The notion of countable well order admits an alternative definition in terms of embeddings between initial segments. We use the framework of reverse mathematics to investigate the logical strength of this definition and its connection with Fraïssé’s conjecture, which has been proved by Laver. We also fill a small gap in Shore’s proof that Fraïssé’s conjecture implies arithmetic transfinite recursion over $\mathbf {RCA}_0$, by giving a new proof of $\Sigma ^0_2$ -induction.

Two countable well orderings are weakly comparable if there is an order preserving injection of one into the other. We say the well orderings are strongly comparable if the injection is an isomorphism between one ordering and an initial segment of the other. In [5], Friedman announced that the statement “any two countable well orderings are strongly comparable” is equivalent to ATR 0 . Simpson provides a detailed proof of this result in Chapter 5 of [13]. More recently, (...) Friedman has proved that the statement “any two countable well orderings are weakly comparable” is equivalent to ATR 0 . The main goal of this paper is to give a detailed exposition of this result. (shrink)

The larger project broached here is to look at the generally sentence “if X is well-ordered then f is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory Tf whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this (...) theme, we prove in this paper that the statement “if X is well-ordered then εX is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a fragment of. (shrink)

Associate to any linear ordering on the integers the mapping whose value on n is the cardinality of {kn; kn}: a purely combinatorial characterization for the mappings associated to the well-orderings is established.

We introduce a new simple first-order framework for theories whose objects are well-orderings (lists). A system ALT (axiomatic list theory) is presented and shown to be equiconsistent with ZFC (Zermelo Fraenkel Set Theory with the Axiom of Choice). The theory sheds new light on the power set axiom and on Gs axiom of constructibility. In list theory there are strong arguments favoring Gs axiom, while a bare analogon of the set theoretic power set axiom looks artificial. In fact, there is (...) a natural and attractive modification of ALT where every object is constructible and countable. In order to substantiate our foundational interest in lists, we also compare sets and lists from the perspective of finite objects, arguing that lists are, from a certain point of view, conceptually simpler than sets. (shrink)

It is shown that the boldface maximality principle for subcomplete forcing,, together with the assumption that the universe has only set many grounds, implies the existence of a well‐ordering of definable without parameters. The same conclusion follows from, assuming there is no inner model with an inaccessible limit of measurable cardinals. Similarly, the bounded subcomplete forcing axiom, together with the assumption that does not exist, for some, implies the existence of a well‐ordering of which is Δ1‐definable without parameters, (...) and ‐definable using a subset of ω1 as a parameter. This well‐order is in. Enhanced versions of bounded forcing axioms are introduced that are strong enough to have the implications of mentioned above, and along the way, a bounded forcing axiom for countably closed forcing is proposed. (shrink)

We provide a general criterion for Fraenkel–Mostowski models of $${\textsf{ZFA}}$$ (i.e. Zermelo–Fraenkel set theory weakened to permit the existence of atoms) which implies “every linearly ordered set can be well ordered” ( $${\textsf{LW}}$$ ), and look at six models for $${\textsf{ZFA}}$$ which satisfy this criterion (and thus $${\textsf{LW}}$$ is true in these models) and “every Dedekind finite set is finite” ( $${\textsf{DF}}={\textsf{F}}$$ ) is true, and also consider various forms of choice for well-ordered families of well orderable sets in these (...) models. In Model 1, the axiom of multiple choice for countably infinite families of countably infinite sets ( $${\textsf{MC}}_{\aleph _{0}}^{\aleph _{0}}$$ ) is false. It was the open question of whether or not such a model exists (from Howard and Tachtsis “On metrizability and compactness of certain products without the Axiom of Choice”) that provided the motivation for this paper. In Model 2, which is constructed by first choosing an uncountable regular cardinal in the ground model, a strong form of Dependent choice is true, while the axiom of choice for well-ordered families of finite sets ( $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ ) is false. Also in this model the axiom of multiple choice for well-ordered families of well orderable sets fails. Model 3 is similar to Model 2 except for the status of $${\textsf{AC}}^{{\textsf{WO}}}_{{\textsf{fin}}}$$ which is unknown. Models 4 and 5 are variations of Model 3. In Model 4 $${\textsf{AC}}_{\textrm{fin}}^{{\textsf{WO}}}$$ is true. The construction of Model 5 begins by choosing a regular successor cardinal in the ground model. Model 6 is the only one in which $$2{\mathfrak {m}} = {\mathfrak {m}}$$ for every infinite cardinal number $${\mathfrak {m}}$$. We show that the union of a well-ordered family of well orderable sets is well orderable in Model 6 and that the axiom of multiple countable choice is false. (shrink)

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality \. We provide some examples of pairwise non-isomorphic countably saturated linear orders of cardinality \, under different set-theoretic assumptions. We give a new proof of the old theorem of Harzheim, that the class of countably saturated linear orders has a uniquely determined one-element basis. From (...) our proof it follows that this minimal linear order is a Fraïssé limit of certain Fraïssé class. In particular, it is homogeneous with respect to countable subsets. Next we prove the existence and uniqueness of the uncountable version of the random graph. This graph is isomorphic to \,\in \cup \ni )\), where \\) is the set of hereditarily countable sets, and two sets are connected if one of them is an element of the other. In the last section, an example of a prime countably saturated Boolean algebra is presented. (shrink)

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)

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.

The main theorem of this article is that every countable model of set theory 〈M, ∈M〉, including every well-founded model, is isomorphic to a submodel of its own constructible universe 〈LM, ∈M〉 by means of an embedding j : M → LM. It follows from the proof that the countable models of set theory are linearly pre-ordered by embeddability: if 〈M, ∈M〉 and 〈N, ∈N〉 are countable models of set theory, then either M is isomorphic to a (...) submodel of N or conversely. Indeed, these models are pre-well-ordered by embeddability in order-type exactly ω1+ 1. Specifically, the countable well-founded models are ordered under embeddability exactly in accordance with the heights of their ordinals; every shorter model embeds into every taller model; every model of set theory M is universal for all countable well-founded binary relations of rank at most OrdM; and every ill-founded model of set theory is universal for all countable acyclic binary relations. Finally, strengthening a classical theorem of Ressayre, the proof method shows that if M is any nonstandard model of PA, then every countable model of set theory — in particular, every model of ZFC plus large cardinals — is isomorphic to a submodel of the hereditarily finite sets 〈 HFM, ∈M〉 of M. Indeed, 〈 HFM, ∈M〉 is universal for all countable acyclic binary relations. (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)

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)

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)

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)

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)

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)

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)

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)

There is an emerging view according to which countability is not an integral part of the lexical meaning of singular count nouns, but is ‘added on’ or ‘made available’, whether syntactically, semantically or both. This view has been pursued by Borer and Rothstein among others in order to deal with classifier languages such as Chinese as well as challenges to standard views of the mass-count distinction such as object mass nouns such as furniture. I will discuss a range of data, (...) partly from German, that such a grammar-based view of countability receives support when applied to verbs with respect to the event argument position. Verbs themselves fail to specify events as countable in English and related languages; instead countability is made available only by the use of the event classifier time or else particular lexical items, such as frequency expressions, German beides ‘both’, or the nominalizing light noun -thing. The paper will not adopt or elaborate a particular version of the grammar-based view of countability, but rather critically discuss existing versions and present two semantic options of elaborating the view. (shrink)

A wide family of many-valued logics—for instance, those based on the weak Kleene algebra—includes a non-classical truth-value that is ‘contaminating’ in the sense that whenever the value is assigned to a formula φ⁠, any complex formula in which φ appears is assigned that value as well. In such systems, the contaminating value enjoys a wide range of interpretations, suggesting scenarios in which more than one of these interpretations are called for. This calls for an evaluation of systems with multiple contaminating (...) values. In this paper, we consider the countably infinite family of multiple-conclusion consequence relations in which classical logic is enriched with one or more contaminating values whose behavior is determined by a linear ordering between them. We consider some motivations and applications for such systems and provide general characterizations for all consequence relations in this family. Finally, we provide sequent calculi for a pair of four-valued logics including two linearly ordered contaminating values before defining two-sided sequent calculi corresponding to each of the infinite family of many-valued logics studied in this paper. (shrink)

In this article I investigate the phenomenon of minimum and minimal models of second-order set theories, focusing on Kelley–Morse set theory KM, Gödel–Bernays set theory GB, and GB augmented with the principle of Elementary Transfinite Recursion. The main results are the following. (1) A countable model of ZFC has a minimum GBC-realization if and only if it admits a parametrically definable global well order. (2) Countable models of GBC admit minimal extensions with the same sets. (3) There is (...) no minimum transitive model of KM. (4) There is a minimum β-model of GB+ETR. The main question left unanswered by this article is whether there is a minimum transitive model of GB+ETR. (shrink)

It is easy to prove in ZF− that a relation R satisfies the maximal condition if and only if its transitive hull R* does; equivalently: R is well-founded if and only if R* is. We will show in the following that, if the maximal condition is replaced by the chain condition, as is often the case in Algebra, the resulting statement is not provable in ZF− anymore . More precisely, we will prove that this statement is equivalent in ZF− to (...) the countable axiom of choice ACω. Moreover, applying this result we will prove that the axiom of dependent choices, restricted to partial orders as used in Algebra, already implies the general form for arbitrary relations as formulated first by Teichmüller and, independently, some time later by Bernays and Tarski. (shrink)

The Well-Ordered Universe argues that Cavendish's natural philosophy, social and political philosophy, and medical theory share an underlying concern with order. This reveals interesting connections among Cavendish's natural philosophy and her views on gender, animals and the environment, and human health, and explains her commitment to monarchy and social hierarchy.

This paper uses the framework of reverse mathematics to analyze the proof theoretic content of several statements concerning multiplication of countable well-orderings. In particular, a division algorithm for ordinal arithmetic is shown to be equivalent to the subsystem ATR0.

A well-ordered society faces a crisis whenever a sufficient number of noncompliers enter into the political system. This has the potential to destabilize liberal democratic political order. This article provides a formal analysis of two competing solutions to the problem of political stability offered in the public reason liberalism literature—namely, using public reason or using convergence discourse to restore liberal democratic political order in the well-ordered society. The formal analyses offered in this article show that using public reason fails completely, (...) and using convergent discourse, although doing better, has its own critical limitations that have not been previously recognized properly. (shrink)

If there is no inner model with ω many strong cardinals, then there is a set forcing extension of the universe with a projective well-ordering of the reals.

This article agrees with Philip Kitcher that we should aim for a well-ordered science, one that answers the right questions in the right ways. Crucial to this is to address questions of use: Which scientific account is right for which system in which circumstances? This is a difficult question: evidence that may support a scientific claim in one context may not support it in another. Drawing on examples in physics and other sciences, this article argues that work on the warrant (...) of theories in philosophy of science needs to change. Emphasis should move from the warrant of theories in the abstract to questions of evidence for use. (shrink)

Kitcher has proposed an ideal-theory account—well-ordered science (WOS)— of the collective good that science’s research agenda should promote. Against criticism regarding WOS’s action-guidance, Kitcher has advised critics not to confuse substantive ideals and the ways to arrive at them, and he has defended WOS as a necessary and useful ideal for science policy. I provide a distinction between two types of ideal-theories that helps clarifying WOS’s elusive nature. I use this distinction to argue that the action-guidance problem that WOS faces (...) remains even under the aims/means distinction, because the WOS’s failure is more basic than critics have suggested. (shrink)

Assuming ${2^{{N_0}}}$ = N₁ and ${2^{{N_1}}}$ = N₂, we build a partial order that forces the existence of a well-order of H(ω₂) lightface definable over ⟨H(ω₂), Є⟩ and that preserves cardinal exponentiation and cofinalities.

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our (...) main results are in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and there is no such sequence of length $$\omega _1$$ ω 1. (shrink)

Building, restoring and maintaining well-placed trust between scientists and the public is a difficult yet crucial social task requiring the successful cooperation of various social actors and institutions. Kitcher’s takes up this challenge in the context of liberal democratic societies by extending his ideal model of “well-ordered science” that he had originally formulated in his. However, Kitcher nowhere offers an explicit account of what it means for the public to invest epistemic trust in science. Yet in order to understand how (...) his extended model and its implementation in the actual world address the problem of trust as well as to evaluate it critically, an explicit account of epistemic public trust in science needs to be given first. In this article we first present such an account and then scrutinize his project of building public trust in science in light of it. We argue that even though Kitcher’s ideal model and his proposals for its implementation in the real world face a number of problems, they can be addressed with the resources of our account. (shrink)

In the absence of Woodin cardinals, fine structural inner models for mild large cardinal hypotheses admit forcing extensions where bounded forcing axioms hold and yet the reals are projectively well-ordered.

In previous work, the author has shown that $\Pi ^1_1$ -induction along $\mathbb N$ is equivalent to a suitable formalization of the statement that every normal function on the ordinals has a fixed point. More precisely, this was proved for a representation of normal functions in terms of Girard’s dilators, which are particularly uniform transformations of well orders. The present paper works on the next type level and considers uniform transformations of dilators, which are called 2-ptykes. We show that $\Pi (...) ^1_2$ -induction along $\mathbb N$ is equivalent to the existence of fixed points for all 2-ptykes that satisfy a certain normality condition. Beyond this specific result, the paper paves the way for the analysis of further $\Pi ^1_4$ -statements in terms of well ordering principles. (shrink)

We present well-ordering proofs for Martin-Löf's type theory with W-type and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in Setzer show that the proof theoretical strength of the type theory is precisely ψΩ1Ω1 + ω, which is slightly more than the strength of Feferman's theory T0, classical set theory KPI and the subsystem of analysis + . The strength of intensional and extensional version, of the version à (...) la Tarski and à la Russell are shown to be the same. (shrink)

The debate over the use of genetically-modified (GM) crops is one where the heat to light ratio is often quite low. Both proponents and opponents of GM crops often resort more to rhetoric than argument. This paper attempts to use Philip Kitcher’s idea of a “well-ordered science” to bring coherence to the debate. While I cannot, of course, here decide when and where, if at all, GM crops should be used I do show how Kitcher’s approach provides a useful framework (...) in which to evaluate the desirability of using GM crops. At the least Kitcher’s approach allows us to see that the current state of research in to, and use of, GM crops is very far from the ideal of a well-ordered science and gives us a goal to work towards if we wish to achieve a more well-ordered agricultural policy. (shrink)

This essay defends the view that “modern science,” as with modernity in general, is a polycentered phenomenon, something that appears in different forms at different times and places. It begins with two ideas about the nature of rational scientific inquiry: Karin Knorr Cetina's idea of “epistemic cultures,” and Philip Kitcher's idea of science as “a system of public knowledge,” such knowledge as would be deemed worthwhile by an ideal conversation among the whole public under conditions of mutual engagement. This account (...) of the nature of scientific practice provides us with a new perspective from which to understand key elements in the philosophical project of Jaina logicians in the seventh, eighth, and ninth centuries c.e. Jaina theory seems exceptionally well targeted onto two of the key constituents in the ideal conversation—the classification of all human points of view and the representation of end states of the deliberative process. The Buddhist theory of the Kathāvatthu contributes to Indian epistemic culture in a different way: by supplying a detailed theory of how human dialogical standpoints can be revised in the ideal conversation, an account of the phenomenon Kitcher labels “tutoring.” Thus science in India has its own history, one that should be studied in comparison and contrast with the history of science in Europe. In answer to Joseph Needham, it was not ‘modern science’ which failed to develop in India or China but rather non-well-ordered science, science as unconstrained by social value and democratic consent. What I argue is that this is not a deficit in the civilisational histories of these countries, but a virtue. (shrink)

A forcing poset of size 221 which adds no new reals is described and shown to provide a Δ22 definable well-order of the reals . The encoding of this well-order is obtained by playing with products of Aronszajn trees: some products are special while other are Suslin trees. The paper also deals with the Magidor–Malitz logic: it is consistent that this logic is highly noncompact.

We use a reverse Easton forcing iteration to obtain a universe with a definable well-order, while preserving the GCH and proper classes of a variety of very large cardinals. This is achieved by coding using the principle ◊ $_{k^ - }^* $ at a proper class of cardinals k. By choosing the cardinals at which coding occurs sufficiently sparsely, we are able to lift the embeddings witnessing the large cardinal properties without having to meet any non-trivial master conditions.