Let {: i ∈I } be a family of compact spaces and let X be their Tychonoff product. [MATHEMATICAL SCRIPT CAPITAL C] denotes the family of all basic non-trivial closed subsets of X and [MATHEMATICAL SCRIPT CAPITAL C]R denotes the family of all closed subsets H = V × Πmath imageXi of X, where V is a non-trivial closed subset of Πmath imageXi and QH is a finite non-empty subset of I. We show: Every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT (...) CAPITAL C]R extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ if and only if every family H ⊂ [MATHEMATICAL SCRIPT CAPITAL C] with the finite intersection property extends to a maximal [MATHEMATICAL SCRIPT CAPITAL C] family F with the fip. The proposition “if every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ, then X is compact” is not provable in ZF. The statement “for every family {: i ∈ I } of compact spaces, every filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R, Y = Πi ∈IYi, extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem. The statement “for every family {: i ∈ ω } of compact spaces, every countable filterbase ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C]R, X = Πi ∈ωXi, extends to a [MATHEMATICAL SCRIPT CAPITAL C]R-ultrafilter ℱ” is equivalent to Tychonoff's compactness theorem restricted to countable families. The countable Axiom of Choice is equivalent to the proposition “for every family {: i ∈ ω } of compact topological spaces, every countable family ℋ ⊂ [MATHEMATICAL SCRIPT CAPITAL C] with the fip extends to a maximal [MATHEMATICAL SCRIPT CAPITAL C] family ℱ with the fip”. (shrink)

We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.

This is a continuation of [2]. We study the Tychonoff Compactness Theorem for various definitions of compactness and for various types of spaces . We also study well ordered Tychonoff products and the effect that the multiple choice axiom has on such products.

We establish the following results: 1. In ZF (i.e., Zermelo-Fraenkel set theory minus the Axiom of Choice AC), for every set I and for every ordinal number α ≥ ω, the following statements are equivalent: (a) The Tychonoff product of| α| many non-empty finite discrete subsets of I is compact. (b) The union of| α| many non-empty finite subsets of I is well orderable. 2. The statement: For every infinite set I, every closed subset of the Tychonoff product (...) [0, 1] I which consists of functions with finite support is compact, is not provable in ZF set theory. 3. The statement: For every set I, the principle of dependent choices relativised to I implies the Tychonoff product of countably many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰ (i.e., ZF minus the Axiom of Regularity). 4. The statement: For every set I, every ℵ₀-sized family of non-empty finite subsets of I has a choice function implies the Tychonoff product of ℵ₀ many non-empty finite discrete subsets of I is compact, is not provable in ZF⁰. (shrink)

In this paper, we discuss some questions about compactness in MV-topological spaces. More precisely, we first present a Tychonoff theorem for such a class of fuzzy topological spaces and some consequence of this result, among which, for example, the existence of products in the category of Stone MV-spaces and, consequently, of coproducts in the one of limit cut complete MV-algebras. Then we show that our Tychonoff theorem is equivalent, in ZF, to the Axiom of Choice, classical (...) class='Hi'>Tychonoff theorem, and Lowen’s analogous result for lattice-valued fuzzy topology. Last, we show an extension of the Stone–Čech compactification functor to the category of MV-topological spaces, and we discuss its relationship with previous works on compactification for fuzzy topological spaces. (shrink)

We study the Tychonoff Compactness Theorem for several different definitions of a compact space.

We investigate in ZF conditions that are necessary and sufficient for countable products ∏m∈ℕXm of finite Hausdorff spaces Xm resp. Hausdorff spaces Xm with at most n points to be compact resp. Baire. Typica results: Countable products of finite Hausdorff spaces are compact if and only if countable products of non-empty finite sets are non-empty. Countable products of discrete spaces with at most n + 1 points are compact if and only if countable products of (...) non-empty sets with at most n points are non-empty. (shrink)

We study the role that the axiom of choice plays in Tychonoff's product theorem restricted to countable families of compact, as well as, Lindelöf metric spaces, and in disjoint topological unions of countably many such spaces.

We show that for every we ordered cardinal number m the Tychonoff product 2m is a compact space without the use of any choice but in Cohen's Second Mode 2ℝ is not compact.

We show that the property of sequential compactness for subspaces of.

We show that the Axiom of Choice is equivalent to each of the following statements: A product of closures of subsets of topological spaces is equal to the closure of their product ; A product of complete uniform spaces is complete.

If a locale is presented by a “flat site”, it is shown how its frame can be presented by generators and relations as a dcpo. A necessary and sufficient condition is derived for compactness of the locale . Although its derivation uses impredicative constructions, it is also shown predicatively using the inductive generation of formal topologies. A predicative proof of the binary Tychonoff theorem is given, including a characterization of the finite covers of the product by basic opens. The (...) discussion is then related to the double powerlocale. (shrink)

In intuitionistic generalized predicative systems as constructive set theory, or constructive type theory, two categories have been proposed to play the role of the category of locales: the category FSp of formal spaces, and its full subcategory FSpi of inductively generated formal spaces. Considered in impredicative systems as the intuitionistic set theory IZF, FSp and FSpi are both equivalent to the category of locales. However, in the mentioned predicative systems, FSp fails to be closed under basic constructions such as that (...) of finite products, while FSpi is complete , but does not contain some naturally occurring classes of formal spaces. Moreover, completeness of FSpi only holds under the assumption of strong principles for the existence of inductively defined sets.In this paper, working in the context of constructive set theory, the category ImLoc of imaginary locales is introduced. ImLoc is complete already over weak formulations of constructive set theory, contains FSp and FSpi as full subcategories, and, as FSp and FSpi, is equivalent to the category of locales in IZF.Applications of the concept of imaginary locale to a proof of a point-free version of Tychonoff embedding theorem, and to set-representation theorems, are then presented. (shrink)

Imagination is important for many things in science: solving problems, interpreting data, designing studies, etc. Philosophers of imagination typically account for the productive role played by imagination in science by focusing on how imagination is constrained, e.g., by using self-imposed rules to infer logically, or model events accurately. But the constraints offered by these philosophers either constrain too much, or not enough, and they can never account for uses of imagination that are needed to break today’s constraints in order to (...) make progress tomorrow. Thus, epistemology of imagination needs to make room for an element of epistemological anarchy. (shrink)

A doctrine of intelligent design through evolution is not going to find many friends. It is destined to encounter opposition on all sides. Among scientists the backlog of evolution will have little patience for intelligent design. Among religiousists, many who form intelligent design have their doubts about evolution. In the general public s mind there is a diametrical opposition between evolution and intelligent design: one excludes the other. This book will argue that this view of the matter is not correct, (...) and that in actuality one can regard evolution itself as a pathway to intelligent design. We would do well to go beyond The Origin of Species and taking as our guide such works as W. Wentworth Thomson s On Growth and Form acknowledging that evolutionary adaptation can result in solutions of a sort that intelligence could readily ratify. Accordingly, what the present book seeks is a naturalization of Intelligent Design that sees such design as itself the result of natural and evolutionary processes.". (shrink)

We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a (...) further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations. (shrink)

We study the preservation of certain properties under products of classes of finite structures. In particular, we examine indivisibility, definable self-similarity, the amalgamation property, and the disjoint n-amalgamation property. We explore how each of these properties interacts with the lexicographic product, full product, and free superposition of classes of structures. Additionally, we consider the classes of theories which admit configurations indexed by these products. In particular, we show that, under mild assumptions, the products considered in this article (...) do not yield new classes of theories. (shrink)

Louis Althusser argued that Marx initiated a transformation of philosophy, a new way of doing philosophy. This book follows that provocation to examine the way in which central Marxist concepts and problems from primitive accumulation to real abstraction animate and inform philosophers from Theodor Adorno to Paolo Virno. While also examining the way in which reading Marx casts new light on such philosophers as Spinoza. At the centre of this transformation is the production of subjectivity, the manner in which relations (...) of production produces ways of thinking and living. (shrink)

This paper criticises the view that, according to Marx, “productive forces” determine “relations of production” and that the growth of the former basically determines the course of history. The particular version of this account discussed is that to be found in G.A. Cohen’s Karl Marx's Theory of History: A Defence. The main part of this criticism in:-volves a presentation of what, it is suggested, was in fact Marx's conception of "productive forces", "relations of production" and their relations, and an identification (...) of class-struggle as the primary factor at least for the historically significant periods with which Marx was concerned. (shrink)

There has been much discussion of so-called teleosemantic approaches to the naturalization of content. Such discussion, though, has been largely confined to simple, innate mental states with contents such as ?There is a fly here.? Even assuming we can solve the issues that crop up at this stage, an account of the content of human mental states will not get too far without an account of productivity: the ability to entertain indefinitely many thoughts. The best-known teleosemantic theory, Millikan's biosemantics, offers (...) an account of productivity in thought. This paper raises a basic worry about this account: that the use of mapping functions in the theory is unacceptable from a naturalistic point of view. (shrink)

Volume 47, Issue 3, November 2018, Page 467-472.

The 2023 edition of the Venice Architecture Biennial Time Space Existence will draw attention to the emerging expressions of sustainability in their numerous forms, ranging from a focus on the environment and urban landscape to the unfolding conversations on innovation, reuse, community, and inclusion. In response to climate change, exhibited projects will investigate new technologies and construction methods that reduce energy consumption through circular design and develop innovative, organic, and recycled building materials. Participants will also address social justice by presenting (...) living solutions envisioned for displaced communities and minorities, while others will examine the tensions between the built urban environment and the nature surrounding it, identifying opportunities for coexistence. Time Space Existence not only aims to be a display but also a living workshop where established and emerging practitioners can work together and present their own visions of the status of architecture and the built environment nowadays. By bringing together ideas from across the world, the exhibition presents itself as a platform for exchange and experimentation and strives to stimulate an engaging dialogue that is inclusive of all voices. (shrink)

This bookis about the roots of managerial rationality. A theoretical base, founded on the concept of 'memetics' is developed in order to explain human thinking and human reason as products of cultural evolution. Cultural change and development are explained by simple, value-driven memetic mechanisms like 'ritualization' and 'extremization'.

Henri Lefebvre has considerable claims to be the greatest living philosopher. His work spans some sixty years and includes original work on a diverse range of subjects, from dialectical materialism to architecture, urbanism and the experience of everyday life. The Production of Space is his major philosophical work and its translation has been long awaited by scholars in many different fields. The book is a search for a reconciliation between mental space and real space. In the course of his exploration, (...) Henri Lefebvre moves from metaphysical and ideological considerations of the meaning of space to its experience in the everyday life of home and city. He seeks, in other words, to bridge the gap between the realms of theory and practice, between the mental and the social, and between philosophy and reality. In doing so, he ranges through art, literature, architecture and economics, and further provides a powerful antidote to the sterile and obfuscatory methods and theories characteristic of much recent continental philosophy. This is a work of great vision and incisiveness. It is also characterized by its author's wit and by anecdote, as well as by a deftness of style which Donald Nicholson-Smith's sensitive translation precisely captures. (shrink)