We apply Mints’ technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.

We propose a new method for constructing Turing ideals satisfying principles of reverse mathematics below the Chain–Antichain (CAC) Principle. Using this method, we are able to prove several new separations in the presence of Weak König’s Lemma (WKL), including showing that CAC+WKL does not imply the thin set theorem for pairs, and that the principle “the product of well-quasi-orders is a well-quasi-order” is strictly between CAC and the Ascending/Descending Sequences principle, even in the presence of WKL.

Hindman's Theorem is a prototypical example of a combinatorial theorem with a proof that uses the topology of the ultrafilters. We show how the methods of this proof, including topological arguments about ultrafilters, can be translated into second order arithmetic.

We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA0, ATR0, and [Formula: see text].

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from RT, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below RT has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that ADS is not equivalent to CAC and that EM is not equivalent to RT.

The technique of using infinitary rules in an ordinal analysis has been one of the most productive developments in ordinal analysis. Unfortunately, one of the most advanced variants, the Buchholz Ωμ rule, does not apply to systems much stronger than -comprehension. In this paper, we propose a new extension of the Ω rule using game-theoretic quantifiers. We apply this to a system of inductive definitions with at least the strength of a recursively inaccessible ordinal.

In reverse mathematics, it is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on how to weaken the assumption while preserving the conclusion (other than reducing all the way to the tautology of assuming the conclusion). A main cause of this phenomenon is the proof of a $\Pi^1_2$ sentence from the theory $\mathbf{\Pi^{\textbf{1}}_{\textbf{1}}-CA_{\textbf{0}}}$. Using methods based on the functional interpretation, we introduce a family of weakenings of $\mathbf{\Pi^{\textbf{1}}_{\textbf{1}}-CA_{\textbf{0}}}$ (...) and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs. (shrink)

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several examples of colorings of the integers which do not have computable witnesses to Hindman's Theorem.

We present a realizability interpretation for classical analysis–an association of a term to every proof so that the terms assigned to existential formulas represent witnesses to the truth of that formula. For classical proofs of Π2 sentences ∀x∃yA(x,y), this provides a recursive type 1 function which computes the function given by f(x)=y iff y is the least number such that A(x,y).

We characterize the completely determined Borel subsets of HYP as exactly the [Formula: see text] subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this (...) answers a question of Astor, Dzhafarov, Montalbán, Solomon and the third author. (shrink)

The \-substitution method is a technique for giving consistency proofs for theories of arithmetic. We use this technique to give a proof of the consistency of the impredicative theory \ using a variant of the cut-elimination formalism introduced by Mints.

We show that any countable model of a model complete theory has an elementary extension with a “pseudofinite-like” quasi-dimension that detects dividing.

Extending Gödel's Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees.

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemer\'edi's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength (...) of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the $\omega^{\omega^\omega}$th level. (shrink)

We describe an Aldous–Hoover-type characterization of random relational structures that are exchangeable relative to a fixed structure which may have various equivalence relations. Our main theorem gives the common generalization of the results on relative exchangeability due to Ackerman \)-invariant measures: part I, 2015. arXiv:1509.06170) and Crane and Towsner and hierarchical exchangeability results due to Austin and Panchenko :809–823, 2014).

So much for the Aesthetic. We can now proceed to the Analytic, the philosophical importance of which is much greater. Kant's main contentions in this part of his work can be summed up in; two propositions: human understanding contains certain a priori concepts, and on these are based certain non-empirical principles; these concepts are only general concepts of a phenomenal object, and therefore the principles in question are only prescriptive to sense-experience. As has already been said, interest in the first (...) proposition has distracted attention from the fact that the important thing Kant has to say is contained in the second. (shrink)

Henri Bergson (1859-1941) is one of the truly great philosophers of the Modernist period, and there is currently a major renaissance of interest in his unduly neglected texts and ideas amongst philosophers, literary theorists, and social theorists. Mind-Energy is a collection of essays and lectures from the period 1901-13 and has long been out of print. It features essays on life and consciousness, soul and body, mind and brain, and on dreams, memory and the phenomenon of false recognition; the insights (...) Bergson develops in them remain highly pertinent to contemporary work in the philosophy of mind. (shrink)

Cet ouvrage n'est ni un commentaire ni une explication d'obedience historiciste ou deterministe. Par questionnement, lecture et interpretation, l'auteur a voulu produire une semantique philosophique. La doctrine des idees est ainsi reexaminee a la convergence des problemes du langage, de la science et de la cite (logos, episteme, polis). Elle ne peut plus des lors etre interpretee sous l'hypothese de l' idealisme. Elle se manifeste comme une serie de questions de sens ou s'indique une philosophie de la raison. Partout ou (...) le logos nous porte, c'est la qu'il faut aller, tel en serait le mot d'ordre, mais avec la precision qu'a l'articulation du mythique et du theorique, du tragique et du philosohique, et plus profondement de la deraison et du logos, le rationalisme platonicien n'est fait que de raison. (shrink)

Henry Home (1696-1782) has been called "perhaps the most complete 'Enlightenment man' among the eighteenth-century Scottish thinkers." Kinsman and friend of David Hume, mentor and patron of Adam Smith, John Millar, and Thomas Reid, he was a key figure in that circle of luminaries. He read law, was called to the bar in 1723, was raised to the Bench of the Court of Session in 1752, with the title Lord Kames (the name of his family estate), and joined the (...) High Court of the Justiciary in 1763. Publishing broadly in law, history, philosophy, and criticism, Kames made significant contributions to the Enlightenment's science of human nature. The Essays is commonly considered Kames's most important philosophical work. In the first part, he sets forth the principles and foundations of morality and justice, attacking Hume's moral skepticism and addressing the controversial issue of the freedom of human will. In the second part, Kames focuses on questions of metaphysics and epistemology to offer a natural theology in which the authority of the external senses is an important basis for belief in the Deity. The text of this volume is based on the third edition of 1779, while the appendix presents substantial variant readings in the first and second editions. (shrink)

This article identifies a novel challenge to standard understandings of responsibility practices, animated by experimental studies of biases and heuristics. It goes on to argue that this challenge illustrates a general methodological challenge for theorizing about responsibility. That is, it is difficult for a theory to give us both guidance in real world contexts and an account of the metaphysical and normative foundations of responsibility without treating wide swaths of ordinary practice as defective. The general upshot is that theories must (...) either hew more closely to actual practice than they appear to, or they must provide some normative foundation for responsibility that does not go through actual practice. (shrink)

Cet ouvrage n'est ni un commentaire ni une explication d'obedience historiciste ou deterministe. Par questionnement, lecture et interpretation, l'auteur a voulu produire une semantique philosophique. La doctrine des idees est ainsi reexaminee a la convergence des problemes du langage, de la science et de la cite (logos, episteme, polis). Elle ne peut plus des lors etre interpretee sous l'hypothese de l' idealisme. Elle se manifeste comme une serie de questions de sens ou s'indique une philosophie de la raison. Partout ou (...) le logos nous porte, c'est la qu'il faut aller, tel en serait le mot d'ordre, mais avec la precision qu'a l'articulation du mythique et du theorique, du tragique et du philosohique, et plus profondement de la deraison et du logos, le rationalisme platonicien n'est fait que de raison. (shrink)

This book constitutes one of the most important contributions to recent Kant scholarship. In it, one of the pre-eminent interpreters of Kant, Henry Allison, offers a comprehensive, systematic, and philosophically astute account of all aspects of Kant's views on aesthetics. The first part of the book analyses Kant's conception of reflective judgment and its connections with both empirical knowledge and judgments of taste. The second and third parts treat two questions that Allison insists must be kept distinct: the normativity (...) of pure judgments of taste, and the moral and systematic significance of taste. The fourth part considers two important topics often neglected in the study of Kant's aesthetics: his conceptions of fine art, and the sublime. (shrink)

A collection of essays and lectures from Henry Steele Commager, one of America's eminent historians. The focus is on the age of enlightenment, with particular attention to Thomas Jefferson, and how enlightenment philosophy shaped the birth of the United States.

Since its publication in 1918, Oswald Spengler's The Decline of the West has been the object of academic controversy and opprobrium. In their efforts to dispose of it, scholars have resorted to a variety of tactics: bitter invective, icy scorn, urbane mockery, or simply pretending that the book is not there. Yet generations of readers have refused to be warned off, finding in Spengler a prophetic voice and a source of profound intellectual excitement. H. Stuart Hughes's Oswald Spengler offers a (...) judicious and objective reading of Spengler's works that admirably fills the gap between hypercritical invective and naive enthusiasm. This pioneering volume makes clear why Spengler's pessimistic reading of the fate of European civilization continues to resonate with contemporary anxieties. Despite the author's self-imposed intellectual and social isolation, Spengler's work was as Hughes demonstrates, a part of the enormous effort of intellectual reevaluation that has characterized the early twentieth century. Viewing Spengler in the broadest possible perspective, the author places his thought in its cultural relationship to that of such predecessors as Giambattista Vico, Friedrich Nietzsche, and Nikolai Danilevsky and contemporaries including Benedetto Croce, Henri Bergson, and Vilfredo Pareto. A chapter of Hughes's book is devoted to Spengler's influence on later cyclical thinkers such as Arnold Toynbee and Pitirim Sorokin. Another chapter clarifies the essentially antagonistic relationship between his thought and Nazi ideology. Throughout, Hughes is carefully attuned to the complex and often bewildering shifts of Spengler's ideas and manner, providing a unified picture of the sober historian; the lofty seer; the cool, detached observer; and the impassioned participant. In his introduction to this new edition, Hughes comments on the timeliness of Spengler's message with respect to technology and environmental issues and draws some unexpected and fascinating parallels between Spengler's thought and that of Ludwig Wittgenstein. Oswald Spengler offers an illuminating view of the achievements and limitations of one of the most influential and representative figures of the twentieth century. It will be of concern to intellectual historians, philosophers, political scientists, and sociologists. (shrink)

Few books have so firmly established their place in American literature as The Education of Henry Adams. When it was first published in 1918, it became an instant bestseller and went on to win the Pulitzer Prize. More than eighty years later, in an age of self-reflection and exhaustive memoirs, The Education still stands as perhaps the greatest American autobiography. The son of a diplomat, the grandson and great-grandson of two American presidents, a man of extraordinary gifts and learning (...) in his own right, Henry Adams recounts his life from his birth in 1838 and upbringing as a Boston Brahmin, through the Civil War, the nation's industrial expansion, and its emergence as a world power. In the process, he gives us a brilliant history of a changing country as well as a thoughtful, humane, often tender exploration of himself. From the original publisher, this edition of The Education of Henry Adams, newly introduced by Donald Hall, celebrates and honors this classic work on what it means to be an American. (shrink)