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 and study some local versions of o-minimality, requiring that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighbourhood of every point. Motivating examples are the expansions of the ordered reals by sine, cosine and other periodic functions.

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.

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)

In the first centenary of Cantor's death, we discuss how to introduce his life, his works and his theories about mathematical infinity to today's students. Keywords: proper and improper infinite, cardinal number, countable set, continuum, continuum hypothesis. Sunto Nel primo centenario della scomparsa di Cantor, si discute come presentare la sua vita, le sue opere e le sue teorie sull’infinito agli studenti di oggi. Parole chiave: infinito proprio e improprio, numero cardinale, numerabile, continuo, ipotesi del continuo.

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 consider the decision problem for modules over a group ring ℤ[G], where G is a cyclic group of prime order. We show that it reduces to the same problem for a class of certain abelian structures, and we obtain some partial decidability results for this class.

We provide a first order axiomatization of the expansion of the complex field by the exponential function restricted to the subring of integers modulo the first order theory of (Z, +, ·).

We extend the analysis of the decision problem for modules over a group ring ℤ[G] to the case when G is a cyclic group of squarefree order. We show that separated ℤ[G]-modules have a decidable theory, and we discuss the model theoretic role of these modules within the class of all ℤ[G]-modules. The paper includes a short analysis of the decision problem for the theories of modules over ℤ[ζm], where m is a positive integer and ζm is a primitive mth (...) root of 1. (shrink)

We propose a definition of weak o-minimality for structures expanding a Boolean algebra. We study this notion, in particular we show that there exist weakly o-minimal non o-minimal examples in this setting.

We prove that, if V is an effectively given commutative valuation domain such that its value group is dense and archimedean, then the theory of all V-modules is decidable.

For arbitrary finite group $G$ and countable Dedekind domain $R$ such that the residue field $R/P$ is finite for every maximal $R$ -ideal $P$ , we show that the localizations at every maximal ideal of two $RG$ -lattices are isomorphic if and only if the two lattices satisfy the same first order sentences. Then we investigate generalizations of the above results to arbitrary $R$ -torsion-free $RG$ -modules and we apply the previous results to show the decidability of the theory of (...) ${\vec Z}C(2)^2$ -lattices. Eventually, we show that ${\vec Z} [i] C(2)^2$ -lattices have undecidable theory. (shrink)

On the basis of the Klingler–Levy classification of finitely generated modules over commutative noetherian rings we approach the old problem of classifying finite commutative rings R with a decidable theory of modules. We prove that if R is wild, then the theory of all R-modules is undecidable, and verify decidability of this theory for some classes of tame finite commutative rings.

Discrete weakly o-minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o-minimal structures with infinite definable discrete subsets and we study the Boolean algebra of definable sets of its countable models.

We consider the sets definable in the countable models of a weakly o-minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic , in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within (...) expansions of Boolean lattices, every weakly o-minimal theory is p-ω-categorical. (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.

The study of pairs of modules (over a Dedekind domain) arises from two different perspectives, as a starting step in the analysis of tuples of submodules of a given module, or also as a particular case in the analysis of Abelian structures made by two modules and a morphism between them. We discuss how these two perspectives converge to pairs of modules, and we follow the latter one to obtain an alternative approach to the classification of pairs of torsionfree objects. (...) Then we restrict our attention to pairs of free modules. Our main results are that the theory of pairs of free Abelian groups is co-recursively enumerable, and that a few remarkable extensions of this theory are decidable. (shrink)

Let G be a finite group. For every formula ø in the language of groups, let K denote the class of groups H such that ø is a normal abelian subgroup of H and the quotient group H;ø is isomorphic to G. We show that if G is nilpotent and its order is not square-free, then there exists a formula ø such that the theory of K is undecidable.

Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ℤ[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree.

Let G be the direct sum of the noncyclic groupof order four and a cyclic groupwhoseorderisthe power pn of some prime p. We show that ℤ2G-lattices have a decidable theory when the cyclotomic polynomia equation image is irreducible modulo 2ℤ for every j ≤ n. More generally we discuss the decision problem for ℤ2G-lattices when G is a finite group whose Sylow 2-subgroups are isomorphic to the noncyclic group of order four.

A module is weakly minimal if and only if every pp-definable subgroup is either finite or of finite index. We study weakly minimal modules over several classes of rings, including valuation domains, Prüfer domains and integral group rings.

We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains whose localizations at maximal ideals have dense value groups. For Bézout domains, these conditions are also necessary.

We study the R‐torsion‐free part of the Ziegler spectrum of an order Λ over a Dedekind domain R. We underline and comment on the role of lattices over Λ. We describe the torsion‐free part of the spectrum when Λ is of finite lattice representation type.

Minding Time: A Philosophical and Theoretical Approach to the Psychology of Time offers an innovative philosophical account of the most fundamental kinds of time representation.

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last (...) century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

Many of the computing systems programmed using Machine Learning are opaque: it is difficult to know why they do what they do or how they work. Explainable Artificial Intelligence aims to develop analytic techniques that render opaque computing systems transparent, but lacks a normative framework with which to evaluate these techniques’ explanatory successes. The aim of the present discussion is to develop such a framework, paying particular attention to different stakeholders’ distinct explanatory requirements. Building on an analysis of “opacity” from (...) philosophy of science, this framework is modeled after accounts of explanation in cognitive science. The framework distinguishes between the explanation-seeking questions that are likely to be asked by different stakeholders, and specifies the general ways in which these questions should be answered so as to allow these stakeholders to perform their roles in the Machine Learning ecosystem. By applying the normative framework to recently developed techniques such as input heatmapping, feature-detector visualization, and diagnostic classification, it is possible to determine whether and to what extent techniques from Explainable Artificial Intelligence can be used to render opaque computing systems transparent and, thus, whether they can be used to solve the Black Box Problem. (shrink)

In his new book, Carlos Alberto Torres, an internationally renowned critical theorist of education, explores the early writings of Paulo Freire whose ideas have had a tremendous and long-lasting impact on the world of pedagogy and politics. Torres analyzes Freire's works, from the 1960s and 1970s, before Freire gained worldwide recognition for his Pedagogy of the Oppressed. Offering an in-depth look into the formative thinking of Freire, Torres identifies how his ideas produced frameworks for educating global citizens, building community and (...) mutual respect, creating social responsibility, instilling an appreciation for diversity, promoting multiple literacies, and social justice education. This volume is the result of more than 3 decades of research with access to Freire's personal library and the archives of the Paulo Freire Institute, as well as the author's extensive conversations with Paulo Freire over two decades--Dr. Torres was Freire's adviser during his tenure as Secretary of Education in the Municipality of São Paulo, Brazil, 1989-1991. (shrink)

This paper extends earlier work by its authors on formal aspects of the processes of contracting a theory to eliminate a proposition and revising a theory to introduce a proposition. In the course of the earlier work, Gardenfors developed general postulates of a more or less equational nature for such processes, whilst Alchourron and Makinson studied the particular case of contraction functions that are maximal, in the sense of yielding a maximal subset of the theory (or alternatively, of one of (...) its axiomatic bases), that fails to imply the proposition being eliminated. In the present paper, the authors study a broader class, including contraction functions that may be less than maximal. Specifically, they investigate "partial meet contraction functions", which are defined to yield the intersection of some nonempty family of maximal subsets of the theory that fail to imply the proposition being eliminated. Basic properties of these functions are established: it is shown in particular that they satisfy the Gardenfors postulates, and moreover that they are sufficiently general to provide a representation theorem for those postulates. Some special classes of partial meet contraction functions, notably those that are "relational" and "transitively relational", are studied in detail, and their connections with certain "supplementary postulates" of Gardenfors investigated, with a further representation theorem established. (shrink)