its semisimple members is closed under isomorphic images, subalgebras and products, but it is not closed under homomorphic images, and so it is not a variety. In this paper we study varieties of bounded residuated lattices whose semisimple members form a variety, and we give an equational presentation for them. We also study locally representable varieties whose semisimple members form a variety. Finally, we analyze the (...) relationship with the property “to have radical term”, especially for k-radical varieties, and for the hierarchy of varieties k>0 defined in Cignoli and Torrens. (shrink)

The Artin‐Schreier theorem says that every formally real field has orders. Friedman, Simpson and Smith showed in [6] that the Artin‐Schreier theorem is equivalent to over. We first prove that the generalization of the Artin‐Schreier theorem to noncommutative rings is equivalent to over. In the theory of orderings on rings, following an idea of Serre, we often show the existence of orders on formally real rings by extending pre‐orders to orders, where Zorn's lemma is used. We then (...) prove that “pre‐orders on rings not necessarily commutative extend to orders” is equivalent to. (shrink)

In spite of differences the thought of Bernays, Dooyeweerd and Gödel evinces a remarkable convergence. This is particularly the case in respect of the acknowledgement of the difference between the discrete and the continuous, the foundational position of number and the fact that the idea of continuity is derived from space (geometry – Bernays). What is furthermore similar is the recognition of what is primitive (and indefinable) as well as the account of the coherence of what is unique, such (...) as when Gödel observes something quasi-spatial character of sets. It is shown that Dooyeweerd’s theory of modal aspects provides a philosophical framework that exceeds his own restrictive understanding of infinity )to the potential infinite) and at the same time makes it possible to account for key insights found in the thought of Bernays and Gödel. When Laugwitz says that discreteness rules within the sphere of the numerical, he says nothing more than what Dooyeweerd had in mind with his idea that discrete quantity, as the meaning-nucleus of the arithmetical aspect, qualifies every element within the structure of the quantitative aspect. And when Bernays says that analysis expresses the idea of the continuum in arithmetical language his mode of speech is equivalent to saying that mathematical analysis could seen as being founded upon the spatial anticipation within the modal structure of the arithmetical aspect. The view of the actual infinite (the at once infinite) in terms of an “as if” approach (Bernays), that is, as appreciated as a regulative hypothesis through which every successively infinite multiplicity of numbers could be envisaged as being giving all at once, as an infinite totality, provides a sound understanding of the at once infinite and makes it plain why every form of arithmeticism fails. Such attempts have to call upon Cantor’s proof on the non-denumerability of the real numbers – and this proof pre-supposes the use of the at once infinite which, in turn, pre-supposes the (irreducibile) spatial order of simultaneity and the patial whole-parts relation. (shrink)

We consider the problem of constructing first-order definitions in the language of rings of holomorphy rings of one-variable function fields of characteristic 0 in their integral closures in finite extensions of their fraction fields and in bigger holomorphy subrings of their fraction fields. This line of questions is motivated by similar existential definability results over global fields and related questions of Diophantine decidability.

This new edition of Introduction to Lattices and Order presents a radical reorganization and updating, though its primary aim is unchanged. The explosive development of theoretical computer science in recent years has, in particular, influenced the book's evolution: a fresh treatment of fixpoints testifies to this and Galois connections now feature prominently. An early presentation of concept analysis gives both a concrete foundation for the subsequent theory of complete lattices and a glimpse of a methodology for data analysis that is (...) of commercial value in social science. Classroom experience has led to numerous pedagogical improvements and many new exercises have been added. As before, exposure to elementary abstract algebra and the notation of set theory are the only prerequisites, making the book suitable for advanced undergraduates and beginning graduate students. It will also be a valuable resource for anyone who meets ordered structures. (shrink)

By RCA 0 , we denote a subsystem of second order arithmetic based on Δ0 1 comprehension and Δ0 1 induction. We show within this system that the real number system R satisfies all the theorems (possibly with non-standard length) of the theory of real closed fields under an appropriate truth definition. This enables us to develop linear algebra and polynomial ring theory over real and complex numbers, so that we particularly obtain Hilbert’s Nullstellensatz in RCA 0.

We prove that every convexly ordered valuation ring has a unique completion as a uniform space, which furthermore is a convexly ordered valuation ring. In addition, we give a model theoretic characterisation of complete convexly ordered valuation rings, and give a necessary and sufficient condition for the completion of a convexly ordered valuation ring to be a real closed ring.

Nelson Goodman's new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper's notion of degrees of testability is linked to (...) minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham's razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman's riddle is just one example. (shrink)

Let G be a finite group, T denote the theory of Z[G]-lattices . It is shown that T is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p4, or p is odd and S is non-cyclic of order p2, or p = 2 and S is a non-cyclic abelian group of order 8 . More precisely, first we prove that T is undecidable because it interprets the word problem (...) for finite groups; then we lift undecidability from T to T. (shrink)

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: (...) the lattice of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

Glaucon¿s story about the ring of invisibility in Republic 359d-60b is examined in order to assess the wider role of fictional fabrication in Plato¿s philosophical argument. The first part of the article (I) looks at the close connections this tale has to the account of Gyges in Herodotus (1.8-12). It is argued that Plato exhibits a specific dependence on Herodotus, which suggests Glaucon¿s story might be an original invention: the assumption that there must be a lost ¿original¿ to inspire Plato¿s (...) story of the ring has never accommodated the possibility of Plato drawing, perhaps quite directly, from Herodotus. The next section (II) considers the function of that fable within the larger philosophical and aesthetic structure of the Republic. Appreciation of the entire dialogue as an exercise in fiction, as well as philosophy, helps to reveal the ways in which philosophical argument and fictional invention are closely bound up in the formation of Glaucon¿s fabulous anecdote. Finally (III), a reading of Cicero¿s treatment of the story in De Officiis confirms the degree to which philosophical reasoning and fiction can be quite generally interdependent. Although the arguments in Sections II and III are consistent with the opening contention that the ring story was invented by Plato, they do not presuppose it. (shrink)