This article describes well quasi orders as a category, focusing on limits and colimits. In particular, while quasi orders with monotone maps form a category which is finitely complete, finitely cocomplete, and with exponentiation, the full subcategory of well quasi orders is finitely complete and cocomplete, but with no exponentiation. It is interesting to notice how finite antichains and finite proper descending chains interact to induce this structure in the category: in fact, the full subcategory (...) of quasi orders with finite antichains has finite colimits but no products, while the full subcategory of well founded quasi orders has finite limits but no coequalisers. Moreover, the article characterises when exponential objects exist in the category of well quasi orders and well founded quasi orders. This completes the systematic description of the fundamental constructions in the categories of quasi orders, well founded quasi orders, quasi orders with finite antichains, and well quasi orders. (shrink)

This book bridges the gaps between logic, mathematics and computer science by delving into the theory of well-quasi orders, also known as wqos. This highly active branch of combinatorics is deeply rooted in and between many fields of mathematics and logic, including proof theory, commutative algebra, braid groups, graph theory, analytic combinatorics, theory of relations, reverse mathematics and subrecursive hierarchies. As a unifying concept for slick finiteness or termination proofs, wqos have been rediscovered in diverse contexts, and proven (...) to be extremely useful in computer science. The book introduces readers to the many facets of, and recent developments in, wqos through chapters contributed by scholars from various fields. As such, it offers a valuable asset for logicians, mathematicians and computer scientists, as well as scholars and students. (shrink)

A quasi-order Q induces two natural quasi-orders on $${\mathcal{P}(Q)}$$, but if Q is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq (Proceedings of the 22nd Annual IEEE Symposium 4 on Logic in Computer Science (LICS’07), pp. 453–462, 2007) showed that moving from a well-quasi-order Q to the quasi-orders on $${\mathcal{P}(Q)}$$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper (...) topologies of the induced quasi-orders on $${\mathcal{P}(Q)}$$ are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form “if Q is a well-quasi-order then a certain topology on (a subset of) $${\mathcal{P}(Q)}$$ is Noetherian” in the style of reverse mathematics, proving that these theorems are equivalent to ACA 0 over RCA 0. To state these theorems in RCA 0 we introduce a new framework for dealing with second-countable topological spaces. (shrink)

We prove that the Wadge order on the Borel subsets of the Scott domain is not a well-quasi-order, and that this feature even occurs among the sets of Borel rank at most 2. For this purpose, a specific class of countable 2-colored posets$\mathbb{P}_{emb} $equipped with the order induced by homomorphisms is embedded into the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain. We then show that$\mathbb{P}_{emb} $admits both infinite strictly decreasing chains and (...) infinite antichains with respect to this notion of comparison, which therefore transfers to the Wadge order on the$\Delta _2^0 $-degrees of the Scott domain. (shrink)

In this paper we prove general exact unprovability results that show how a threshold between provability and unprovability of a finite well-quasi-orderedness assertion of a combinatorial class is transformed by the sequence-construction, multiset-construction, cycle-construction and labeled-tree-construction. Provability proofs use the asymptotic pigeonhole principle, unprovability proofs use Weiermann-style compression techniques and results from analytic combinatorics.

In this paper, we give a step by step introduction to the theory of well quasi-ordered transition systems. The framework combines two concepts, namely (i) transition systems which are monotonic wrt. a well-quasi ordering ; and (ii) a scheme for symbolic backward reachability analysis. We describe several models with infinite-state spaces, which can be analyzed within the framework, e.g., Petri nets, lossy channel systems, timed automata, timed Petri nets, and multiset rewriting systems. We will also present (...) better quasi-ordered transition systems which allow the design of efficient symbolic representations of infinite sets of states. (shrink)

We define a quasi-order on Borel functions from a zero-dimensional Polish space into another that both refines the order induced by the Baire hierarchy of functions and generalises the embeddability order on Borel sets. We study the properties of this quasi-order on continuous functions, and we prove that the closed subsets of a zero-dimensional Polish space are well-quasi-ordered by bi-continuous embeddability.

We provide partial answers to the following problem: Is the class of Borel linear orders well-quasi-ordered under embeddability? We show that it is indeed the case for those Borel orders which are embeddable in R ω , with the lexicographic ordering. For Borel orders embeddable in R 2 , our proof works in ZFC, but it uses projective determinacy for Borel orders embeddable in some $\mathbf{R}^n, n , and hyperprojective determinacy for the general case.

Excerpt: IT will lead into my subject most conveniently to contrast and separate two divergent types of mind, types which are to be distinguished chiefly by their attitude toward time, and more particularly by the relative importance they attach and the relative amount of thought they give to the future. The first of these two types of mind, and it is, I think, the predominant type, the type of the majority of living people, is that which seems scarcely to think (...) of the future at all, which regards it as a sort of blank non-existence upon which the advancing present will presently write events. The second type, which is, I think, a more modern and much less abundant type of mind, thinks constantly and by preference of things to come, and of present things mainly in relation to the results that must arise from them. The former type of mind, when one gets it in its purity, is retrospective in habit, and it interprets the things of the present, and gives value to this and denies it to that, entirely with relation to the past. The latter type of mind is constructive in habit, it interprets the things of the present and gives value to this or that, entirely in relation to things designed or foreseen. While from that former point of view our life is simply to reap the consequences of the past, from this our life is to prepare the future. The former type one might speak of as the legal or submissive type of mind, because the business, the practice, and the training of a lawyer dispose him toward it; he of all men must constantly refer to the law made, the right established, the precedent set, and consistently ignore or condemn the thing that is only seeking to establish itself. The latter type of mind I might for contrast call the legislative, creative, organizing, or masterful type, because it is perpetually attacking and altering the established order of things, perpetually falling away from respect for what the past has given us. It sees the world as one great workshop, and the present is no more than material for the future, for the thing that is yet destined to be. It is in the active mood of thought, while the former is in the passive; it is the mind of youth, it is the mind more manifest among the western nations, while the former is the mind of age, the mind of the oriental. Things have been, says the legal mind, and so we are here. The creative mind says we are here because things have yet to be. Now I do not wish to suggest that the great mass of people belong to either of these two types. Indeed, I speak of them as two distinct and distinguishable types mainly for convenience and in order to accentuate their distinction. There are probably very few people who brood constantly upon the past without any thought of the future at all, and there are probably scarcely any who live and think consistently in relation to the future. The great mass of people occupy an intermediate position between these extremes, they pass daily and hourly from the passive mood to the active, they see this thing in relation to its associations and that thing in relation to its consequences, and they do not even suspect that they are using two distinct methods in their minds. As well known as HG Wells is for his science fiction, this is simply HG Wells discussing our view of the present and the future and various nuances related to both. Very interesting. (shrink)

The centrally recognized theoretical achievement that enabled the Higgs boson discovery in 2012 was the hypothesis of its existence, made by Peter Higgs in 1964. Nevertheless, there is a significant body of comparably important theoretical work prior to and after the Higgs boson hypothesis. In this article we present an additional perspective of how crucial theory work was to the genesis of the Higgs boson hypothesis, especially emphasizing its roots in Landau's theory of phase transitions and subsequent theoretical work on (...) superconductivity. A detailed description is then given of the opposition to the Higgs boson hypothesis by many researchers, giving evidence to its speculative nature. And finally, it is discussed the importance of theory work in the decades after the hypothesis in order to make possible the experimental discovery of the Higgs boson. (shrink)

In Improvisation, Samuel Wells defines improvisation in the theater as "a practice through which actors seek to develop trust in themselves and one another in order that they may conduct unscripted dramas without fear." Sounds a lot like life, doesn't it? Building trust, overcoming fear, conducting relationships, and making choices--all without a script. Wells establishes theatrical improvisation as a model for Christian ethics, a matter of "faithfully improvising on the Christian tradition." He views the Bible not as a "script" (...) but as a "training school" that shapes the habits and practices of the Christian community. Drawing on scriptural narratives and church history, Wells explains six practices that characterize both improvisation and Christian ethics. His model of improvisation reinforces the goal of Christian ethics--to teach Christians to "embody their faith in the practices of discipleship all the time."--Publisher description. (shrink)

It is widely held that, in his pre-Critical works, Kant endorsed a necessitation account of laws of nature, where laws are grounded in essences or causal powers. Against this, I argue that the early Kant endorsed the priority of laws in explaining and unifying the natural world, as well as their irreducible role in in grounding natural necessity. Laws are a key constituent of Kant’s explanatory naturalism, rather than undermining it. By laying out neglected distinctions Kant draws among types (...) of natural law, grounding relations, and ontological levels, I show that his early works present a coherent and sophisticated laws-first account of the natural order. (shrink)

Rachel Wells turns to the examination of three recent artistic practices, which integrate facts in their work not as an antagonistic other but as a constitutive element to their efficacy and ethics. She argues, that in introducing news, factual actions, or objects with traces of factual events, Alfredo Jaar, Jeremy Deller and Martin Creed use facts in order to retract from the position of art as an expression of artistic freedom and subjectivity and thus as the opposite of fact. (...) Instead, she states that by introducing the factual the artists underline each in their own way the instability of given eptistemological and ethical frameworks. Far from being a mere relativist pose, Wells understands this denial of a stable subjectivist position as a reconfigured sense of “decision”—perhaps in the sense of Nancy’s articulation of a “decision of existence—that lets the factual take precedence over the control in and of the artwork as a heightened form of responsiveness and responsibility. Whereas Jaar uses the factual to engage overt political action, Deller presents facts that avoid taking an overtly critical perspective forcing the viewer to think about political events. Creed in contrast seeks for the interpretation of the past altogether, would avoid to take over the responsibility of taking a position. Whereas David Hume stated famously, that reasoning concerning matter of fact is founded in causality and Immanuel Kant concludes, that responsibility and freedom begins where causality ends, Wells understands the positions of Jaar, Deller and Creed as an attempt to reconcile the realm of the factual and the realm of the moral. Responsibility would then arise exactly out of the insight in the impossibility to ground moral stances in rationality and causality and in an attempt to use causality to demonstrate this impossibility. (shrink)

We are pleased to annouce that God’s Companions by Samuel Wells has been shortlisted for the 2007 Michael Ramsey Prize for theological writing. www.michaelramseyprize.org.uk Grounded in Samuel Wells’ experience of ordinary lives in poorer neighborhoods, this book presents a striking and imaginative approach to Christian ethics. It argues that Christian ethics is founded on God, on the practices of human community, and on worship, and that ethics is fundamentally a reflection of God's abundance. Wells synthesizes dogmatic, liturgical, ethical, scriptural, and (...) pastoral approaches to theology in order to make a bold claim for the centrality of the local church in theological reflection. He considers the abundance of gifts God gives through the practices of the Church, particularly the Eucharist. His central thesis, which governs his argument throughout, is that God gives his people everything they need to worship him, be his friends, and eat with him. Wells engages with serious scholarly material, yet sets out the issues lucidly for a student audience. (shrink)

This book covers the author's conception of God aside from any religion. He does not come from a religious view in order to transmit the truest conception of God that he is capable of because any religion, whatever it might be, always claims God for itself in an exclusionary fashion. In other words, you must be a follower of the chosen faith before God will accept you into his kingdom. Wells rejects this view. Any man or woman who accepts (...) God's love as one person connecting to their own creator, is fine. An intermediary, despite what they might crow from whatever pulpits they use, is not necessary. We are all God's children. We are all seeking the same God, so to be dogmatic within a certain exclusionary faith is small-minded and petty. If you want to grow up, spiritually, then this book is for you. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:When Is Writing Already Quotation?A Developmental Perspective on a Postmodern QuestionRebecca Wells-Jopling (bio)IntroductionPostmodern literary-critical thinking introduced into many disciplines in the 1950s and 1960s the quite peculiar, yet intellectually engaging, idea that what is written is always already-quoted. This idea is a logical derivation from the concurrent idea that writing is "prior to history"1 ; thus, what was written and what is written were simply always there, and someone (...) wrote it long before the reader held the written document in hand. The semiotician Roland Barthes claimed that "Every text, being itself the intertext of another text, belongs to the intertextual, which must not be confused with a text's origins: to search for the 'sources of' and 'influence upon' a work is to satisfy the myth of filiation. The quotations from which a text is constructed are anonymous, irrecoverable, and yet already read: they are quotations without quotation marks."2 In a similar vein, the cognitive psychologist David Olson argues that writing may be conceived not as transcription of speech, nor as a completely different mode of expression from speech, but rather as a type of reflexive language that points to itself as language.3 Writing is thus "quotation" that calls for "a distinctive mode of interpretation."4 The purpose of this article is to offer some reasons why conceiving of what is written as always already-quoted is likely not accurate as an ontological claim, but it may hold in particular circumstances and at a particular moment in human development. The phrase "already-quoted" will refer to this postmodern notion that entire texts, and components thereof, routinely call attention to their own opacity. Sometimes this opacity is achieved through the writer's use of actual quotation marks, but sometimes the opacity is only in the reader's experience of detecting a phrase or other unit in the text that summons up rather directly a perspective or another text in a way that blocks the transparent referential power that the words might otherwise have. [End Page 59]Quotation and the Questioning of AuthorityFirst, a bit of history. Pierre de la Ramée (also known as Peter Ramus) was a French philosopher and logician whose 1543 tract Dialecticae Partitiones in Latin was the first philosophical work to appear in French (in 1555 under the title Dialectique), and, more pertinently for this discussion of quotation, was one of the first writers to use proto-quotation marks in his writings.5 The earlier Latin text "is printed entirely in roman type, except the preface, which is entirely in italics. But in his French Dialectique all citations of foreign texts are distinguished by typography: italics are used for poetry, inverted commas enclose prose."6 His motivation, according to Compagnon,7 was the desire to bypass the kind of univocal authority suggested by a text in which voices cannot be distinguished from one another, and to point out the status of quoted passages as "reasoned examples or illustrations."8 If this was in fact de la Ramée's motivation, one wonders if he would have preferred that his entire text be put in yet a different typographical face in order to suggest that his own words in the text are not authoritative, but, contrary to Barthes's belief, not anonymous, utterly recoverable, and admittedly perspectival in nature.This detour through publishing history highlights an important difficulty in viewing what is written as always already-quoted: someone has to choose the quotations and arrange them (and, in de la Ramée's case, have the courage to counter the academy and point them out as citations). Is this act of choosing quotations also a quotation of a former chooser of quotations? In de la Ramée's case, obviously not, since he was the first to use different typefaces for different authoritative voices. This is one of the reasons why we cannot conceive of an entire written text as always already derivative. And if not derivative, then, by definition, not quoted. Assembling quotations and gluing them together with argumentation or poetry, as the case may be, is a creative act. This is so even if the words... (shrink)

The story of God -- The story of the church -- The story of ethics -- The story of Christian ethics -- Universal ethics -- Subversive ethics -- Ecclesial ethics -- Good order -- Good life -- Good relationships -- Good beginnings and endings -- Good earth.

Attempting to trace the intellectual history of any political movement is, at best,problematic. Humans construct political movements and the intellectual, philosophical underpinnings of those movements, and, in general, it is not one person who is doing the creating, but rather a multitude of people are involved; the circumstance of how politics is created is a web, which makes it difficult for researchers to trace the historical roots of movements. Nazi Germany has been the focus of numerous research projects to understand (...) the intellectual roots of Nazism and the how and why they were successful in gaining and consolidating power. In line with popular theories in Sociology and History, earlier researchers have traced the intellectual roots of the Nazis in order to situate Nazi Germany as anti-modern, which by extension would situate their crimes against humanityand fascism in the same camp. In particular, Romanticism has been the movement that some historians have cited as a possible root for Nazism. The primary goal of this paper will be to disrupt the historical continuation argument, deconstruct the main parts of each of the camps, and provide support for the appropriation argument. This goal is designed to connect to the much larger debate of the state of anti-modern/modern of Nazism, and aid in showing Nazism as a modern movement. It is through researching and analyzingthe how and why the Nazis appropriated Romanticism that allows academics to study the influences from the past in the development of National Socialism, while accounting for the frame that the Nazis used to read the Romantics and the purpose for the way that Romantic literature was framed within Nazi-Germany. (shrink)

We show that the maximal linear extension theorem for well partial orders is equivalent over RCA0 to ATR0. Analogously, the maximal chain theorem for well partial orders is equivalent to ATR0 over RCA0.

The expectation is fulfilled, but in an unexpected way. 'The first studies toward this book were addressed to topics in the field of ethics' ; but our author, like Wagner composing 'Der Ring des Nibelungen', found himself becoming preoccupied with prolegomena. To these the present volume is wholly devoted. In order to establish its fundamental thesis that valuation is a form of empirical knowledge, two preparatory discussions are called for. An analysis of empirical knowledge in general is one of (...) these; but there is an inquiry properly prior even to this, an examination of the topic of meaning. The articulation of the book is accordingly tripartite; and Book I Meaning and Analytic Truth, Book II Empirical Knowledge, and Book III Valuation are of roughly equal length. (shrink)

In this work we investigate the Weihrauch degree of the problem Decreasing Sequence of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem Bad Sequence of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf {DS}$, despite being hard to solve, is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a (...) Weihrauch degree. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$, $\boldsymbol {\Sigma }^1_1$, $\boldsymbol {\Pi }^1_1$. We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the Baire hierarchy and show that they do not collapse at any finite level. (shrink)

The demise of the two federal communist states, the Soviet Union and Yugoslavia, was the major cause of the appearance of a new kind of state entity, unheard of since the end of feudalism in Europe. While the 15 Union Republics of the Soviet Union became independent from the old center (Moscow) as well as detached from each other, some ethnically different or mixed parts of the newly independent republics, deprived of federal protection, felt uncomfortable within the new framework (...) and emerged as separate or detached entities. It should be remembered that this separatist movement predates the disintegration of the Soviet Union and is parallel to Gorbachev's ‘perestroika’. In the Trans-Caucasus, the separatist forces within the republics began to act as soon as Moscow loosened its grip and the national republics began to assert themselves. The half-hearted and limited use of force by Moscow in order to prevent the disintegration of the Soviet Empire (in Baku, Tbilisi, Vilnius) succeeded no better than attempts by individual republics to stop separatism within their own borders. Georgia and Azerbaijan were most affected, with several ‘quasi-states’ appearing within their borders: Nagorno-Karabagh in Azerbaijan, Abkhazia and Ossetia in Georgia. With the direct support of the Armenian republic and covert Russian assistance, the Armenian majority in the former Nagorno-Karabagh Autonomous Region split from Azerbaijan, expelled the Azeri minority, occupied a substantial chunk of Azeri territory, and created a ‘state’ of its own. The Abkhazian minority (less than one-fifth of the total population), with Russian assistance, split Abkhazia from Georgia, expelled the Georgian majority, and established its own ‘state’. South Ossetia declared itself a ‘sovereign republic’, while Adjaria and parts of Georgia proper became practically independent from Tbilisi. Political mistakes and chauvinist acts on the part of the Azeri and Georgian Popular Front governments that initially came to power contributed to local separatism. (shrink)

We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the well-known Kirby-Paris' Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the well-quasi-ordering of finite sequences of natural numbers with respect to Friedman's gapembeddability. A characterization of gap-embeddability in terms of provability logic due (...) to Lev Beklemishev is also presented. (shrink)

This paper focuses on the interpretation of the Italian approximative adverb quasi 'almost' by primarily looking at cases in which it modifies temporal connectives, a domain which, to our knowledge, has been largely unexplored thus far. Consideration of this domain supports the need for a scalar account of the semantics of quasi (close in spirit to Hitzeman's semantic analysis of almost, in: Canakis et al. (eds) Papers from the 28th regional meeting of the Chicago Linguistic Society, 1992). When (...) paired with suitable analyses of temporal connectives, such an account can provide a simple explanation of the patterns of implication that are observed when quasi modifies locational (e. g. quando 'when'), directional (e. g. fino 'until' and da 'since'), and event-sequencing temporal connectives (e. g. prima 'before' and dopo ' after'). A challenging empirical phenomenon that is observed is a contrast between the modification of fino and da by quasi, on the one hand, and the modification of prima and dopo by the same adverb, on the other. While quasi fino and quasi da behave symmetrically, a puzzling asymmetry is observed between quasi prima and quasi dopo. To explain the asymmetry, we propose an analysis of prima and dopo on which the former has the meaning of the temporal comparative più presto 'earlier', while the latter is seen as an atomic predicate denoting temporal succession between events (Del Prete, Nat Lang Semantics 16: 157-203, 2008). We show that the same pattern of implication observed for quasi prima is attested when quasi modifies overt comparatives, and propose a pragmatic analysis of this pattern that uniformly applies to both cases, thus providing new evidence for the claim that prima is underlyingly a comparative. A major point of this paper is a discussion of the notion of scale which is relevant for the semantics of quasi', in particular, we show that the notion of Horn (entailment-based) scale is not well-suited for handling modification of temporal connectives, and that a more general notion of scale is required in order to provide a uniform analysis of quasi as a cross-categorial modifier. (shrink)

We show that the variety of n -dimensional weakly higher order cylindric algebras, introduced in Németi [9], [8], is finitely axiomatizable when n > 2. Our result implies that in certain non-well-founded set theories the finitization problem of algebraic logic admits a positive solution; and it shows that this variety is a good candidate for being the cylindric algebra theoretic counterpart of Tarski’s quasi-projective relation algebras.

The situational context within which an activity takes place, as well as the personality characteristics of individuals shape the types of strategies people choose in order to regulate their emotions, especially when confronted with challenging or undesirable situations. Taking self-regulation as the framework to study emotions in relation to learning and performing chamber music canon repertoire, this quasi-experimental and intra-individual study focused on the self-rated emotional states of a professional classical cellist during long-term sustained practice across 100-weeks. (...) This helped to develop greater awareness of different emotions and how they vary over artistic events. Data analysis included traditional psychometric measurements to test the internal consistency of the time series data as well as the relationship between variables. The study mapped the cellist’s flexible regulation of 17 different positive and negative emotions empirically linked to learning and achievement while practicing within the social context of performing music publicly at a high level. Findings arising from the study help with understandings of how to support musicians to maximize their artistic potential by reducing emotion dysregulation and strengthening the types of adaptive methods that enable them to manage their own emotions. (shrink)

For a complete theory of Boolean algebras T, let MT denote the class of countable models of T. For B1, B2 ∈ MT, let B1 ≤ B2 mean that B1 is elementarily embeddable in B2. Theorem 1. For every complete theory of Boolean algebras T, if T ≠ Tω, then $\langle M_T, \leq\rangle$ is well-quasi-ordered. ■ We define Tω. For a Boolean algebra B, let I(B) be the ideal of all elements of the form a + s such (...) that $B\upharpoonright a$ is an atomic Boolean algebra and $B\upharpoonright s$ is an atomless Boolean algebra. The Tarski derivative of B is defined as follows: B(0) = B and B(n + 1) = B(n)/I(B(n)). Define Tω to be the theory of all Boolean algebras such that for every n ∈ ω, B(n) ≠ {0}. By Tarski [1949], Tω is complete. Recall that $\langle A, \leq\rangle$ is a partial well-quasi-ordering, if it is a partial quasi-ordering and for every $\{a_i\mid i \in \omega\} \subseteq A$ , there are $i < j < \omega$ such that ai ≤ aj. Theorem 2. $\langle M_{T_\omega}, \leq\rangle$ contains a subset M such that the partial orderings $\langle M, \leq \upharpoonright M \rangle$ and $\langle\mathscr{P}(\omega), \subseteq\rangle$ are isomorphic. ■ Let M'0 denote the class of all countable Boolean algebras. For B1, B2 ∈ M'0, let B1 ≤' B2 mean that B1 is embeddable in B2. Remark. $\langle M'_0, \leq'\rangle$ is well-quasi-ordered. ■ This follows from Laver's theorem [1971] that the class of countable linear orderings with the embeddability relation is well-quasi-ordered and the fact that every countable Boolean algebra is isomorphic to a Boolean algebra of a linear ordering. (shrink)

Organisational Ethics could be more effectively taught if organisational agency could be better distinguished from activity in other group entities, and defended against criticisms. Some criticisms come from the side of what is called “methodological individualism”. These critics argue that, strictly speaking, only individuals really exist and act, and organisations are not individuals, real things, or agents. Other criticisms come from fear of the possible use of alleged “corporate personhood” to argue for a possible radical expansion of corporate rights e.g. (...) to claim a right to vote. Others say that if corporate businesses were persons, and judged as such, they would be sociopaths (Bakan 2004). Large corporate businesses are one kind of organisation, and if organisations were allowed to claim to be full persons, this would mean that the latter criticisms were valid. The proposed account goes between the horns of the above dilemma: impersonal aggregates or full persons. It avoids both the reductive individualism and expansive, full personhood account. It gives grounds for attributing and distributing praise and blame to organisations as whole complex entities, and to their parts. Primarily, responsibility belongs to Board incumbents within them, on the basis of their goals; their direction of the acts of the other main internal role incumbents, managers and staff, and enablers, in the role structure; their own ethical decision methods; the impact of their acts on external stakeholders like end-users, other players in the practice or local community, and on the state. These factors are picked out on the basis of a theory of meaning Aristotle’s theory of analogy of attribution, some architectonic features of his Philosophical Anthropology; and Neo-Aristotelian Virtue Ethics (NAVE). The paper expounds a quasi-person model of organisations, (QPM), previously described in this journal (Ardagh 2011, 2012, 2013). This model identifies functional similarities at three levels, between capacities within natural persons (NPs) on the one hand and the roles/ incumbents within the structure of organisational artificial/quasi-persons (APs on the other). In each entity, we find an architectonic direction, sub-ordered operational entities, and enablers (a DOE structure) in a communicative structure or reporting chain of authority. The common elements between natural persons (NP) and artificial persons (AP) and their ethically relevant attributes are captured in an acronym –GRAEOS. Both persons and organisations have a “top” entity capable of having/choosing worthwhile ethically permissible goals, G; of architectonically directing the use of capacities/repertoires or incumbents in designed architectonic roles, R, within a DOE-type structure, well or badly; of using ethical decision making procedures, E; of execution of good choices for the person or organisation in action, A, with good or bad impacts on others, or with others, O, in the context practice or domain; and of having good relations with the law and the state, S. Application to the 1989 Exxon Valdez disaster, and public sector organisations, is briefly discussed to illustrate the potential to deal with short or long case study narratives. (shrink)

In (Fund Math 60:175–186 1967), Wolk proved that every well partial order (wpo) has a maximal chain; that is a chain of maximal order type. (Note that all chains in a wpo are well-ordered.) We prove that such maximal chain cannot be found computably, not even hyperarithmetically: No hyperarithmetic set can compute maximal chains in all computable wpos. However, we prove that almost every set, in the sense of category, can compute maximal chains in all computable (...) wpos. Wolk’s original result actually shows that every wpo has a strongly maximal chain, which we define below. We show that a set computes strongly maximal chains in all computable wpo if and only if it computes all hyperarithmetic sets. (shrink)

Competitive athletes act within cultures of risk in sports and often decide to return to sport despite having acute health problems. The outcomes of such risky return-to-play decisions can not only negatively affect their future health, but also limit their sports performance or even upset their career paths. Following risk-management-decision theory with its focus on active risk defusing, we developed a model for understanding the process of return-to-play decision making from an athlete’s perspective. Based on the method of active information (...) search, a quasi-naturalistic return-to-play decision scenario was created in order to assess amateur team sport athletes’ decision making strategies. The main goals were to identify different information acquisition patterns and to analyze the influence of social pressure to play hurt on decision making. A total of 72 competitive team sport athletes (36 females, 36 males, m= 25.7 years of age, 3rd to 6th league level) out of three disciplines (volleyball, basketball, and handball) participated in the experimental study. Facing the same medical scenario (a partial tear of the supraspinatus tendon), athletes show different approaches to return-to-play decision making. The main focus is on potential sporting consequences of withdrawal from competition due to injury, while only few players favoring well-informed decisions based on thorough risk analysis. The athletes who chose the medically risky alternative to play hurt mostly employed strategies of active risk defusing, which got activated when severe sporting consequences were perceived. Those who chose to withdraw from competition primarily referred to maximin heuristic, particularly when social pressure to play was reduced. The findings can be used to improve rehabilitation. (shrink)

Population ethics contains several principles that avoid the repugnant conclusion. These rules rank all possible alternatives, leaving no room for moral ambiguity. Building on a suggestion of Parfit, this paper characterizes principles that provide incomplete but ethically attractive rankings of alternatives with different population sizes. All of them rank same-number alternatives with generalized utilitarianism.

We study the provability in subsystems of second-order arithmetic of two theorems of Harrington and Shelah [6] about Borel quasi-orderings of the reals. These theorems turn out to be provable in ATR0, thus giving further evidence to the observation that ATR0 is the minimal subsystem of second-order arithmetic in which significant portion of descriptive set theory can be developed. As in [6] considering the lightface versions of the theorems will be instrumental in their proof and the main (...) techniques employed will be the reflection principles and Gandy forcing. (shrink)

We prove that the poset algebra of every scattered poset with finite width is embeddable in the poset algebra of a well ordered poset.

Here we investigate the classes RCA $^\uparrow_\alpha$ of representable directed cylindric algebras of dimension α introduced by Nemeti[12]. RCA $^\uparrow_\alpha$ can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, "purely cylindric algebraic" proof for the following theorems of Nemeti: (i) RCA $^\uparrow_\alpha$ is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one (...) can obtain a strong representation theorem for RCA $^\uparrow_\alpha$ if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories. (shrink)

In this paper we give a proof of the II12-completeness of the set of countable better quasi orderings . This result was conjectured by Clote in [2] and proved by the author in his Ph.d. thesis [6] . Here we prove it using Simpson's definition of better quasi ordering and as little bqo theory as possible.

Normative theories suggest that inconsistencies be pointed out to the Decision Maker who is thus given the chance to modify his/her judgments. In this paper, we suggest that the inconsistencies problem be transferred from the Decision Maker to the Analyst. With the Mixture of Maximal Quasi Orders, rather than pointing out incoherences for the Decision Maker to change, these inconsistencies may be used as new source of information to model his/her preferences.

In this article we characterize a countable ordinal known as the big Veblen number in terms of natural well-partially ordered tree-like structures. To this end, we consider generalized trees where the immediate subtrees are grouped in pairs with address-like objects. Motivated by natural ordering properties, extracted from the standard notations for the big Veblen number, we investigate different choices for embeddability relations on the generalized trees. We observe that for addresses using one finite sequence only, the embeddability coincides with (...) the classical tree-embeddability, but in this article we are interested in more general situations. We prove that the maximal order type of some of these new embeddability relations hit precisely the big Veblen ordinal ϑΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega}}$$\end{document}. Somewhat surprisingly, changing a little bit the well-partially ordered addresses, the maximal order type hits an ordinal which exceeds the big Veblen number by far, namely ϑΩΩΩ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\vartheta \Omega^{\Omega^\Omega}}$$\end{document}. Our results contribute to the research program on classifying properties of natural well-orderings in terms of order-theoretic properties of the functions generating the orderings. (shrink)

Two forms of continuity are defined for Pareto representations of preferences. They are designated “continuity” and “coordinate continuity.” Characterizations are given of those Pareto representable preferences that are continuously representable and, in dimension two, of those that are coordinate-continuously representable.

The aim of the present paper is to provide an axiomatic analysis of incomplete social judgments. In particular, we clarify the underlying power structure of Arrovian collective choice rules when social preferences are allowed to be incomplete. We propose the concept of quasi-decisiveness and investigate the properties of the collection of quasi-decisive sets associated with an Arrovian collective choice rule. In the course of this, we offer a series of applications.

We prove a finite model theorem and infinitary completeness result for the propositional -calculus. The construction establishes a link between finite model theorems for propositional program logics and the theory of well-quasi-orders.

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 725–731] , Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. of (...) the LMS 14 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory .In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.For these independence principles we classify their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles.As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results.This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem. (shrink)

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain (CAC) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of (...) WKL. (shrink)

We simultaneously generalize Silver’s perfect set theorem for co-analytic equivalence relations and Harrington-Marker-Shelah’s Dilworth-style perfect set theorem for Borel quasi-orders, establish the analogous theorem at the next definable cardinal, and give further generalizations under weaker definability conditions.

We investigate the extent to which structures consisting of sequences of forests on the same underlying set are well-quasi-ordered under embeddings.

We introduce the appropriate iterated version of the system of ordinal notations from [G1] whose order type is the familiar Howard ordinal. As in [G1], our ordinal notations are partly inspired by the ideas from [P] where certain crucial properties of the traditional Munich' ordinal notations are isolated and used in the cut-elimination proofs. As compared to the corresponding “impredicative” Munich' ordinal notations (see e.g. [B1, B2, J, Sch1, Sch2, BSch]), our ordinal notations arearbitrary terms in the appropriate simple (...) term algebra based on the notion of collapsing functions (which we would rather identify as projective functions). In Sect. 1 below we define the systems of ordinal notationsPRJ( ), for any primitive recursive limit wellordering . In Sect. 2 we prove the crucial well-foundness property by using the appropriate well-quasi-ordering property of the corresponding binary labeled trees [G3]. In Sect. 3 we interprete inPRJ( ) the familiar Veblen-Bachmann hierarchy of ordinal functions, and in Sect. 4 we show that the corresponding Buchholz's system of ordinal notationsOT( ) is a proper subsystem ofPRJ( ), although it has the same order type according to [G3] together with the interpretation from Sect. 2 in the terms of labeled trees. In Sect. 5 we use Friedman's approach in order to obtain an appropriate purely combinatorial statement which is not provable in the theory of iterated inductive definitions ID< λ, for arbitrarily large limit ordinalλ. Formal theories, axioms, etc. used below are familiar in the proof theory of subsystems of analysis (see [BFPS, T, BSch]). (shrink)