Set theory, it has been contended, developed from its beginnings through a progression ofmathematicalmoves, despite being intertwined with pronounced metaphysical attitudes and exaggerated foundational claims that have been held on its behalf. In this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership (...) distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his well-known paradox, an early expression of our motif. The motif becomes fully manifest through the study of functionsof the power set of a set into the set in the fundamental work of Zermelo on set theory. His first proof in 1904 of his Well-Ordering Theoremis a central articulation containing much of what would become familiar in the subsequent development of set theory. Afterwards, the motif is cast by Kuratowski as a fixed point theorem, one subsequently abstracted to partial orders by Bourbaki in connection with Zorn's Lemma. Migrating beyond set theory, that generalization becomes cited as the strongest of fixed point theorems useful in computer science.Section 1 describes the emergence of our guiding motif as a line of development from Cantor's diagonal proof to Russell's Paradox, fueled by the clarification of the inclusion vs. membership distinction. Section 2 engages the motif as fully participating in Zermelo's work on the Well-Ordering Theorem and as newly informing on Cantor's basic result that there is no bijection. Then Section 3 describes in connection with Zorn's Lemma the transformation of the motif into an abstract fixed point theorem, one accorded significance in computer science. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...) are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (shrink)

Ernst Friedrich Ferdinand Zermelo transformed the set theory of Cantor and Dedekind in the first decade of the 20th century by incorporating the Axiom of Choice and providing a simple and workable axiomatization setting out generative set-existence principles. Zermelo thereby tempered the ontological thrust of early set theory, initiated the delineation of what is to be regarded as set-theoretic, drawing out the combinatorial aspects from the logical, and established the basic conceptual framework for the development of modern set theory. Two (...) decades later Zermelo promoted a distinctive cumulative hierarchy view of models of set theory and championed the use of infinitary logic, anticipating broad modern developments. In this paper Zermelo's published mathematical work in set theory is described and analyzed in its historical context, with the hindsight afforded by the awareness of what has endured in the subsequent development of set theory. Elaborating formulations and results are provided, and special emphasis is placed on the to and fro surrounding the Schröder-Bernstein Theorem and the correspondence and comparative approaches of Zermelo and Gödel. Much can be and has been written about philosophical and biographical issues and about the reception of the Axiom of Choice, and we will refer and defer to others, staying the course through the decidedly mathematical themes and details. (shrink)

Azriel Levy did fundamental work in set theory when it was transmuting into a modern, sophisticated field of mathematics, a formative period of over a decade straddling Cohen’s 1963 founding of forcing. The terms “Levy collapse”, “Levy hierarchy”, and “Levy absoluteness” will live on in set theory, and his technique of relative constructibility and connections established between forcing and definability will continue to be basic to the subject. What follows is a detailed account and analysis of Levy’s work and contributions (...) to set theory. (shrink)

Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions cast in set-theoretic terms and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, (...) set theory has proceeded in the opposite direction, from a web of intensions to a theory of extensionpar excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical structures and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression ofmathematicalmoves, whatever and sometimes in spite of what has been claimed on its behalf.What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by current set theory. The whole transfinite landscape can be viewed as the result of Cantor's attempt to articulate and solve the Continuum Problem. (shrink)

Kurt Gödel with his work on the constructible universeLestablished the relative consistency of the Axiom of Choice and the Continuum Hypothesis. More broadly, he ensured the ascendancy of first-order logic as the framework and a matter of method for set theory and secured the cumulative hierarchy view of the universe of sets. Gödel thereby transformed set theory and launched it with structured subject matter and specific methods of proof. In later years Gödel worked on a variety of set theoretic constructions (...) and speculated about how problems might be settled with new axioms. We here chronicle this development from the point of view of the evolution of set theory as a field of mathematics. Much has been written, of course, about Gödel's work in set theory, from textbook expositions to the introductory notes to his collected papers. The present account presents an integrated view of the historical and mathematical development as supported by his recently published lectures and correspondence. Beyond the surface of things we delve deeper into the mathematics. What emerges are the roots and anticipations in work of Russell and Hilbert, and most prominently the sustained motif of truth as formalizable in the “next higher system”. We especially work at bringing out how transforming Gödel's work was for set theory. It is difficult now to see what conceptual and technical distance Gödel had to cover and how dramatic his re-orientation of set theory was. (shrink)

We consider natural strengthenings of H. Friedman's Borel diagonalization propositions and characterize their consistency strengths in terms of the n -subtle cardinals. After providing a systematic survey of regressive partition relations and their use in recent independence results, we characterize n -subtlety in terms of such relations requiring only a finite homogeneous set, and then apply this characterization to extend previous arguments to handle the new Borel diagonalization propositions.

In this study, we discuss the psychological acceptability of an utterance strategy used by the Driving Support Agent. Previous literature regarding DSA suggests that the adoption of a small robot as a form will increase acceptability. However, the agent’s utterance has been reported as a problem faced by the user. Therefore, in this study, we designed the agent’s utterance using politeness strategy as described by Brown and Levinson’s famous sociolinguistics and pragmatics theory and analyzed its acceptability through a participant-based experiment. (...) In this experiment, we used positive and negative politeness strategies. In general, PPS is utilized to reflect the desire to be liked/recognized by others, whereas NPS is utilized to reflect the need for not wanting to be disturbed by others. Based on our results, PPS was rated high compared to NPS. Therefore, many participants highly evaluated PPS. However, there was a group of participants who appreciated NPS. There were also participants who evaluated the two strategies equally. The number of participants in these three groups was observed at 4:1:1. This result contributes as an index on the utterance design of the DSA. (shrink)

Greek, Indian and Arabic Logic marks the initial appearance of the multi-volume Handbook of the History of Logic. Additional volumes will be published when ready, rather than in strict chronological order. Soon to appear are The Rise of Modern Logic: From Leibniz to Frege. Also in preparation are Logic From Russell to Gödel, The Emergence of Classical Logic, Logic and the Modalities in the Twentieth Century, and The Many-Valued and Non-Monotonic Turn in Logic. Further volumes will follow, including Mediaeval and (...) Renaissance Logic and Logic: A History of its Central. In designing the Handbook of the History of Logic, the Editors have taken the view that the history of logic holds more than an antiquarian interest, and that a knowledge of logic's rich and sophisticated development is, in various respects, relevant to the research programmes of the present day. Ancient logic is no exception. The present volume attests to the distant origins of some of modern logic's most important features, such as can be found in the claim by the authors of the chapter on Aristotle's early logic that, from its infancy, the theory of the syllogism is an example of an intuitionistic, non-monotonic, relevantly paraconsistent logic. Similarly, in addition to its comparative earliness, what is striking about the best of the Megarian and Stoic traditions is their sophistication and originality. Logic is an indispensably important pivot of the Western intellectual tradition. But, as the chapters on Indian and Arabic logic make clear, logic's parentage extends more widely than any direct line from the Greek city states. It is hardly surprising, therefore, that for centuries logic has been an unfetteredly international enterprise, whose research programmes reach to every corner of the learned world. Like its companion volumes, Greek, Indian and Arabic Logic is the result of a design that gives to its distinguished authors as much space as would be needed to produce highly authoritative chapters, rich in detail and interpretative reach. The aim of the Editors is to have placed before the relevant intellectual communities a research tool of indispensable value. Together with the other volumes, Greek, Indian and Arabic Logic, will be essential reading for everyone with a curiosity about logic's long development, especially researchers, graduate and senior undergraduate students in logic in all its forms, argumentation theory, AI and computer science, cognitive psychology and neuroscience, linguistics, forensics, philosophy and the history of philosophy, and the history of ideas. (shrink)

A long-awaited edition of Zermelo’s works Content Type Journal Article Pages 1-4 DOI 10.1007/s11016-011-9548-y Authors José Ferreirós, Instituto de Filosofia, CCHS-CSIC, Albasanz, 26-28, 28037 Madrid, Spain Journal Metascience Online ISSN 1467-9981 Print ISSN 0815-0796.

When the Canonical Ramsey’s Theorem by Erdős and Rado is applied to regressive functions, one obtains the Regressive Ramsey’s Theorem by Kanamori and McAloon. Taylor proved a “canonical” version of Hindman’s Theorem, analogous to the Canonical Ramsey’s Theorem. We introduce the restriction of Taylor’s Canonical Hindman’s Theorem to a subclass of the regressive functions, the $$\lambda $$ λ -regressive functions, relative to an adequate version of min-homogeneity and prove some results about the Reverse Mathematics of this Regressive Hindman’s Theorem (...) and of natural restrictions of it. In particular we prove that the first non-trivial restriction of the principle is equivalent to Arithmetical Comprehension. We furthermore prove that the well-ordering-preservation principle for base-$$\omega $$ ω exponentiation is reducible to this same principle by a uniform computable reduction. (shrink)

The contributions to this volume are from participants of the international conference "Kreisel's Interests - On the Foundations of Logic and Mathematics", which took place from 13 to 14 2018 at the University of Salzburg in Salzburg, Austria. The contributions have been revised and partially extended. Among the contributors are Akihiro Kanamori, Göran Sundholm, Ulrich Kohlenbach, Charles Parsons, Daniel Isaacson, and Kenneth Derus. The contributions cover the discussions between Kreisel and Wittgenstein on philosophy of mathematics, Kreisel's Dictum, proof theory, (...) the discussions between Kreisel and Gödel on philosophy of mathematics, some bibliographical facts, and a collection of extracts from Kreisel's letters. (shrink)

We classify a sharp phase transition threshold for Friedman’s finite adjacent Ramsey theorem. We extend the method for showing this result to two previous classifications involving Ramsey theorem variants: the Paris–Harrington theorem and the Kanamori–McAloon theorem. We also provide tools to remove ad hoc arguments from the proofs of phase transition results as much as currently possible.

Generalizing Woodin’s extender algebra, cf. e.g. Steel (in: Kanamori (ed) Handbook of set theory, Springer, Berlin, 2010), we isolate the long extender algebra as a general version of Bukowský’s forcing, cf. Bukovský (Fundam Math 83:35–46, 1973), in the presence of a supercompact cardinal.

When the question is posed as to when Chinese thought influenced Western philosophy, people often turn to the philosophy of the German rationalist Christian Wolff, whose 1721 speech on the virtues of Confucianism led to his academic indictment and eventual ousting from the University of Halle in 1723. In his speech, Wolff lauds the Chinese for attaining virtues by natural revelation rather than appealing to Christian revelation, which made their accomplishments all the more impressive in his eyes (Fuchs 2006). According (...) to Kanamori (1997), the audience of Wolff’s speech “criticized it as the product of atheistic thoughts because they thought that the speaker valued Chinese morality as highly as Christian morals, which were formulated by divine revelation” (p. 299). The worry, in other words, was that Wolff was corrupted by Chinese thought since the atheism of Confucianism may have influenced Wolff’s own views. Almost invariably, Gottfried Wilhelm von Leibniz’s name is involved in conversations that question the Chinese influence on Western philosophy, but it is typically tempered by the standard scholarly interpretation of the Leibniz-China connection, which is that Leibniz respected Chinese thought but was not influenced by it. I will argue against this standard scholarly interpretation by showing how Leibniz could have been influenced by Chinese thought. In particular, I propose that Leibniz’s conception of monads, which is something that we only see after his engagement with Chinese thought, may have been influenced by the Huayan Buddhist concept of li, which creeps into the Neo-Confucianism with which Leibniz had exposure. Should my argument be persuasive, this means that Chinese thought influenced the Western world sooner than has been previously believed. (shrink)

Analyzes the origin, structure and resolution of Kant's antinomies of reason from a systematic rather than a historical perspective, exploring the relationship between the theoretical antinomies and the practical antinomy in order to indicate their similarities and differences and to suggest the dependence of the latter on the former.

Professor Solmsen's interpretation is orthodox; his comprehensive account builds on recent more specialized studies, including his own, and those of Jaeger, Ross, and Cherniss. If in some ways the book contains no large surprises, it nevertheless makes a major contribution by its treatment of Plato. The author has skillfully disengaged Plato's observations about nature from the customary ethical, epistemic, or, as the case may be, metaphysical contexts. He demonstrates that Plato was toward the end of his career a more serious (...) investigator of physical phenomena than is commonly supposed. Moreover, at almost every turn he finds Platonic precedents for Aristotle's thinking in this field. He does not detract from Aristotle's originality, but illumines it, and Plato's as well. (shrink)

A brief presentation of the major conclusions contained in the recently published monograph entitled "The Ascent to Law: Searches and Solutions," written over a period of many years. Chapter headings are: The law - an objective reality. The dogma of the law. The drama of scholarship. Searching. "The entire" substance of the law. Juridical constructs. The logic of laws. The secret of the law. Law - the highest purpose. Law in the life and fate of mankind.

The purpose of the present study was to address the shortcomings of Cronbach’s alpha concerning the semantic overlap between items. Using an example from a motivational measure, the correction of Cronbach’s alpha was applied by partialing out the effects due to conceptual overlap. The significance of Cronbach’s alpha was tested using simulated random data derived from the measure and by estimating the confidence intervals with known and unknown distributions. The results indicated that the uncorrected conceptual overlap coefficient alpha was equal (...) to 0.89 and 0.66 following the correction. After simulating the corrected statistical results, the distribution of alpha with random numbers had an estimate of 95%, equal to 0.41. The lower bound of the corrected alpha distribution was equal to 0.41, suggesting that the corrected alpha could easily belong to the distribution of alpha developed from simulated random numbers. Thus, the semantic overlap between items on a measure represents a significant threat to the validity of the alpha coefficient. (shrink)

In this article I try to articulate a defensible argumentation against the idea of an ontology of infinity. My position is phenomenologically motivated and in this virtue strongly influenced by the Husserlian reduction of the ontological being to a process of subjective constitution within the immanence of consciousness. However taking into account the historical charge and the depth of the question of infinity over the centuries I also include a brief review of the platonic and aristotelian views and also those (...) of Locke and Hume on the concept to the extent that they are relevant to my own discussion of infinity both in a purely philosophical and epistemological context. Concerning the latter context, I argue against Kanamori’s position, in The Infinite as Method in Set Theory and Mathematics, that the mathematical infinite can be accounted for solely in terms of epistemological articulation, that is, in the way it is approached, assimilated, and applied in the course of the construction of mathematical hierarchies. Instead I point to a subjectively constituted immanent ‘infinity’ in virtue of the a priori as well as factual characteristics of subjective constitution, underlying and conditioning any talk of infinity in an epistemological sense. From this viewpoint I also address some other positions on the question of a possible ontology of the mathematical infinite. My whole approach to the question of the infinite in an epistemological sense hinges on the assumption that the mathematical infinite subsumes the infinite of physical theories to the extent that physics and science in general deal with the infinite in terms of the corresponding mathematical language and the specific techniques involved. (shrink)

Machine generated contents note: 1. Introduction Juliette Kennedy and Roman Kossak; 2. Historical remarks on Suslin's problem Akihiro Kanamori; 3. The continuum hypothesis, the generic-multiverse of sets, and the [OMEGA] conjecture W. Hugh Woodin; 4. [omega]-Models of finite set theory Ali Enayat, James H. Schmerl and Albert Visser; 5. Tennenbaum's theorem for models of arithmetic Richard Kaye; 6. Hierarchies of subsystems of weak arithmetic Shahram Mohsenipour; 7. Diophantine correct open induction Sidney Raffer; 8. Tennenbaum's theorem and recursive reducts James (...) H. Schmerl; 9. History of constructivism in the 20th century A. S. Troelstra; 10. A very short history of ultrafinitism Rose M. Cherubin and Mirco A. Mannucci; 11. Sue Toledo's notes of her conversations with Gödel in 1972-1975 Sue Toledo; 12. Stanley Tennenbaum's Socrates Curtis Franks; 13. Tennenbaum's proof of the irrationality of [the square root of] 2́. (shrink)

Part of the accepted scholarly lore about Kierkegaard is that he holds that “existence”—human existence—and “the System” are mutually incompatible. For Kierkegaard, human being cannot be understood in terms of a nice, neat, complete systematic package; he shows, on this view, that the Hegelian attempt to grasp all of reality in terms of a philosophical system will always fail to grasp the reality of at least one thing: the concrete, living, existing individual human being.

This is the first scholarly study of Atlas Shrugged, covering in detail the historical, literary, and philosophical aspects of Ayn Rand's magnum opus. Topics explored in depth include the history behind the novel's creation, publication, and reception; its nature as a romantic novel; and its presentation of a radical new philosophy.

The biblical image of Wisdom as the power who “orders all things well” in nature and in human life can be read in the light of contemporary information theory. Some current scientific speculation offers an interpretation of reality as a vast information‐processing system, in which informational situations are continuously transformed through algorithmic operations. This interpretation finds a metaphysical counterpart in the distinction between “nature natured” and “nature naturing” in the philosophical theology of Samuel Taylor Coleridge. This confluence of religious, metaphysical, (...) and scientific imagery suggests a picture of the world in which the processes of “nature naturing,”“human humaning,” and “God Godding” inform and recur in each other. (shrink)

Popper proposes a technical concept of 'verisimilitude' as a test of the progressiveness of scientific theories. The paper attempts to show its uselessness and inapplicability on mathematical and practical grounds, As well as raising doubts about the value of any such attempt to give a mechanical test of scientific progress.

Augustine’s thoughts on human salvation not only influenced early Protestant theology but also dominated the conceptualisation of the missio Dei from the perspective of the 1952 Willingen Conference. His doctrine of synergism arguably only manifested much later in the conception of the missio Dei, which anticipated human obedience or active participation in the mission to attain the goal of human salvation. The idea of synergism in this regard, or in the context of the missio Dei, is that while salvation remains (...) an unmerited gift of grace through faith in Jesus Christ, God prescribes human cooperative involvement in its attainment. In this way people do not become unresponsive recipients of God’s grace, but active and willing participants in a mission of salvation. With this in mind, this article examined the available literature on the topic of synergism in order to provide further insights into the conceptualisation of the missio Dei. The account of Noah in Genesis 5:32–10:1 and the Great Commission in Matthew 28:16–20 were used to redefine the doctrine of synergism or its relevance in the context of the missio Dei. Contribution: This article recalled the doctrine of synergism in order to enrich the complex understanding of the missio Dei, essentially adding new knowledge in the field of missiology. (shrink)

A t first glance you might not noorder, which afflicts about 0.5 percent of tice anything odd on meeting a American children. Neither researcher young boy with autism. But if had any knowledge of the other’s work, you try to talk to him, it will and yet by an uncanny coincidence each quickly become obvious that gave the syndrome the same name: autism, something is seriously wrong. He may not which derives from the Greek word autos, make eye contact with (...) you; instead he may meaning “self.” The name is apt, because avoid your gaze and fidget, rock his body the most conspicuous feature of the disorto and fro, or bang his head against the der is a withdrawal from social interacwall. More disconcerting, he may not be tion. More recently, doctors have adopted able to conduct anything remotely resemthe term “autism spectrum disorder” to bling a normal conversation. Even though make it clear that the illness has many rehe can experience emotions such as fear, lated variants that range widely in severity rage and pleasure, he may lack genuine but share some characteristic symptoms. empathy for other people and be oblivious Ever since autism was identified, reto subtle social cues that most children searchers have struggled to determine would pick up effortlessly. what causes it. Scientists know that sus- In the 1940s two physicians—Americeptibility to autism is inherited, although.. (shrink)