We study the relative strength of the two axioms Every Pell equation has a nontrivial solution Exponentiation is total over weak fragments, and we show they are equivalent over IE1. We then define the graph of the exponential function using only existentially bounded quantifiers in the language of arithmetic expanded with the symbol #, where # = x[log2y]. We prove the recursion laws of exponentiation in the corresponding fragment.

Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We (...) also show that if R is a countable recursively saturated real closed ordered field, then there is an integer part I such that R = RC(I) and I is a nonstandard model of PA. (shrink)

A generalization is given of McAloon's result on initial segments ofmodels of GlΔ0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

Gauss used quadratic forms in his second proof of quadratic reciprocity. In this paper we begin to develop a theory of binary quadratic forms over weak fragments of Peano Arithmetic, with a view to reproducing Gauss’ proof in this setting.

We consider the theory and its weak fragments in the language of arithmetic expanded with the functional symbol . We prove that and its weak fragments, down to and , are subject to the Tennenbaum phenomenon with respect to , , and . For the last two theories it is still unknown if they may have nonstandard recursive models in the usual language of arithmetic.

In [4] it is shown that only using exponentiation can one prove the existence of non trivial solutions of Pell equations in IΔ 0 . However, in this paper we will prove that any Pell equation has a non trivial solution modulo m for every m in IΔ 0.

We give a valuation theoretic characterization for a real closed field to be recursively saturated. This builds on work in [9], where the authors gave such a characterization forκ-saturation, for a cardinal$\kappa \ge \aleph _0 $. Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.

In this paper we prove, modulo Schanuel's Conjecture, that there are algorithms which decide if two exponential polynomials in π are equal in ℝ and if two exponential polynomials in π and i coincide in ℂ.

1. Etica e coscienza. Etica, scienza e tecnica -- 2. Etica e cultura. Etica e politica -- 3. Etica ed economia. Problemi de etica.

Let R be a real closed field. An integer part I for R is a discretely ordered subring such that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${r \in R}$$\end{document}, there exists an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${i \in I}$$\end{document} so that i ≤ r < i + 1. Mourgues and Ressayre (J Symb Logic 58:641–647, 1993) showed that every real closed field has an integer part. The procedure of Mourgues and (...) Ressayre appears to be quite complicated. We would like to know whether there is a simple procedure, yielding an integer part that is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document} —limit computable relative to R. We show that there is a maximal Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} which is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2(R)}$$\end{document}. However, this I may not be an integer part for R. By a result of Wilkie (Logic Colloquium ’77), any Z-ring can be extended to an integer part for some real closed field. Using Wilkie’s ideas, we produce a real closed field R with a Z-ring \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${I \subseteq R}$$\end{document} such that I does not extend to an integer part for R. For a computable real closed field, we do not know whether there must be an integer part in the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document}. We know that certain subclasses of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^0_2}$$\end{document} are not sufficient. We show that for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \in \omega}$$\end{document}, there is a computable real closed field with no n-c.e. integer part. In fact, there is a computable real closed field with no n-c.e. integer part for any n. (shrink)

In this second paper of the series we do a local analysis of quadratic forms over completions of a non-standard model of IΔ0+Ω1.

In [4] the authors studied the residue field of a model M of IΔ0 + Ω1 for the principal ideal generated by a prime p. One of the main results is that M/ has a unique extension of each finite degree. In this paper we are interested in understanding the structure of any quotient field of M, i.e. we will study the quotient M/I for I a maximal ideal of M. We prove that any quotient field of M satisfies the (...) property of having a unique extension of each finite degree. We will use some of Cherlin's ideas from [3], where he studies the ideal theory of non standard algebraic integers. (shrink)

We prove that if G is a Δ 0 -definable function on the natural numbers and F = Π i = 0 n G , then F is also Δ 0 -definable. Moreover, the inductive properties of F can be proved inside the theory IΔ 0.

La réception de l’Idéologie en Argentine au début du XIXe siècle a joué un rôle essentiel dans la construction politique et culturelle du pays à travers l’action de son président, Bernardino Rivadavia. Les professeurs de philosophie de la nouvelle université de Buenos Aires furent chargés d’adapter en espagnol et de diffuser les idées des philosophes français dans un contexte politique souvent difficile. L’Idéologie est ainsi devenue un enjeu important dans les débats sur les modèles d’organisation politique de la jeune République (...) succédant à l’organisation coloniale. (shrink)

Lex necessitatis vel contingentiae Cet article se propose de reconsidérer de manière critique deux présupposés spécifiques à l’interprétation gilsonienne de Thomas d’Aquin : (i) le fait que, dans l’univers thomasien, tous les étants différents de Dieu soient marqués par une « contingence radicale » ; (ii) l’idée qu’en dernière analyse cette contingence est fondée sur la distinction entre être et essence – et, plus généralement, (iii) la conviction que la conception thomiste de la providence se présente comme une alternative radicale (...) au déterminisme et au nécessitarisme de la tradition gréco-arabe. À cette fin, nous examinerons successivement les points suivants : les significations les plus communes que Thomas confère aux notions de nécessité et de contingence ; le rapport entre fortune, sort et providence ; la réfutation conditionnée du déterminisme astral ; le mode sur lequel la providence dispose les événements nécessaires aussi bien que les événements contingents ; les réactions opposées de Siger de Brabant et de Jean Duns Scot au modèle par lequel Thomas tente de sauver la contingence au sein de l’ordre providentiel ; l’affirmation par Thomas de l’existence de créatures formellement nécessaires. En conclusion, on suggère que Thomas affaiblit certains présupposés fondamentaux de la stratégie anti-déterministe mise en œuvre par Aristote dans le livre VI de la Métaphysique, que l’alternative entre nécessité et contingence ne représente pas chez Thomas le véritable départ entre le créateur et la créature, et que, pour cette raison, la distinction entre être et essence n’a pas pour premier et principal objectif de rendre compte de la contingence de tous les étants créés. (shrink)

Rather unusually, Henry of Ghent includes ‘totality’ in the list of divine attributes discussed in his Summa quaestionum ordinariarum. Availing himself of the different philosophical definitions of totality given by Boethius and Avicenna, Henry concludes that God cannot be considered a totum universale, a totum numerale and a totum virtuale or potestativum, but concedes that He may be considered a totality insofar as His being comprehends the exemplary perfections of all created beings.

Context. The identification of the sources that reionized the Universe and their specific contribution to this process are key missing pieces of our knowledge of the early Universe. Faint star-forming galaxies may be the main contributors to the ionizing photon budget during the epoch of reionization, but their escaping photons cannot be detected directly due to inter-galactic medium opacity. Hence, it is essential to characterize the properties of faint galaxies with significant Lyman continuum photon leakage up to z 4 to (...) define indirect indicators allowing analogues to be found at the highest redshift. Aims. Long gamma-ray bursts explode typically in star-forming regions of faint, star-forming galaxies. Through LGRB afterglow spectroscopy it is possible to detect directly LyC photons. Our aim is to use LGRBs as tools to study LyC leakage from faint, star-forming galaxies at high redshift. Methods. Here we present the observations of LyC emission in the afterglow spectra of GRB 191004B at z = 3:5055, together with those of the other two previously known LyC-emitting LGRB, to determine their LyC escape fraction and compare their properties. Results. From the afterglow spectrum of GRB 191004B we determine a neutral hydrogen column density at the LGRB redshift of og =17:2 0:15, and negligible extinction. The only metal absorption lines detected are C iv and Si iv. In contrast to GRB 050908 and GRB 060607A, the host galaxy of GRB 191004B displays significant Ly emission. From its Ly emission and the non-detection of Balmer emission lines we constrain its star-formation rate to 1 SFR 4:7 M yr. 1. We fit the Ly emission with a shell model and find parameters values consistent with the observed ones. The absolute LyC escape fractions we find for GRB 191004B, GRB 050908 and GRB 060607A are of 0:35+0:10.0:11, 0:08+0:05.0:04 and :20+0:05.0:05, respectively. We compare the LyC escape fraction of LGRBs to the values of other LyC emitters found from the literature, showing that LGRB afterglows can be powerful tools to study LyC escape for faint high-redshift star-forming galaxies. Indeed we could push LyC leakage studies to much higher absolute magnitudes. The host galaxies of the three LGRB presented here have all M1600 >.19:5 mag, with the GRB 060607A host at M1600 >.16 mag. LGRB hosts may therefore be particularly suitable for exploring the ionizing escape fraction in galaxies that are too faint or distant for conventional techniques. Furthermore the time investment is very small compared to galaxy studies. (shrink)

Searching for better ways to inspire people to pursue wisdom, Frederick D. Aquino argues that teachers and researchers should focus less on state-of-the-art techniques and learning outcomes and instead pay more attention to the intellectual formation of their students. We should, Aquino contends, encourage the development of an integrative habit of mind, which entails cultivating the capacity to grasp how various pieces of data and areas of inquiry fit together and to understand how to apply this information to (...) new situations. To fully explore this notion, _An Integrative Habit of Mind_ brings the work of the great religious figure and educator John Henry Newman into fruitful conversation with recent philosophical developments in epistemology, cognition, and education. Aquino unearths some crucial but neglected themes from Newman’s writings and carries them forward into the contemporary context, revealing how his ideas can help us broaden our horizons, render apt judgments, and better understand our world and how we think about it. (shrink)

In this article, we argue that John Henry Newman was right to think that our passional nature can play a legitimate epistemic role. First, we unpack the standard objection to Newman’s understanding of the relationship between our passional nature and the evidential basis of faith. Second, we argue that the standard objection to Newman operates with a narrow definition of evidence. After challenging this notion, we then offer a broader and more humane understanding of evidence. Third, we survey recent scholarship (...) arguing that emotions, a key aspect of our passional nature, are cognitive. In this light, they plausibly have a proper epistemic role. Fourth, we defend Newman’s reliance on the passional nature in epistemic matters by showing how reasonable it is in light of this recent work on evidence and the nature of emotions. Newman’s insistence that the formation of a right state of heart and mind is crucial for epistemic success is far from untenable. (shrink)

It considers the epistemology of theology and features 42 chapters, divided into 4 sections on 'Theology Relative Epistemic Concepts' and 'General Epistemic Concepts as Related to Theology', and on studies of individual theologians from St Paul through to Hans Urs von Balthasar and of contemporary movements such as Liberation Theology and Feminism.

In recent years, Duncan Pritchard has developed a position in religious epistemology called quasi‐fideism that he claims traces back to John Henry Newman's treatment of the rationality of religious belief. In this paper, we give three reasons to think that Pritchard's reading of Newman as a quasi‐fideist is mistaken. First, Newman's parity argument does not claim that religious and non‐religious beliefs are on a par because both are groundless; instead, for Newman, they are on a par because both often stem (...) from implicit rather than explicit reasoning. Second, pace Pritchard, Newman's distinction between simple and complex assent does not map onto the Wittgensteinian distinction between groundless hinge commitments and beliefs that flow from these hinges. For Newman, simple and complex assent differ in terms of the believer's level of awareness of their grounds. Third, and finally, Newman does not reject Locke's evidentialism in toto. Instead, he argues that certitude is not restricted to beliefs stemming from intuition and demonstration but often rightly includes probabilistically supported (or fallibly evidenced) beliefs. (shrink)