This book excavates the rhetorical devices that Nietzsche habitually uses and explains how they constitute a distinctive form of philosophical argumentation. Through a sustained analysis of Nietzsche’s rhetorical style, stratagems, and didactic aims in two of his early works (‘On Truth and Lies in a Non-Moral Sense’ and Daybreak) and two of his later works (Beyond Good and Evil and Twilight of the Idols), the book assesses the extent to which Nietzsche's substantial rhetorical arsenal undermines the philosophical claims he is (...) seeking to advance. The four case studies also bring to the fore some of the less palatable aspects of Nietzsche’s thought such as racial-stereotyping, the essentialising of a so-called slave mentality, and the ranking of human beings based on a highly idiosyncratic and prejudiced view of what qualifies as noble and ignoble. (shrink)

In this essay I argue that Down by Law (Jarmusch, 1986) is about how the encounter with otherness renews freedom and American identity. I first develop the idea of renewal through otherness by way of a discussion of Levinas' philosophy of freedom and Arendt's notion natality, contrasting it with the idea of negative liberty, which I explicate through a discussion of Hobbes, Locke, Hegel, and Tocqueville. Next, I show how negative liberty is engrained in the idea of America through a (...) discussion of the formative American motifs of the road and the prison, which play structuring roles in Down by Law . I then try to show how the immigrant character of Bob (Roberto Benigni) is an other who liberates the American characters of Jack (John Lurie) and Zack (Tom Waits), who are existentially 'imprisoned' by their negative liberty. (shrink)

It is common in accounts of the secularization of Western thought to make reference to the name of Nietzsche. Nietzsche is undeniably a critic of religion, but he is equally a critic of the secular. It is for this reason that I propose thinking about Nietzsche’s philosophy as postsecular. This term is one that has evolved over the last couple decades in response to the so-called “return of the religious” in society, social theory, and philosophy and suggests that secularity and (...) religiosity are not antithetical and therefore that we must move beyond the forms of thinking that would regard them as such. First, I focus on his concepts of will to power, interpretation, perspectivism, and history, and sketch how they are all interrelated and how these ground his interpretation of religion. Based on this view, I show, second, that while religion is, for Nietzsche, a historical perspective, it is also, for him, part of the historical genealogy of secularity. This point leads to my final one on the relationship between secularity and religion in Nietzsche’s thought, where I argue that, on his own terms, Nietzsche’s anti-Christianity is a hyper-Christianity, even while rejecting facets of Christianity. (shrink)

In recent continental philosophy of religion there has been significant attention paid to the Abrahamic doctrines of creation ex nihilo and divine omnipotence, especially by deconstructive thinkers such as Derrida, Caputo, and Keller. For these thinkers, the doctrine represents a form of agency that does violence to various forms of alterity. While broadly supportive of their fundamental philosophical and ethico-political views, especially about the primordiality of alterity, I differ from them in that I argue that creation ex nihilo articulates the (...) very structure of the alterity they are concerned with. The essay proceeds through a reading of Derrida’s representation of the doctrine and a “deconstruction” of his view by means of a reading of Augustine and Anselm. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Otherwise than Laïcité? Toward an Agonistic Secularism in LevinasMark Cauchi (bio)Levinas and SecularismAlong with the so-called “return of the religious” in contemporary Western philosophy and politics, there has been a renewed effort in recent years to rethink secularism, the political doctrine of the separation of religion and politics.1 It would not be difficult to show that Emmanuel Levinas has been a substantial force in the resurgence of interest in (...) religion among philosophers, but his impact on thinking about secularism is less evident. Part of the reason for this uncertainty, and part of the reason it is not taken up by scholars, is doubtless the fact that Levinas’s conception of secularism is difficult to discern.2 Although he often discusses religion, politics, and even their relationship, he rarely addresses the question of secularism directly. He certainly uses the term laïcité, and, especially in Difficult Freedom, often discusses both the place of Judaism in secular society and what he calls the “Christian atmosphere” of modernity, but he never articulates, either here or elsewhere, a complete account of the concept.In addition to the paucity of Levinas’s writings on secularism, a further difficulty is that what Levinas does say about secularism can often seem contradictory and unclear. Consider, for example, his view [End Page 187] of the most basic issue of secularism: the separation of church and state. Is such a separation wholly compatible with Levinas’s thinking? On the one hand, Levinas is unambiguous about his endorsement of the separation. He writes in “How Is Judaism Possible?” (1959) that the separation “is the citadel that protects us [French Jews] against injustice,” and therefore that “we must beware of tampering with it.” And yet, as he will immediately add, the “luminous and incontestable formula” separating church and state must be given “its best content,” as if in its common understanding it does not possess its best meaning (DF 245). This specific double move, where he simultaneously endorses and criticizes the separation of church and state, is indicative of Levinas’s more general attitude from the 1950s onward toward the whole of the standard liberal conception of secularism.What I mean by the latter conception will be fleshed out in the following sections, but it can be defined now, in a preliminary fashion, as the entire liberal enterprise of creating a culturally neutral state and public sphere for the sake of protecting the rights and liberties of the individual citizen. Because the individual citizen is defined as possessing the right to, or liberty of, thought, expression, and association—all of which together, in effect, produce freedom of religion, whether explicitly proclaimed or not—the state and public sphere must be constituted in particular ways to guarantee these liberties; in particular, they must be culturally neutralized so that any citizen, from whichever cultural background, can participate in public institutions without bias and thus without the violation of their liberties. Such neutralization is achieved by the reciprocal move wherein participants in these institutions—that is to say, individual citizens—are not allowed to bring their cultural frameworks with them into the constitutions and operations of these institutions, leading to the so-called privatization of religion.Because Levinas criticizes this model of secularism while seeming to endorse some notion of it, what I would like to do in this essay is to determine what is precisely Levinas’s criticism of secularism and, in addition, what a “Levinasian secularism” might be. My central contention will be that the latter (Levinasian secularism) is not equivalent [End Page 188] to the former (his stated views about secularism). The reason for this discrepancy is that Levinas critiques standard liberal secularism on the basis of his more general philosophy of ethical alterity, but then reinstates liberal secularism (or an unspecified version of it) as the best way to protect institutionally ethical alterity. This double movement raises the question, how can ethical alterity at once be consistent and inconsistent with liberal secularism? The question is naturally related to the much-discussed one about the relation of ethics and politics in Levinas’s thinking. Levinas’s approach to the latter relation is consistently... (shrink)

La fondation d'une revue comme Dialogue s'accompagne de crainte et d'espoir. Crainte de ne pouvoir la faire vivre et progresser, crainte « inavouée » que la production philosophique ne soit pas à la hauteur de l'opinion que nous avons de nous-mêmes ou de ce que le monde extérieur attend de nous… Espoir cependant de consolider les structures d'une Association canadienne de philosophie à peine naissante, chancelante, mais vivant déjà d'un idéal exigeant et dont la revue porte le nom.

Using Kierkegaard’s birthday as my starting point, my essay contends that in order to celebrate Kierkegaard’s birth we have to bring him into our present age, which task involves understanding how his thought is related to modernity. I first explain how, from Kierkegaard’s point of view, any celebration risks being mere celebrity and nostalgia, and discuss the conception of temporality that Kierkegaard identifies as undergirding both concepts. To counteract the temporality of celebrity and nostalgia, I next argue that we must (...) begin with the fact of Kierkegaard’s death and absence from our present age and establish, through an “unconditioned relation to the unconditioned,” a relationship of contemporaneity with him. Regarding Kierkegaard as a contemporary requires understanding his relationship to modernity and the present age and thus requires a critique of his own critique of modernity. I conclude by acknowledging the problems and questions regarding the relationship between religion and secularism, for Kierkegaard and for ourselves, which result from situating Kierkegaard in modernity. (shrink)

Les premiers travaux réalisés sur les effets du confinement du printemps 2020 sur les enfants ont établi que ceux qui étaient en résidence alternée l’avaient mieux vécu que les autres. L’objectif de cet article est de comprendre pourquoi. À partir de l’enquête Sapris et plus particulièrement du sous-échantillon des enfants âgés de 8-9 ans des cohortes Elfe et Epipage2, nous comparons les conséquences du confinement sur les enfants selon la structure familiale dans laquelle ils vivaient. Continu dans les familles nucléaires, (...) le temps familial est habituellement beaucoup plus fractionné entre les résidences des parents pour les enfants de familles séparées. Le confinement est venu transformer les expériences habituelles du temps familial pour les enfants, aussi bien dans son organisation que dans sa texture. Il a modifié leur cadre de vie ordinaire, en réorganisant leurs occupations au quotidien et en modifiant leurs relations avec leurs parents. En observant particulièrement les différences apparues entre les résidences alternées qui ont continué et celles qui ont été interrompues, nous montrons que si les mouvements réguliers des enfants ont pu constituer une respiration bienvenue pour les différents acteurs concernés, les alternances interrompues ont ouvert des espaces exceptionnels dans lesquels il a été possible de renouer avec des formes passées de continuité familiale. (shrink)

Gemins 2–8 and Unr‐interacting protein (UNRIP) are intimate partners of the survival motor neuron (SMN) protein, which is the determining factor for the neuromuscular disorder spinal muscular atrophy (SMA). The most documented role of SMN, Gemins and UNRIP occurs within the large macromolecular SMN complex and involves the cytoplasmic assembly of spliceosomal uridine‐rich small nuclear ribonucleoproteins (UsnRNPs), a housekeeping process critical in all cells. Several reports detailing alternative functions for SMN in either motor neurons or skeletal muscles may, however, hold (...) the answer to the extreme neuromuscular tissue specificity observed in SMA. Recent discoveries indicate that collaboration between SMN and Gemins also extends to these non‐canonical functions, hence raising the possibility that mutations in Gemin genes may be the cause of unlinked neuromuscular hereditary syndromes. This review evaluates the functions of Gemins and UNRIP inside the SMN complex and discusses whether these less notorious SMN complex members are capable of acting independently of SMN. (shrink)

Much philosophy of the last few decades has witnessed a turn toward otherness and a corresponding calling into question of the autonomy of the agent. In my paper I attempt to re-conceive what agency is in light of this emphasis placed on otherness. I undertake this reconsideration through an analysis of the concepts of unconditionality in Kant and of conditioning by the other in Levinas. Through these analyses I arrive at a new concept: the unconditioning of the agent by the (...) other. I then provide some description of this concept by considering the interpretation of the theological concept of creation in Augustine, Kant, and Levinas. (shrink)

This book contains the fruits of decades of reflection and teaching on basic philosophical problems, historical, epistemological, metaphysical, and ethical. It does not purport to be a systematic, tightly demonstrative treatise, but a meditation in which opinion and critical irony sometimes have the upper hand. Though the author strongly leans towards Cartesian, Kantian, Hegelian or phenomenological viewpoints, it seems when the cards are down, that the fundamental inspiration of his philosophical thinking remains on the whole Aristotelian and Thomistic.

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type τ is naturally measured by the smallest integer q such that a sentence of quantifier rank q - 1 is satisfied by A but not by B. In this way the space $\operatorname{Str}^\tau$ of structures of type τ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$ . Using the rudiments of the theory of uniform spaces, in this elementary note (...) we prove the convergence of every Cauchy net of structures, for any type τ. (shrink)

The inverse of the distance between two structures $\mathscr{A} \not\equiv \mathscr{B}$ of finite type $\tau$ is naturally measured by the smallest integer $q$ such that a sentence of quantifier rank $q - 1$ is satisfied by $\mathscr{A}$ but not by $\mathscr{B}$. In this way the space $\operatorname{Str}^\tau$ of structures of type $\tau$ is equipped with a pseudometric. The induced topology coincides with the elementary topology of $\operatorname{Str}^\tau$. Using the rudiments of the theory of uniform spaces, in this elementary note we (...) prove the convergence of every Cauchy net of structures, for any type $\tau$. (shrink)

We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy—Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

This manuscript investigates fixed point of single-valued Hardy-Roger’s type F -contraction globally as well as locally in a convex b -metric space. The paper, using generalized Mann’s iteration, iterates fixed point of the abovementioned contraction; however, the third axiom of the F -contraction is removed, and thus the mapping F is relaxed. An important approach used in the article is, though a subset closed ball of a complete convex b -metric space is not necessarily complete, the convergence of the Cauchy (...) sequence is confirmed in the subset closed ball. The results further lead us to some important corollaries, and examples are produced in support of our main theorems. The paper most importantly presents application of our results in finding solution to the integral equations. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s (...) proof makes, Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)

Notoriously, the Einstein equations of general relativity have solutions in which closed timelike curves occur. On these curves time loops back onto itself, which has exotic consequences: for example, traveling back into one's own past becomes possible. However, in order to make time travel stories consistent constraints have to be satisfied, which prevents seemingly ordinary and plausible processes from occurring. This, and several other "unphysical" features, have motivated many authors to exclude solutions with CTCs from consideration, e.g. by conjecturing a (...) chronology protection law. In this contribution we shall investigate the nature of one particular class of exotic consequences of CTCs, namely those involving unexpected cases of indeterminism or determinism. Indeterminism arises even against the backdrop of the usual deterministic physical theories when CTCs do not cross spacelike hypersurfaces outside of a limited CTC-region|such hypersurfaces fail to be Cauchy surfaces. We shall compare this CTC-indeterminism with four other types of indeterminism that have been discussed in the philosophy of physics literature: quantum indeterminism, the indeterminism of the hole argument, non-uniqueness of solutions of differential equations and lack of predictability due to insuffcient data. By contrast, a certain kind of determinism appears to arise when an indeterministic theory is applied on a CTC: things cannot be different from what they already were. Again we shall make comparisons, this time with other cases of determination in physics. We shall argue that on further consideration both this indeterminism and determinism on CTCs turn out to possess analogues in other, familiar areas of physics. CTC- indeterminism is close to the epistemological indeterminism we know from statistical physics, while the "fixedness" typical of CTC-determinism is pervasive in physics. CTC- determinism and CTC-indeterminism therefore do not provide incontrovertible grounds for rejecting CTCs as conceptually inadmissible. (shrink)

The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.

In this paper, we introduce the notion of S ∗ p -partial metric spaces which is a generalization of S-metric spaces and partial-metric spaces. Also, we give some of the topological properties that are important in knowing the convergence of the sequences and Cauchy sequence. Finally, we study a new common fixed point theorems in this spaces.

In 1996, W. Veldman and F. Waaldijk present a constructive (intuitionistic) proof for the homogeneity of the ordered structure of the Cauchy real numbers, and so this result holds in any topos with natural number object. However, it is well known that the real numbers objects obtained by the traditional constructions of Cauchy sequences and Dedekind cuts are not necessarily isomorphic in an arbitrary topos with natural numbers object. Consequently, Veldman and Waaldijk's result does not apply to the ordered structure (...) of Dedekind real numbers in toposes. The main result to be proved in the present paper is that the ordered structure of the Dedekind real numbers object is homogeneous, in any topos with natural numbers object. This result is obtained within the framework of local set theory. (shrink)

We give a new account of Björling’s contribution to uniform convergence in connection with Cauchy’s theorem on the continuity of an infinite series. Moreover, we give a complete translation from Swedish into English of Björling’s 1846 proof of the theorem. Our intention is also to discuss Björling’s convergence conditions in view of Grattan-Guinness’ distinction between history and heritage. In connection to Björling’s convergence theory we discuss the interpretation of Cauchy’s infinitesimals.

In this paper, we first present a generalization of the Cauchy-Schwarz inequality. As an application of our result, we obtain a new sufficient condition for the stability of a class of nonlinear impulsive control systems. We end up this note with a numerical example which shows the effectiveness of our method.

We investigate how nonstandard reals can be established constructively as arbitrary infinite sequences of rationals, following the classical approach due to Schmieden and Laugwitz. In particular, a total standard part map into Richman's generalised Dedekind reals is constructed without countable choice.

This paper aims to propose an improved learning algorithm for feature selection, termed as binary superior tracking artificial bee colony with dynamic Cauchy mutation. To enhance exploitation capacity, a binary learning strategy is proposed to enable each bee to learn from the superior individuals in each dimension. A dynamic Cauchy mutation is introduced to diversify the population distribution. Ten datasets from UCI repository are adopted as test problems, and the average results of cross-validation of BSTABC-DCM are compared with other seven (...) popular swarm intelligence metaheuristics. Experimental results demonstrate that BSTABC-DCM could obtain the optimal classification accuracy and select the best representative features for the UCI problems. (shrink)

One of the most influential scientific treatises in Cauchy's era was J.-L. Lagrange's Mécanique Analytique, the second edition of which came out in 1811, when Cauchy was barely out of his teens. Lagrange opens his treatise with an unequivocal endorsement of infinitesimals. Referring to the system of infinitesimal calculus, Lagrange writes:Lorsqu'on a bien conçu l'esprit de ce système, et qu'on s'est convaincu de l'exactitude de ses résultats par la méthode géométrique des premières et dernières raisons, ou par la méthode analytique (...) des fonctions dérivées, on peut employer les infiniment petits comme un instrument sûr et commode pour abréger et simplifier les demonstrations.Lagrange's renewed enthusiasm for .. (shrink)

Cauchy’s contribution to the foundations of analysis is often viewed through the lens of developments that occurred some decades later, namely the formalisation of analysis on the basis of the epsilon-delta doctrine in the context of an Archimedean continuum. What does one see if one refrains from viewing Cauchy as if he had read Weierstrass already? One sees, with Felix Klein, a parallel thread for the development of analysis, in the context of an infinitesimal-enriched continuum. One sees, with Emile Borel, (...) the seeds of the theory of rates of growth of functions as developed by Paul du Bois-Reymond. One sees, with E. G. Björling, an infinitesimal definition of the criterion of uniform convergence. Cauchy’s foundational stance is hereby reconsidered. (shrink)

It is consistent with constructive set theory (without Countable Choice, clearly) that the Cauchy reals (equivalence classes of Cauchy sequences of rationals) are not Cauchy complete. Related results are also shown, such as that a Cauchy sequence of rationals may not have a modulus of convergence, and that a Cauchy sequence of Cauchy sequences may not converge to a Cauchy sequence, among others.

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

We study computability of the abstract linear Cauchy problem equation image)where A is a linear operator, possibly unbounded, on a Banach space X. We give necessary and sufficient conditions for A such that the solution operator K: x ↦ u of the problem is computable. For studying computability we use the representation approach to computable analysis developed by Weihrauch and others. This approach is consistent with the model used by Pour-El/Richards.

George Soros makes an important analytical contribution to understanding the concept of reflexivity in social science by explaining reflexivity in terms of how his cognitive and manipulative causal functions are connected to one another by a pair of feedback loops (Soros, 2013). Fallibility, reflexivity and the human uncertainty principle. Here I put aside the issue of how the natural sciences and social sciences are related, an issue he discusses, and focus on how his thinking applies in economics. I argue that (...) standard economics assumes a ‘classical’ view of the world in which knowledge and action are independent, but that we live in a complex reflexive world in which knowledge and action are interdependent. I argue that Soros's view provides a reflexivity critique of the efficient market hypothesis seen as depending on untenable claims about the nature of random phenomena and the nature of economic agents. Regarding the former, I develop this critique in terms of Cauchy distributions; regarding the latter I develop it in terms of rational expectations and rational addiction reasoning. (shrink)

This is a study of the set of the Malitz-completions of a given infinite first-order structure, put in relation with properties of directed sets.

In his note [5] Hausner states a simple combinatorial principle, namely: $$(H)\left\{ {\begin{array}{*{20}c} {if f is a function a non - empty finite set \sigma into itself, p a} \\ {prime, f^p = i_\sigma and \sigma _0 the set of fixed points of f, then } \\ {\left| \sigma \right| \equiv \left| {\sigma _0 } \right|(mod p).} \\\end{array}} \right.$$ .He then shows how this principle can be used to prove:Fermat's little theorem,Cauchy's theorem for finite groups,Lucas' theorem for binomial numbers.Letε=(0,1, ...),ℱ (...) 1−1 the family of all one-to-one functions from a subset ofε intoε andℳ 1−1 the family of all p.r. (i.e., partial recursive) one-to-one functions from a subset ofε intoε. A subsetα of ε is finite, ifα is not equivalent to a proper subset under a function inℱ 1−1. The setα is calledisolated, if it is not equivalent to a proper subset under a function inℳ 1−1. An isolated set can also be defined as a subset ofε which has no infinite r.e. (i.e., recursively enumerable) subset. While every finite set is isolated, there are $c = 2^{\aleph _0 }$ infinite sets which are isolated; these sets are calledimmune. It is the purpose of this paper to generalize (H) to a principle (H*) for isolated sets and to show how (H*) can be used to prove generalizations of Fermat's little theorem and Cauchy's theorem for finite groups. We have been unable to generalize Lucas' theorem in a similar manner. (shrink)

We investigate the provability or nonprovability of certain ordinary mathematical theorems within certain weak subsystems of second order arithmetic. Specifically, we consider the Cauchy/Peano existence theorem for solutions of ordinary differential equations, in the context of the formal system RCA 0 whose principal axioms are ▵ 0 1 comprehension and Σ 0 1 induction. Our main result is that, over RCA 0 , the Cauchy/Peano Theorem is provably equivalent to weak Konig's lemma, i.e. the statement that every infinite {0, 1}-tree (...) has a path. We also show that, over RCA 0 , the Ascoli lemma is provably equivalent to arithmetical comprehension, as is Osgood's theorem on the existence of maximum solutions. At the end of the paper we digress to relate our results to degrees of unsolvability and to computable analysis. (shrink)