We prove that there is a -ideal of compact sets which is strictly above in the Tukey order. Here is the collection of all compact nowhere dense subsets of the Cantor set. This answers a question of Louveau and Veličković asked in [Alain Louveau, Boban Veličković, Analytic ideals and cofinal types, Ann. Pure Appl. Logic 99 171–195].

We prove several trichotomy results for ideals of compact sets. Typically, we show that a "sufficiently rich" universally Baire ideal is either $\Pi _{3}^{0}$-hard, or $\Sigma _{3}^{0}$-hard, or else a σ-ideal.

Given a compact Polish space E and the hyperspace of its compact subsets , we consider Gδσ-ideals of compact subsets of E. Solecki has shown that any σ-ideal in a broad natural class of Gδ ideals can be represented via a compact subset of ; in this article we examine the behaviour of Gδ subsets of E with respect to the representing set. Given an ideal I in this class, we construct a representing set (...) that recognises a compact subset of E as being “small” precisely when it is in I, and recognises a Gδ subset of E as being “small” precisely when it is covered by countably many compact sets from I. (shrink)

Reviewed Works:Murray G. Bell, J. Streprans, S. Watson, Spaces of Ideals of Partial Functions.Alan Dow, Compact Spaces of Countable Tightness in the Cohen Model.Peter J. Nyikos, Classes of Compact Sequential Spaces.Franklin D. Tall, Topological Problems for Set-Theorists.

This is a survey paper on the descriptive set theory of hereditary families of closed sets in Polish spaces. Most of the paper is devoted to ideals and σ-ideals of closed or compact sets.

We extract some properties of Mahlo’s operation and show that some other very natural operations share these properties. The weakly compact sets form a similar hierarchy as the stationary sets. The height of this hierarchy is a large cardinal property connected to saturation properties of the weakly compact ideal.

We show that a first category homogeneous zero-dimensional Borel set X can be embedded in as an ideal on ω if and only if X is homeomorphic to X × X if and only if X is Wadge-equivalent to X × X. Furthermore, we determine the Wadge classes of such X, thus giving a complete picture of the possible descriptive complexity of Borel ideals on ω. We also discuss the connection with ideals of compact sets.

We investigate the Tukey order in the class of Fσ ideals of subsets of ω. We show that no nontrivial Fσ ideal is Tukey below a Gδ ideal of compact sets. We introduce the notions of flat ideals and gradually flat ideals. We prove a dichotomy theorem for flat ideals isolating gradual flatness as the side of the dichotomy that is structurally good. We give diverse characterizations of gradual flatness among flat ideals using (...) Tukey reductions and games. For example, we show that gradually flat ideals are precisely those flat ideals that are Tukey below the ideal of density zero sets. (shrink)

We study the structure of analytic ideals of subsets of the natural numbers. For example, we prove that for an analytic ideal I, either the ideal {X (Ω × Ω: En X ({0, 1,…,n} × Ω } is Rudin-Keisler below I, or I is very simply induced by a lower semicontinuous submeasure. Also, we show that the class of ideals induced in this manner by lsc submeasures coincides with Polishable ideals as well as analytic P-ideals. We (...) study this class of ideals and characterize, for example, when the ideals in it are Fσ or when they carry a locally compact group topology. We apply these results to Borel partial orders to rederive a theorem of Todorcevic and to Borel equivalence relations to answer a question of Kechris and Louveau. As another application we give a characterization of σ-ideals of μ-zero sets for Maharam submeasures μ on the Cantor set which is to a large extent analogous to a characterization of the meager ideal due to Kechris and the author. (shrink)

§1. Introduction. Ideals and filters of subsets of natural numbers have been studied by set theorists and topologists for a long time. There is a vast literature concerning various kinds of ultrafilters. There is also a substantial interest in nicely definable ideals—these by old results of Sierpiński are very far from being maximal— and the structure of such ideals will concern us in this announcement. In addition to being interesting in their own right, Borel and analytic (...) class='Hi'>ideals occur naturally in the investigations of compact subsets of the space of all Baire class 1 functions on a Polish space, see [12, 18]. Also, certain objects associated with such ideals are of considerable interest and were quite extensively studied by several authors. Let us list here three examples; in all three of them I stands for an analytic or Borel ideal.1. The partial order induced by I on P: X ≥I Y iff X \ Y ϵ I and the partial order.2. Boolean algebras of the form P/I and their automorphisms.3. The equivalence relation associated with I: XEI Y iff X Δ ϵ I.In Section 4, we will have an opportunity to state some consequences of our results for equivalence relations as in 3. (shrink)

With every σ-ideal I on a Polish space we associate the σ-ideal I* generated by the closed sets in I. We study the forcing notions of Borel sets modulo the respective σ-ideals I and I* and find connections between their forcing properties. To this end, we associate to a σ-ideal on a Polish space an ideal on a countable set and show how forcing properties of the forcing depend on combinatorial properties of the ideal. We also study (...) the 1—1 or constant property of σ-ideals, i.e., the property that every Borel function defined on a Borel positive set can be restricted to a positive Borel set on which it either 1—1 or constant. We prove the following dichotomy: if I is a σ-ideal generated by closed sets, then either the forcing P I adds a Cohen real, or else I has the 1—1 or constant property. (shrink)

We start by studying the relationship between two invariants isolated by Shelah, the sets of good and approachable points. As part of our study of these invariants, we prove a form of “singular cardinal compactness” for Jensen's square principle. We then study the relationship between internally approachable and tight structures, which parallels to a certain extent the relationship between good and approachable points. In particular we characterise the tight structures in terms of PCF theory and use our characterisation to (...) prove some covering results for tight structures, along with some results on tightness and stationary reflection. Finally, we prove some absoluteness theorems in PCF theory, deduce a covering theorem, and apply that theorem to the study of precipitous ideals. (shrink)

Let κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ + -saturated, i.e., are there κ + stationary subsets of κ with pairwise intersections nonstationary? Our first observation is: Theorem. NS is κ + -saturated iff for every normal ideal J on κ there is a stationary set $A \subseteq \kappa$ such that $J = NS \mid A = \{X \subseteq (...) \kappa:X \cap A \in NS\}$ . Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define "greatly Mahlo" cardinals and obtain the following: Theorem. If κ is greatly Mahlo then NS is not κ + -saturated. Theorem. If κ is ordinal Π 1 1 -indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries a κ-saturated ideal, then κ is greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π 1 1 -indescribable cardinal. These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ + -saturated. (shrink)

Compact set valued iterations generalize classical point iterations quite naturally by replacing the function f with a tube f in the discrete iterations equation. In Section 3, some bifurcation results about logistic tube iterations are given. In Section 4, an analogous dynamical behaviour for the phase response tube involved in the entrainment of the respiratory rhythm is studied.

We introduce a notion of ideal type such that any two ideals with the same ideal type are isomorphic. From this we infer, under the axiom t = h, that each ideal which consists of all nowhere Ramsey sets contained in some family of infinite subsets of natural numbers is isomorphic with the ideal of all nowhere Ramsey sets.

In lieu of an abstract, here is a brief excerpt of the content:The Idealization of Contingency in Traditional Japanese AestheticsRobert Wicks (bio)In many popular writings that date from the initial decades of the twentieth century, and also in recent scholarly studies, "Japanese aesthetics"—insofar as we can speak sweepingly of a complicated, multidimensional, and dynamic historical phenomenon—is characterized with a set of adjectives whose present linguistic entrenchment is clearly evident. Specifically we read that traditional Japanese aesthetics is an aesthetics of imperfection, (...) insufficiency, incompleteness, asymmetry, and irregularity, not to mention perishability, suggestiveness, and simplicity.1 Given this collection of qualities, we are presented with a matching set of paradigmatic Japanese aesthetic experiences and objects, both as illustrations and as legitimations of this close-to-standard portrait. Examples include the suggestive moon covered by drifting clouds, the irregularly formed ceramics of ceremonial teacups, the simple and asymmetrical arrangement of unpolished rocks in the dry landscape garden, the bold and dashing monochromatic strokes typical of Zen calligraphy, transient cherry blossoms, serene and cloud-capped mountain summits, lonely thatched huts, sunsets in the foggy twilight, the call of a crane that breaks through the silence, and the light autumn rain that drizzles upon a secluded pond.Upon further reflection, it becomes noticeable that some familiar Japanese aesthetic objects do not easily conform to this standard picture, and this raises doubts about the typical characterization of Japanese aesthetics. The quintessentially Japanese shōji screens and tatami mats, for instance, are uniformly rectangular rather than irregular in form. They are, moreover, neither asymmetrical, nor imperfect, nor incomplete, nor insufficient. How then, can these perfectly regular screens and mats—items crucial to the aesthetics of Japanese domestic architecture—fit neatly into the usually encountered portrait of traditional Japanese aesthetics, if it is said to be fundamentally "the worship of imperfection"?2 Or similarly if "imperfection" is [End Page 88] the keynote of this aesthetics, how does one explain the meticulously raked surfaces of the dry landscape garden or the impeccably perfected movements requisite for performing the Japanese tea ceremony?3Since such examples do not obviously fit into an "aesthetics of imperfection and insufficiency," some reconsideration of the prevailing characterization of Japanese aesthetics is needed to accommodate these, along with other examples of "perfected" items and arrangements. In short, the main difficulty is that the concept of "perfection" has been underthematized.4 To develop a revaluation of the presently submerged status of "perfection" in most scholarly discussions of Japanese aesthetics, I will highlight the familiar principle of aesthetic complementarity but contextualize the experiential foundations of this principle in a specifically Japanese-philosophic manner, namely, in reference to a fundamental feature of time-consciousness, as described by Dōgen (1200-53). This historical grounding will align the principle of complementarity with some of the acknowledged Zen Buddhist sources of traditional Japanese aesthetics.5 The upshot will be that the above-cited collection of adjectives typically used to describe the prevailing atmosphere of Japanese aesthetics—imperfection, asymmetry, incompleteness, suggestibility, etc.—presents only half of the aesthetic and philosophic story and, moreover, tends to pass over what can be appreciated as a more consequential underlying concept, namely, the idea of "contingency."In what follows, basic reflections on the phenomenology of time-consciousness will lead to the claim that, in many paradigmatic instances of traditional Japanese aesthetics, we have before us a situation that is aptly describable as "the idealization of contingency." This involves the assertion of a philosophic proposition through an aesthetic presentation (i.e., the proposition is expressed using metaphor, symbolism, or analogy). In the case of traditional Japanese aesthetics, the main proposition is the Buddhist-associated one that the foundation of things is contingent, conditional, and nonabsolute (i.e., there are no absolute foundations). The aesthetic presentation of this proposition will, as we shall see, make ample room for "perfection" in understanding Japanese aesthetics: By means of juxtaposing contingent, perishable individuals (which usually have an "imperfect" appearance) against a perfected, polished, and idealized background, the contingency of the individuals is thereby aesthetically highlighted and made more readily appreciable.The resulting criticism of the standard characterization is thus a simple one: The typical group of concepts used to describe traditional Japanese aesthetics has neglected to give due... (shrink)

We provide tools for a concise axiomatization of a broad class of quantifiers in many-valued logic, so-called distribution quantifiers. Although sound and complete axiomatizations for such quantifiers exist, their size renders them virtually useless for practical purposes. We show that for quantifiers based on finite distributive lattices compact axiomatizations can be obtained schematically. This is achieved by providing a link between skolemized signed formulas and filters/ideals in Boolean set lattices. Then lattice theoretic tools such as Birkhoff's representation theorem (...) for finite distributive lattices are used to derive tableau-style axiomatizations of distribution quantifiers. (shrink)

We investigate the covering numbers of some ideals on $${^{\omega }}{2}{}$$ ω 2 associated with tree forcings. We prove that the covering of the Sacks ideal remains small in the Silver and uniform Sacks model, respectively, and that the coverings of the uniform Sacks ideal and the Mycielski ideal, $${\mathfrak {C}_{2}}$$ C 2, remain small in the Sacks model.

For n<ω, we say that theΠn1-reflection principle holds at κ and write Refln if and only if κ is a Πn1-indescribable cardinal and every Πn1-indescribable subset of κ has a Πn1-indescribable proper initial segment. The Πn1-reflection principle Refln generalizes a certain stationary reflection principle and implies that κ is Πn1-indescribable of order ω. We define a forcing which shows that the converse of this implication can be false in the case n=1; that is, we show that κ being Π11-indescribable of (...) order ω need not imply Refl1. Moreover, we prove that if κ is -weakly compact where α<κ+, then there is a forcing extension in which there is a weakly compact set W⊆κ having no weakly compact proper initial segment, the class of weakly compact cardinals is preserved and κ remains -weakly compact. We also formulate several open problems and highlight places in which standard arguments seem to break down. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 26.1 (2002) 173-187 [Access article in PDF] Symposium: On Ken Burns's "Jazz" Jazz After Jazz: Ken Burns and the Construction of Jazz History Theodore Gracyk As all action is by its nature to be figured as extended in breadth and in depth, as well as in length; and so spreads abroad on all hands... so all narrative is, by its nature, of only one dimension; only (...) travels forward towards one, or towards successive points; narrative is linear, action is solid. Alas for our "chains" or chainlets, of "causes and effects." —Thomas Carlyle, "On History," 1830 Jazz and freedom are synonyms. —Mezz Mezzrow I JAZZ, A FILM BY KEN BURNS is a fascinating and monumental achievement. In this essay, I identify and critique some of the project's core assumptions. My principal target is the assumption that a specific essence of jazz underlies most (but not all) of the music that is regularly identified as jazz and that the historical unfolding of this essence is the true subject of a historical investigation of jazz. This assumption accounts for the project's otherwise puzzling suggestion that jazz history ended approximately thirty-five years ago.The most visible element of the historical project is the ten-part, nineteen-hour documentary, Jazz: A Film by Ken Burns. However, the historical project has three other public elements. The first is a book, Jazz: A History of America's Music. 1 Credited to both Geoffrey C. Ward and Ken Burns, the bulk of the book is the film's spoken text, which includes both a unifying narrative and numerous quotations. The book makes it clear that this text is primarily Ward's work. Second, there is a website that makes available extensive transcripts of many of the [End Page 173] seventy-five interviews conducted for use in the film. Third, there is a set of twenty-two compact discs, each giving an overview of an important jazz musician (e.g., one devoted to Count Basie, one devoted to Dizzy Gillespie). Highlights of these discs have, in turn, been assembled into a five-disc box set called Ken Burns Jazz. My aim is to reveal and examine the conceptual scaffolding that supports and organizes this vast, multifaceted project.What is the principal assumption of this historical project? The project consistently emphasizes the music's capacity to give audible embodiment to the idea of American democracy. At the same time, the project is less than forthcoming about the aesthetic assumptions embraced by Ward and Burns. Familiar and predictable arguments are introduced to justify the appellation of "art" and to address the issue of the purported death of jazz in the 1970s. However, these two strands of the larger argument may be at cross-purposes with the principal assumption and the ideology that jointly justify the project's status as jazz history.Let me begin by describing the principal assumption in some detail. While there is no reason to suppose that its source is Ken Burns or his chief collaborator, historian Geoffrey Ward, the following blurb was used to advertise the project: Jazz has been called the purest expression of American democracy; a music built on individualism and compromise, independence and cooperation. Join us for an exploration of jazz, America's greatest cultural achievement. In keeping with this theme, the narration and principal interviewees of the film are positioned to emphasize that jazz is an audible embodiment of the idea of American democracy. Let us call this idea the democracy thesis. It is placed as a frame around the whole film: Wynton Marsalis announces it in the opening minutes of the first episode, and then he reiterates it in fresh words at the end of the last episode. Although it is repeated with numerous variations, the democracy thesis proposes that jazz involves an ideal of democratic America, not the fact of life in America: jazz, Wynton Marsalis assures us, is "a glimpse of what America is going to be when it becomes itself. And this music tells you that it will become itself." 2 Jazz... (shrink)

In this paper, inspired by methods of Bigard, Keimel, and Wolfenstein , we develop an approach to sheaf representations of MV-algebras which combines two techniques for the representation of MV-algebras devised by Filipoiu and Georgescu and by Dubuc and Poveda . Following Davey approach , we use a subdirect representation of MV-algebras that is based on local MV-algebras. This allowed us to obtain: a representation of any MV-algebras as MV-algebra of all global sections of a sheaf of local MV-algebras on (...) the spectruum of its prime ideals; a representation of MV-algebras, having the space of minimal prime ideals compact, as MV-algebra of all global sections of a Hausdorff sheaf of MV-chains on the space of minimal prime ideals, which is a Stone space; an adjunction between the category of all MV-algebras and the category of MV-algebraic spaces, where an MV-algebraic space is a pair , where X is a compact topological space and F is a sheaf of MV-algebras with stalks that are local. (shrink)

This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [Formula: see text] on [Formula: see text] is equivalent to weak compactness. Winning the game of length [Formula: see text] is equivalent to [Formula: see text] being measurable. We show that for games of intermediate length [Formula: see text], II winning implies (...) the existence of precipitous ideals with [Formula: see text]-closed, [Formula: see text]-dense trees. The second part shows the first is not vacuous. For each [Formula: see text] between [Formula: see text] and [Formula: see text], it gives a model where II wins the games of length [Formula: see text], but not [Formula: see text]. The technique also gives models where for all [Formula: see text] there are [Formula: see text]-complete, normal, [Formula: see text]-distributive ideals having dense sets that are [Formula: see text]-closed, but not [Formula: see text]-closed. (shrink)

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. This paper has two parts. The first is concerned with a variant of a family of games introduced by Holy and Schlicht, that we call Welch games. Player II having a winning strategy in the Welch game of length [math] on [math] is equivalent to weak compactness. Winning the game of length [math] is equivalent to [math] being measurable. We show that for games of intermediate length [math], II winning implies (...) the existence of precipitous ideals with [math]-closed, [math]-dense trees. The second part shows the first is not vacuous. For each [math] between [math] and [math], it gives a model where II wins the games of length [math], but not [math]. The technique also gives models where for all [math] there are [math]-complete, normal, [math]-distributive ideals having dense sets that are [math]-closed, but not [math]-closed. (shrink)

We work in set-theory without choice ZF. A set is Countable if it is finite or equipotent with ${\Bbb N}$ . Given a closed subset F of [0, 1] I which is a bounded subset of $\ell ^{1}(I)$ (resp. such that $F\subseteq c_{0}(I)$ ), we show that the countable axiom of choice for finite sets, (resp. the countable axiom of choice AC N ) implies that F is compact. This enhances previous results where AC N (resp. the axiom (...) of Dependent Choices) was required. If I is linearly orderable (for example $I={\Bbb R}$ ), then, in ZF, the closed unit ball of the Hilbert space $\ell ^{2}(I)$ is (Loeb-)compact in the weak topology. However, the weak compactness of the closed unit ball of $\ell ^{2}(\scr{P}({\Bbb R}))$ is not provable in ZF. (shrink)

The Ideal of a Rational Morality collects the most important essays by the distinguished moral philosopher Marcus G. Singer. Its guiding theme is the concept of a morality based in reason, which is presupposed in ordinary moral contexts and provides an ideal for improving ordinary morality and correcting moral judgements. Singer makes compelling claims that certain fundamental presuppositions are inescapable in moral thought, that fundamental moral principles can be proved, and that the concepts of truth and 'common sense' are essential (...) to ethics.Subsequent essays proceed to analyse the nature of value judgements and of moral judgements, emphasising the vital importance of certain basic distinctions. There is a discussion of race and racism, presenting new ideas about the nature of both. The relation of law to morality is considered, as are the relations between moral judgements of individual persons and the actions and moral judgements of institutions, leading to an examination of the relations between moral issues and social problems. A particularly well-known essay on the Golden Rule is reproduced, where Singer's analysis shows it to be applicable and defensible as it stands, and finally the nature of happiness is explored through a discussion of John Stuart Mill's moral philosophy.Each essay is supplemented by a set of Additional Notes and Comments, with supplementary pieces and response to criticisms. The Ideal of a Rational Morality will be fascinating reading for anybody seeking rigour and clarity in ethical issues. (shrink)

We investigate the cofinality of the partial order N κ of functions from a regular cardinal κ into the ideal N of Lebesgue measure zero subsets of R. We show that when add(N) = κ and the covering lemma holds with respect to an inner model of GCH, then cf(N κ ) = max {cf(κ κ ), cf([ cf(N)] κ )}. We also give an example to show that the covering assumption cannot be removed.

The United States Constitution embodies both the real and the ideal. It is a concrete written text that uses particular words, has a history, and possesses certain limits; it is also a statement of the aspirations and dreams of a society. This dual identity requires that the Constitution be understood both as written positive law, and as an expression of a national vision and set of ideals. ;I argue for a conceptual theory of law that is positivistic in the (...) sense that it relies for the identification of law on social conventions that form the foundations for determining what counts as law. My view is drawn from H. L. A. Hart's legal positivist view, but it departs from this view to argue that judicial custom can be, along with Hart's rule of recognition, an additional social custom that provides criteria for identifying and creating law. Judicial custom involves the acceptance over time by judges of certain principles as authoritative in the context of adjudicative decisionmaking. I contend that both rules and principles can be part of law, and that principles attain the status of law through judicial custom. ;I rely on this understanding of law to argue for a theory of constitutional adjudication that is an alternative to both originalism and traditional non-originalist theories. Originalism is the view that judges can decide constitutional cases exclusively on the basis of two grounds: the text or the original intentions or understanding of its framers. The theory of law that I defend accepts that these grounds are important for any constitutional decision, but holds that constitutional decisions can also legitimately be made on the basis of constitutional theories. Such theories explain the often broad and abstract language of constitutional provisions by articulating hypotheses about the principles that underlie the text. I argue that the central constitutional commitment is to each individual's right to self-determination. This requires protection of an aspect of personal liberty that involves the deeply personal and unique nature of each person's aspirations, ideas, and initiatives. (shrink)

For a space X let $\mathcal {K}(X)$ be the set of compact subsets of X ordered by inclusion. A map $\phi :\mathcal {K}(X) \to \mathcal {K}(Y)$ is a relative Tukey quotient if it carries compact covers to compact covers. When there is such a Tukey quotient write $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$, and write $(X,\mathcal {K}(X)) =_T (Y,\mathcal {K}(Y))$ if $(X,\mathcal {K}(X)) \ge _T (Y,\mathcal {K}(Y))$ and vice versa. We investigate the initial structure of pairs $(X,\mathcal (...) {K}(X))$ under the relative Tukey order, focussing on the case of separable metrizable spaces. Connections are made to Menger spaces. Applications are given demonstrating the diversity of free topological groups, and related free objects, over separable metrizable spaces. It is shown a topological group G has the countable chain condition if it is either $\sigma $ -pseudocompact or for some separable metrizable M, we have $\mathcal {K}(M) \ge _T (G,\mathcal {K}(G))$. (shrink)

An ideal in a commutative ring R with unit is a nonempty I Õ R such that for all x,y Œ I, z Œ R, we have x+y and xz Œ I. A set of generators for I is a subset of I such that I is the least ideal containing that subset.

This article offers a study of the early formation and development of the ideal of harmony in ancient Chinese philosophy and ancient Greek philosophy. It shows that, unlike the Pythagorean notion of harmony, which is primarily based on a linear progressive model with a pre-set order, the ancient Chinese concept of harmony is best understood as a comprehensive process of harmonization. It encompasses spatial as well as temporal dimensions, metaphysical as well as moral and aesthetical dimensions. It is a fundamentally (...) open notion in the sense that it does not aim to conform to any pre-set order. This broader, richer, and more liberal understanding of harmony has had a profound influence on Chinese culture as whole in its long history. (shrink)

Let J be any proper ideal of subsets of the real line R which contains all finite subsets of R. We define an ideal J * ∣B as follows: X ∈ J * ∣B if there exists a Borel set $B \subset R \times R$ such that $X \subset B$ and for any x ∈ R we have $\{y \in R: \langle x,y\rangle \in B\} \in \mathscr{J}$ . We show that there exists a family $\mathscr{A} \subset \mathscr{J}^\ast\mid\mathscr{B}$ of power ω (...) 1 such that $\bigcup\mathscr{A} \not\in \mathscr{J}^\ast\mid\mathscr{B}$ . In the last section we investigate properties of ideals of Lebesgue measure zero sets and meager sets in Cohen extensions of models of set theory. (shrink)

Public reason liberalism is a family of theories according to which liberal political institutions, social structures, and/or basic social rules are politically or morally justified if and only if they can be endorsed from the perspective of each and every free and equal "reasonable and rational" person. Let us call these persons "the members of the justificatory public." Public reason liberalism idealizes the members of the justificatory public in three senses. First, the members of the justificatory public are assumed to (...) be free from at least some of the cognitive distortions and biases that often characterize actual people; we suppose that they generally reason in a sound way on the basis of relevant information. Different versions of public reason liberalism press this idealization quite far, while others insist on a "moderate idealization." Secondly, the members of the justificatory public are idealized insofar as it is assumed that each is a good-willed person, concerned with living with others on terms that are mutually acceptable. In Rawls's language, we suppose that they are not simply rational, but "reasonable": they have a form of "moral sensibility" according to which they are ready to propose fair terms of cooperation, and are willing to abide by them "provided others can be relied on to do likewise." The members of the justificatory public are thus idealized insofar as they are assumed to be moved by their sense of justice, or their aim to respect others as free and equal persons. This leads to a third idealization: by assuming that the members of the justificatory public are moved by this moral sensibility, it is supposed that they do not pay attention to their reasons to ignore their sense of justice . More generally, any specific version of public reason liberalism will hold that some reasons of actual people are not relevant to the justificatory question: the members of the justificatory public "bracket" these reasons in their deliberations. (shrink)

Carnap's ideal of explication has become a key concept in analytic philosophy and the basis of a method of analysis which may be considered as an alternative to various forms of naturalism, including Quine's conception of a naturalized epistemology. More recently, new light has been shed on this aspect of the classical Carnap-Quine debate by contemporary philosophers. Whereas Michael Friedman articulated a notion of relativized a priori which owes much to Carnap's internal/external distinction, André Carus attempted to restate Carnap's ideal (...) of explication in a way that bridges the gap between conceptual engineering and naturalism. On the other hand, Mark Wilson argued that concepts develop in unpredictable ways and cannot be planned in the way classical analytic philosophy has assumed from Frege and Russell onwards. This book consists of a series of defences as well as critiques of Carnap's programme, setting it in its historical context, discussing specific cases of explications, and enriching the on-going debate on conceptual engineering and naturalism in analytic philosophy.Carnap's ideal of explication has become a key concept in analytic philosophy and the basis of a method of analysis which may be considered as an alternative to various forms of naturalism, including Quine's conception of a naturalized epistemology. More recently, new light has been shed on this aspect of the classical Carnap-Quine debate by contemporary philosophers. Whereas Michael Friedman articulated a notion of relativized a priori which owes much to Carnap's internal/external distinction, André Carus attempted to restate Carnap's ideal of explication in a way that bridges the gap between conceptual engineering and naturalism. On the other hand, Mark Wilson argued that concepts develop in unpredictable ways and cannot be planned in the way classical analytic philosophy has assumed from Frege and Russell onwards. This book consists of a series of defences as well as critiques of Carnap's programme, setting it in its historical context, discussing specific cases of explications, and enriching the on-going debate on conceptual engineering and naturalism in analytic philosophy. (shrink)

The United Nations Global Compact is one of the largest transnational governance schemes. Its success or failure, however, is a matter of debate. Drawing on research in cognitive linguistics, we argue that when evaluators discuss the UNGC, they apply the metaphorical concept of the family: the UNGC corresponds to the “family,” the UNGC headquarter to the “parent” and the business participants of the UNGC to the “children” of the family. As a corollary, evaluators’ implicit understanding of how a family (...) is best organized sets different benchmarks against which the governance structure of the UNGC is assessed. We describe two ideal models of “educating” UNGC business participants. Critics of the UNGC adopt a “strict father” model of transnational governance based on the idea that the proper education of inherently “bad” business firms necessitates obedience, discipline and punishment in case firms are non-compliant. In contrast, the UNGC’s advocates follow a “nurturant parent” model, which prioritizes empathy, learning and nurturance to support the moral development of “good” business firms. We develop the “UNGC-as-family” metaphor, explore its implications for transnational governance and discuss under what conditions these idealized models can serve as appropriate guidelines for TGSs. Specifically, we posit that following the behavioral prescriptions of the “strict father” model may, under certain conditions, jeopardize the organizational embedding and institutionalization of UNGC principles, and explain when and why it may be in the best interest of the UNGC and civil society to embrace the instructions of the “nurturant parent” model of transnational governance. (shrink)

N´emeti remarked that the notion of compactness of cylindric of algebras corresponds to the notion of universality of models in logic [5]. The purpose of this paper is to formulate this correspondence in a purely algebraic setting.

On Exceeding Determination and the Ideal of Reason: Immanuel Kant, William Desmond, and the Noumenological Principle examines the critical philosophy of Immanuel Kant, as it bears on theological principles. Focusing on the foundational ideas (of self, world, and God) that constitute Kant's metaphysical system, Shaw argues that these ideal projections of the rational structures of the thinking subject only conceal and obfuscate the more robust sense of the real that exists behind all phenomenal appearances. This book aims to critically assess (...) the theological content presented in the philosophy of Kant whilst reconstructing and developing some of Kant's more obscure points on the epistemological limitations of pure reason and the borders of knowledge itself. Shaw draws into dialogue with the writings of the contemporary philosopher William Desmond to demonstrate some of the problems of Kantian thought when it comes to the deeper mysteries of being. Desmondian philosophy proves to be a powerful influence over much of this work, as Shaw advances upon the delicate nexus where philosophy and theology convene. As a bold addition to this work, Shaw lays the groundwork for a new discourse in theology and the philosophy of religion: "Noumenology". Noumenology, which is Shaw's response to the deficiencies in methods concerning many of the modern approaches to phenomenology, is a call for the recalibration of the starting point for eidetic inquiry through the proposal of a non-discursive approach to the sensible, emotional, and intellectual - or perhaps, the all together spiritual. To accomplish this, Shaw reaches back to Kant for a heterodox approach to the language and content surrounding the noumenonal to set forth his theological interpretation of the "Noumenological Principle". (shrink)

Compactness is an important property of classical propositional logic. It can be defined in two equivalent ways. The first one states that simultaneous satisfiability of an infinite set of formulae is equivalent to the satisfiability of all its finite subsets. The second one states that if a set of formulae entails a formula, then there is a finite subset entailing this formula as well. In propositional many-valued logic, we have different degrees of satisfiability and different possible definitions of entailment, hence (...) the questions of compactness is more complex. In this paper we will deal with compactness of Gödel, GödelΔ, and Gödel∼ logics. (shrink)

Let X be an infinite set and let and denote the propositions “every filter on X can be extended to an ultrafilter” and “X has a free ultrafilter”, respectively. We denote by the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, (ℵ) iff (2ℵ). iff “ is a continuous image of ” iff “ has a free open ultrafilter ” iff “every countably infinite subset of has a limit point”. (...) implies “every open filter on extends to an open ultrafilter” implies “has an open ultrafilter” implies It is relatively consistent with that (ω) holds, whereas (ω) fails. In particular, none of the statements given in (2) implies (ω). (shrink)

We introduce a new sheaf-theoretic construction called the ideal completion of a category and investigate its logical properties. We show that it satisfies the axioms for a category of classes in the sense of Joyal and Moerdijk [17], so that the tools of algebraic set theory can be applied to produce models of various elementary set theories. These results are then used to prove the conservativity of different set theories over various classical and constructive type theories.

In this paper, we argue for an instrumental form of existence, inspired by Hilbert’s method of ideal elements. As a case study, we consider the existence of contradictory objects in models of non-classical set theories. Based on this discussion, we argue for a very liberal notion of existence in mathematics.

The ideal companion guide for capturing awe-inspiring photos of Yosemite! Whether using a compact camera or a high-end dSLR, this companion guide provides you with detailed information for taking spectacular shots of some of the most recognizable landmarks in the world. Whether you aim to capture memorable photos of Half Dome, El Capitan, Vernal Fall, Mariposa Grove, or one of Yosemite's other many remarkable attractions, this portable resource goes where you go and walks you through valuable tips and techniques (...) for taking the best shot possible. You'll discover suggested locations for taking photos, recommended equipment, what camera settings to use, best times of day to photograph specific attractions, how to handle weather challenges, and more. In addition, beautiful images of Yosemite's most breathtaking attractions and recognizable landmarks serve to both inspire and assist you as you embark on an amazing photographic adventure! Elevates your photography skills to a new level with photography secrets from professional photographer Lewis Kemper Presents clear, understandable tips and techniques that span all skill levels, using all types of digital cameras, from compact cameras to high end DSLRs Features Yosemite's main attractions in alphabetical order as well as thumb tabs on the pages so you can quickly and easily access the information you are looking for Shares detailed information and insight on critical topics, such as ideal locations to photograph from, the best time of day to shoot, camera equipment to have handy, weather conditions, and optimal camera settings to consider Whether you're a local familiar with the territory or a visitor seeing this beautiful U.S. National Park for the first time, this handy guide will help capture fantastic photos! (shrink)

A compact, full-color companion guide to photographing San Francisco! Whether using a full-featured compact camera or a high-end dSLR, this companion guide provides you with detailed information for taking stunning shots of beautiful San Francisco. Whether you aim to capture breathtaking photos of the majestic Golden Gate Bridge, crooked Lombard Street, infamous Alcatraz, or unique Victorian homes, this portable resource goes where you go and walks you through valuable tips and techniques for taking the best shot possible. You'll (...) discover suggested locations for taking photos, recommended equipment, what camera settings to use, best times of day to photograph specific attractions, how to handle weather challenges, and more. In addition, lovely images of San Francisco's most breathtaking attractions and recognizable landmarks serve to both inspire and assist you as you embark on an amazing photographic adventure! Elevates your photography skills to a new level with photography secrets from professional photographer Bruce Sawle Presents clear, understandable tips and techniques that span all skill levels, using all types of digital cameras, from compacts to high end DSLRs Features San Fransisco's main attractions in alphabetical order as well as thumb tabs on the pages so you can quickly and easily access the information you are looking for Shares detailed information and insight on critical topics, such as ideal locations to photograph from, the best time of day to shoot, camera equipment to have handy, weather conditions, and optimal camera settings to consider Whether you're a local familiar with the territory or a visitor seeing San Francisco for the first time, this handy guide will help you capture fantastic photos! (shrink)

This paper develops a moderate pessimist account of moral deference. I argue that while some pessimist explanations of the puzzle of moral deference have been misguided in matters of detail, they nevertheless share an important insight, namely that there is a justified moral agency ideal grounded in pro tanto reasons against moral deference. This thought is unpacked in terms of a set of values associated with the practice of morality. I conclude by suggesting that the solution to the puzzle of (...) moral deference developed here gives us a plausible recipe for generalizing to certain cases of nonmoral deference as well. (shrink)

In this note we shall show that if L is a balanced pseudocomplemented Ockham algebra then the set ${\fancyscript{I}_{k}(L)}$ of kernel ideals of L is a Heyting lattice that is isomorphic to the lattice of congruences on B(L) where ${B(L) = \{x^* | x \in L\}}$ . In particular, we show that ${\fancyscript{I}_{k}(L)}$ is boolean if and only if B(L) is finite, if and only if every kernel ideal of L is principal.