We study the reverse mathematics of interval orders. We establish the logical strength of the implications among various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2 \oplus 2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it (...) contains neither 2 \oplus 2 nor 3 \oplus 1. (shrink)

There are close similarities between the Weihrauch lattice and the zoo of axiom systems in reverse mathematics. Following these similarities has often allowed researchers to translate results from one setting to the other. However, amongst the big five axiom systems from reverse mathematics, so far $\mathrm {ATR}_0$ has no identified counterpart in the Weihrauch degrees. We explore and evaluate several candidates, and conclude that the situation is complicated.

We study the computability-theoretic complexity and proof-theoretic strength of the following statements: (1) "If X is a well-ordering, then so is ε X ", and (2) "If X is a well-ordering, then so is φ(α, X)", where α is a fixed computable ordinal and φ represents the two-placed Veblen function. For the former statement, we show that ω iterations of the Turing jump are necessary in the proof and that the statement is equivalent to ${\mathrm{A}\mathrm{C}\mathrm{A}}_{0}^{+}$ over RCA₀. To prove the (...) latter statement we need to use ω α iterations of the Turing jump, and we show that the statement is equivalent to ${\mathrm{\Pi }}_{{\mathrm{\omega }}^{\mathrm{\alpha }}}^{0}{-\mathrm{C}\mathrm{A}}_{0}$ . Our proofs are purely computability-theoretic. We also give a new proof of a result of Friedman: the statement "if x is a well-ordering, then so is φ(x, 0)" is equivalent to ATR₀ over RCA₀. (shrink)

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ2, the first fixed point of the ε-function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in +“φ2 is well-ordered” and, over , implies +“φ2 is well-ordered”.

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.

We introduce the notion of τ-like partial order, where τ is one of the linear order types ω, ω*, ω + ω*, and ζ. For example, being ω-like means that every element has finitely many predecessors, while being ζ-like means that every interval is finite. We consider statements of the form “any τ-like partial order has a τ-like linear extension” and “any τ-like partial order is embeddable into τ” . Working in the framework of reverse mathematics, we show that these (...) statements are equivalent either to equation image or to equation image over the usual base system equation image. (shrink)

We investigate the uniform computational content of the open and clopen Ramsey theorems in the Weihrauch lattice. While they are known to be equivalent to $\mathrm {ATR_0}$ from the point of view of reverse mathematics, there is not a canonical way to phrase them as multivalued functions. We identify eight different multivalued functions and study their degree from the point of view of Weihrauch, strong Weihrauch, and arithmetic Weihrauch reducibility. In particular one of our functions turns out to be strictly (...) stronger than any previously studied multivalued functions arising from statements around $\mathrm {ATR}_0$. (shrink)

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 show that the quasi-order of continuous embeddability between finitely branching dendrites (a natural class of fairly simple compacta) is $\Sigma_1^1$ -complete. We also show that embeddability between countable linear orders with infinitely many colors is $\Sigma_1^1$ -complete.

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)

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.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and the class (...) of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete. (shrink)

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to $\hbox{\sf WKL}_0$ ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to (...) $\hbox{\sf ACA}_0$ ; the statement “every Lebesgue space is Atsuji” is provable in $\hbox{\sf RCA}_0$ ; the statement “every Atsuji space is Lebesgue” is provable in $\hbox{\sf ACA}_0$ . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to $\Pi^1_1\hbox{-\sf CA}_0$. (shrink)

We analyze the axiomatic strength of the following theorem due to Rival and Sands [28] in the style of reverse mathematics. Every infinite partial order P of finite width contains an infinite chain C such that every element of P is either comparable with no element of C or with infinitely many elements of C. Our main results are the following. The Rival–Sands theorem for infinite partial orders of arbitrary finite width is equivalent to $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$ (...) over $\mathsf {RCA}_0$. For each fixed $k \geq 3$, the Rival–Sands theorem for infinite partial orders of width $\leq \!k$ is equivalent to $\mathsf {ADS}$ over $\mathsf {RCA}_0$. The Rival–Sands theorem for infinite partial orders that are decomposable into the union of two chains is equivalent to $\mathsf {SADS}$ over $\mathsf {RCA}_0$. Here $\mathsf {RCA}_0$ denotes the recursive comprehension axiomatic system, $\mathsf {I}\Sigma ^0_{2}$ denotes the $\Sigma ^0_2$ induction scheme, $\mathsf {ADS}$ denotes the ascending/descending sequence principle, and $\mathsf {SADS}$ denotes the stable ascending/descending sequence principle. To the best of our knowledge, these versions of the Rival–Sands theorem for partial orders are the first examples of theorems from the general mathematics literature whose strength is exactly characterized by $\mathsf {I}\Sigma ^0_{2} + \mathsf {ADS}$, by $\mathsf {ADS}$, and by $\mathsf {SADS}$. Furthermore, we give a new purely combinatorial result by extending the Rival–Sands theorem to infinite partial orders that do not have infinite antichains, and we show that this extension is equivalent to arithmetical comprehension over $\mathsf {RCA}_0$. (shrink)

When a linear order has an order preserving surjection onto each of its suborders, we say that it is strongly surjective. We prove that the set of countable strongly surjective linear orders is a [Formula: see text]-complete set. Using hypotheses beyond ZFC, we prove the existence of uncountable strongly surjective orders.

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We (...) show that the left to right directions of these theorems are equivalent to ACA0 and ATR0, respectively. On the other hand, the opposite directions are both provable in WKL0, but not in RCA0. We also prove the equivalence with ACA0 of the following result of Erdös and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains. (shrink)

We study Polish spaces for which a set of possible distances $A \subseteq R^+$ is fixed in advance. We determine, depending on the properties of A, the complexity of the collection of all Polish metric spaces with distances in A, obtaining also example of sets in some Wadge classes where not many natural examples are known. Moreover we describe the properties that A must have in order that all Polish spaces with distances in that set belong to a given class, (...) such as zero-dimensional, locally compact, etc. These results lead us to give a fairly complete description of the complexity, with respect to Borel reducibility and again depending on the properties of A, of the relations of isometry and isometric embeddability between these Polish spaces. (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)

Let ⪯R be the preorder of embeddability between countable linear orders colored with elements of Rado's partial order . We show that ⪯R has fairly high complexity with respect to Borel reducibility , although its exact classification remains open.

The aim of this paper is to argue against the received view among Unamuno scholars that Miguel de Unamuno was defending a sort of pragmatic argument for religious faith and that his notion of religious faith as “querer creer” (“wanting to believe”) is to be identified with William James’s “the will to believe”. As I will show in this paper, one of the aspects that makes Unamuno’s reasoning philosophically relevant is his ability to formulate a non-pragmatist defense of religious faith (...) without a prior commitment to the truth of any religious or theological statement and grounded in our longing for an endless existence through God’s Salvation. (shrink)

The purpose of this succinct work is to present N. Machiavelli's classic republican view from his proposition of an inevitable paradox, the founding of an expansionist republic, difficult to govern, or the founding of a stable, but small and weak republic. Such a paradox, according to Machiavelli, should direct and condition all the constitutive devices of the republic when choosing what will be its destiny as a political body. The model of republic preferred by the Florentine will be the expansionist (...) model of Rome, leading him to assume all the devices that gave this republic its power. From this presentation of the Machiavellian proposition, we will analyse the assimilation of republican thought in England from the Elizabethan period, as well as the political-social scenario that exists there. This itinerary will allow us to understand, in general, why classical republicanism was received on English soil from the perspective of establishing a mixed, stable government, thus favouring the spread of the Venice myth as a serene republic and delaying the use, even that mitigated, of the republican presuppositions expressed in the Machiavellian work that directed towards a Roman model. (shrink)

In this brief article we will seek to support the thesis that there is in the thought of Coluccio Salutati, 14th century Florentine chancellor and prominent humanist, a clear republicanism that turns to the issue of the freedom of the republic and the active life of the individuals participating in it. However, in connection with this demonstration, we will also maintain that such republicanism has strong traces of Augustinian influence, mainly in view of the disposition of laws in the political (...) arrangement, as well as the relationship between active life and the search for a human being’s own good. In view of this, we intend to demonstrate that, in general, Salutati’s Renaissance republicanism turns to the defence of the republic, its freedom and its characteristic of the proper scenario for the active life of individuals, however, it still maintains an ethical-political perspective typical of the medieval period, centrally influenced by Augustinian thought. (shrink)

Tencionamos, neste artigo, tratar da questão da imitação no pensamento maquiaveliano. Tomaremos como balizas suas postulações acerca da natureza do homem, as possibilidades de formatação de uma república profícua em vista desta natureza e o papel dos exemplos dos indivíduos que agem em prol do bem da república. Sendo a questão da imitação o fio condutor, elegeremos ainda um tripé para demonstrar nossa hipótese, a saber, o fator antropológico; o fator educacional; o fator político. Por este movimento, pretende-se demonstrar que (...) a aparente negatividade das postulações maquiavelianas sobre a natureza humana não inviabilizam a promulgação de uma república estável, tendo os bons exemplos políticos um papel preponderante neste propósito. (shrink)

Most of the existing studies on Bourdieu and the digital regards the social and class distinctions in the use of digital technologies, thus presupposing a certain transparency of technologies themselves. Our proposal is to refer to this attitude as “Bourdieu outside the digital.” Yet in this paper, another perspective called “Bourdieu inside the digital” is developed, which moves the focus on the effects of some emerging technologies on social distinctions and discrimination. The main hypothesis is that algorithms of machine learning (...) are producers and reproducers of habitus. Although their results present a greater granularity with respect to the standard techniques of the past, these algorithms still reduce individuals to categories, general trends, classes, and behaviors. Such a reduction has flattening effects on the individuals’ self-understanding, especially in terms of identity and interaction with the social world. This is the phenomenon described in the article as the “personalization without personality,” whose consequences are both existential and social-political. This idea will be illustrated through qualitative and comparative analysis between the correspondence analysis and the multiple correspondence analysis used by Bourdieu in his works and some of those techniques that are performed today in big data analytics. (shrink)

Neste breve artigo, argumentaremos que a pandemia da Covid-19, como outras na história humana, colocou em desnudo as bases da estrutura político-econômica moderna. Defenderemos que tais bases apoiam-se em dois esteios, uma noção política de ‘Estado’, ápice da organização humana, e uma concepção racionalista, técnica e produtiva, que promulga a possibilidade de resposta a qualquer crise surgida na sociedade. Como contrapondo a esta estrutura, apresentaremos a teoria maquiaveliana, que aponta para uma relação indelével entre política e fator natural, natureza e (...) sociedade. Nesta teoria, com uma concepção antropológica própria, as necessidades impostas pela natureza frente a ação humana são determinantes para se pensar a própria construção política. Como resultado, consideraremos ser necessário repensar a estrutura político-econômica moderna em seus pretensos mitos de incontestável avanço político, por meio do ‘Estado’, e inabalável base racional, principalmente técnico-produtiva, retomando o lugar de importância do fator natural para constituição de uma sociedade mais ‘humana’. (shrink)

Tencionamos, neste artigo, demonstrar os meandros constitutivos do que seja um formalismo moral em Immanuel Kant, tendo como pressuposto teórico sua compreensão de uma teoria da vontade. Este movimento de análise e demonstração teórica terá como fio condutor a ideia de autonomia que permeia a construção da vontade no formalismo kantiano, ou seja, a pressuposição de uma possível autonomia formal da razão é a base na qual se assenta a teoria moral deste pensador. Para execução de nossa pretensão será necessário (...) reconstituir os pontos nevrálgicos do que venha a ser uma teoria da vontade para em seguida analisar os pressupostos kantianos que realçam a centralidade da autonomia da razão na compreensão de uma vontade livre. Desta maneira, o que se pretende por em desnudo é a originalidade kantiana, em relação à tradição que promulga a vontade como marca da liberdade humana, a partir de seu pressuposto de autonomia formal na constituição moral do indivíduo humano. (shrink)

Neste artigo intentamos empreender uma leitura do judaísmo antigo, ou veterotestamentário, em alguns de seus aspectos principais, à contraluz da visão maquiaveliana do papel da religião na organização e expansão do Estado. O primeiro passo desta empresa será dado ao expormos os traços fundantes e delineadores da tradição judaica antiga, principalmente nos livros que compõe o chamado velho testamento, demonstrando o uso da religião na ordenação e expansão do Estado, bem como na formação do indivíduo. Em um segundo momento demonstraremos (...) como se cria uma nova tradição judaica, influenciada e influenciadora do mundo grego antigo, logo no início da era cristã a qual será assimilada e explorada em seu viés místico no humanismo renascentista. Esta tradição judaico helenístico será a mais conhecida e assimilada nos círculos eruditos europeus da renascença. Por fim, buscaremos demonstrar como os traços constitutivos do judaísmo antigo, em detrimento ao judaísmo helenístico, podem ser lidos à luz da descrição maquiaveliana do papel da religião na organização e expansão do Estado e na formação do indivíduo. (shrink)

Our objective in this brief article is guided by the demonstration of the existence of a theory of political action in Machiavelli's republican thought, with such a theory having its own character that directs it to highlight the action of individuals in the social context. In addition to this objective, we hope to support the thesis that such a theory of political action has a republican scope, not just “republicanist”, in keeping with the Machiavellian preference for institutions that impress on (...) individuals a civic sense based above all on the materiality of political action in the body social. From this assertion, we indicate that our itinerary will be guided by the demonstration of the search for the valorization of political action in Machiavelli's theory, the materiality of such action, to the detriment of its pure intention, the central focus of Florentine's work. This disposition of the centrality of political action in Machiavelli republicanism will underscore its appreciation for outlining the political functions of the search for recognition, glory and especially the benefit of the political body as a whole. (shrink)

Cultural evolution studies are characterized by the notion that culture evolves accordingly to broadly Darwinian principles. Yet how far the analogy between cultural and genetic evolution should be pushed is open to debate. Here, we examine a recent disagreement that concerns the extent to which cultural transmission should be considered a preservative mechanism allowing selection among different variants, or a transformative process in which individuals recreate variants each time they are transmitted. The latter is associated with the notion of “cultural (...) attraction”. This issue has generated much misunderstanding and confusion. We first clarify the respective positions, noting that there is in fact no substantive incompatibility between cultural attraction and standard cultural evolution approaches, beyond a difference in focus. Whether cultural transmission should be considered a preservative or reconstructive process is ultimately an empirical question, and we examine how both preservative and reconstructive cultural transmission has been studied in recent experimental research in cultural evolution. Finally, we discuss how the relative importance of preservative and reconstructive processes may depend on the granularity of analysis and the domain being studied. (shrink)

The first half of this article follows Mario Bunge’s early realist moves, his efforts to articulate the achievements of theoretical physics as gains in the quest for objective truth and understanding, particularly in the context of the fights against the idealist and subjectivist interpretations of quantum mechanics that, at least until the mid-1970s, prevailed in physics. Bunge’s answers to the problems of quantum mechanics provide a good angle for understanding how his realist positions grew on the “battlefield.” The second half (...) discusses Bunge’s general conception of the scientific realist stance and confronts it with some current approaches to realism in the mainstream literature. (shrink)

The aim of this chapter is to analyse the concept of love —understood in the broad and Christian-inspired sense of love as agape-charis love— in relation to the animes Dragon Ball and Dragon Ball Z. I first comment on the character of Piccolo —and how his friendship with Son Gohan— leads to him losing all his original villainous traits. I argue that the evolution of the character of Piccolo through his friendship with Son Gohan illustrates the philosophical claim that a (...) loving and giving of oneself to the other —far from being a sign of weakness— is a courageous and self-affirming exercise. The result may move the individual to an inner transformative experience that leads to the enjoyment of a renewed and authentic life that is worth living. I then argue that Son Goku’s respectful attitude towards his opponents is ultimately a loving attitude. Last, I argue that it is the inspiring and loving example of Son Goku that ultimately explains why characters such as Vegeta —who are at first presented as villains— end up losing their villainous traits after fighting against Son Goku. (shrink)

El objetivo de este artículo es ofrecer una exposición sistemática de la propuesta filosófica de Joaquim Xirau (Figueras, 1895 – Ciudad de México, 1946), señalando así su interés filosófico al tiempo que se vinculan sus dos grandes obras ensayísticas de madurez, Amor y mundo (1940) y Lo fugaz y lo eterno (1942).

Introduction: Jai Mata Di (praise the mother goddess) -- Sutra 1: Samadhi, or yogic ecstasy -- Sutra 2: realization, or the practice of yoga -- Sutra 3: the Siddhis, or the magical powers -- Sutra 4: absolute freedom.

In this paper I will argue that Unamuno was conceiving of God (and ordinary, non-religious fictional characters more generally) in realist, though non-evidentially grounded, terms. I will point out that this way of conceiving of God allowed Unamuno to claim the actual existence of God (though as a fictional, purely humanly created character) and, with this, the possibility of there being an actual relationship between the concrete religious person and God without having to dispense with his own core claim that (...) religious faith is just a subjective, though natural and thus inevitable, human reaction. (shrink)

BackgroundPrivate umbilical cord blood banking is a for-profit industry in which parents pay to store blood for potential future use. Governments have noted the tendency for private banks to oversell the potential for cord blood use, especially in relation to speculative cell therapies not yet supported by clinical evidence. We assessed the regulatory landscape governing private cord bank marketing in Canada.Main bodyBecause the problematic marketing of private cord blood banking for future use often relates to speculative future cell therapies that (...) do not exist and are not being advertised for current clinical use, most private blood bank marketing seems to fall outside Health Canada’s regulatory scope. However, this problematic marketing is regulated by the Competition Bureau pursuant to the Competition Act. While representations relating to future hypothetical treatments may not always be subject to the legal requirement for claim substantiation, the law also prohibits individuals and companies from knowingly or recklessly making representations that are “false or misleading in a material respect.” A representation is materially false or misleading when it could “influence a consumer’s behavior or purchasing decisions,” and consumers are likely to be considered to be “credulous and inexperienced” for the purposes of assessing an advertisement’s general impression. Because all of the potential benefit of the banking is derived from the potential future use of the biological material for health interventions directed toward the customers and their relatives, and because we know the best available medical evidence indicates a very low probability of utility in this context, we can say with confidence that some private cord blood banking claims are materially misleading. Moreover, to the extent that medical professionals are involved in private bank interactions with customers or hold ownership stakes in private banks, they are subject to professional codes, standards of practice, and potentially fiduciary obligations that further prohibit misleading marketing.ConclusionsPrivate cord blood bank marketing that advertises hypothetical future treatments can be misleading and may influence consumer behaviour. This marketing may breach existing advertising law. Regulatory bodies should enforce the law in order to help prevent public health and personal financial harm. (shrink)

In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as `at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) (...) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames. (shrink)