In this paper, we attempt to show that a weak version of Hilbert's metamathematics is compatible with Gödel's Incompleteness Theorems by employing only what are clearly natural prov‐ ability predicates. Defining first “T proves the consistency of a theory S indirectly in one step”, we subsequently prove “PA proves its own consistency indirectly in one step” and sketch the proof for “If S is a recursively enumerable extension of , S proves its own consistency indirectly in one step”. The formalizations (...) of the metatheoretical consistency assertions that occur in these theorems are clearly the natural ones. We conclude the paper with reflections on indirect consistency proofs and soundness proofs. (shrink)

In this paper, the (possible) role of model theory forstructuralism and structuralist definitions of ``reduction'' arediscussed. Whereas it is somewhat undecisive with respect tothe first point – discussing some pro's and con's ofthe model theoretic approach when compared with a syntacticand a structuralist one – it emphasizes that severalstructuralist definitions of ``reducibility'' do not providegenerally acceptable explications of ``reducibility''. This claimrests on some mathematical results proved in this paper.

From early work of N. Goodman to recent approaches by H. Field and D. Lewis, there have been attempts to combine 2nd order languages with calculi of individuals. This paper is a contribution, containing basic definitions and distinctions and some metatheorems, to the development of a general metatheory of such theories.

This paper deals with the treatment of infinity and finiteness in mereology. After an overview of some first-order mereological theories, finiteness axioms are introduced along with a mereological definition of “x is finite” in terms of which the axioms themselves are derivable in each of those theories. The finiteness axioms also provide the background for definitions of “ T makes an assumption of infinity”. In addition, extensions of mereological theories by the axioms are investigated for their own sake. In the (...) final part, a definition of “x is finite” stated in a second-order language is also presented, followed by some concluding remarks on the motivation for the study of the extensions of mereological theories dealt with in the paper. (shrink)

This paper is an attempt to bring together two separated areas of research: classical mathematics and metamathematics on the one side, non-monotonic reasoning on the other. This is done by simulating nonmonotonic logic through antitonic theory extensions. In the first half, the specific extension procedure proposed here is motivated informally, partly in comparison with some well-known non-monotonic formalisms. Operators V and, more generally, U are obtained which have some plausibility when viewed as giving nonmonotonic theory extensions. In the second half, (...) these operators are treated from a mathematical and metamathematical point of view. Here an important role is played by U -closed theories and U -fixed points. The last section contains results on V-closed theories which are specific for V. (shrink)

Hilbert developed his famous finitist point of view in several essays in the 1920s. In this paper, we discuss various extensions of it, with particular emphasis on those suggested by Hilbert and Bernays in Grundlagen der Mathematik (vol. I 1934, vol. II 1939). The paper is in three sections. The first deals with Hilbert's introduction of a restricted ? -rule in his 1931 paper ?Die Grundlegung der elementaren Zahlenlehre?. The main question we discuss here is whether the finitist (meta-)mathematician would (...) be entitled to accept this rule as a finitary rule of inference. In the second section, we assess the strength of finitist metamathematics in Hilbert and Bernays 1934. The third and final section is devoted to the second volume of Grundlagen der Mathematik. For preparatory reasons, we first discuss Gentzen's proposal of expanding the range of what can be admitted as finitary in his esssay ?Die Widerspruchsfreiheit der reinen Zahlentheorie? (1936). As to Hilbert and Bernays 1939, we end on a ?critical? note: however considerable the impact of this work may have been on subsequent developments in metamathematics, there can be no doubt that in it the ideals of Hilbert's original finitism have fallen victim to sheer proof-theoretic pragmatism. (shrink)

This paper is an investigation into what could be a goodexplication of ``theory S is reducible to theory T''''. Ipresent an axiomatic approach to reducibility, which is developedmetamathematically and used to evaluate most of the definitionsof ``reducible'''' found in the relevant literature. Among these,relative interpretability turns out to be most convincing as ageneral reducibility concept, proof-theoreticalreducibility being its only serious competitor left. Thisrelation is analyzed in some detail, both from the point of viewof the reducibility axioms and of modal logic.

In his paper "Finitism", W.W. Tait maintains that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis "The finitist functions are precisely the primitive recursive functions" is disputable and that another, likewise defended by him, (...) is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA. /// En su articulo "Finitism", W.W. Tait sostiene que la dificultad principal para quien quiere comprender la concepción hilbertiana de la matemática finitista es ésta: especificar el sentido de la demostrabilidad de enunciados generales sobre los números naturales sin presuponer totalidades infinitas. Además, Tait argumenta que todo razonamiento finitista es esencialmente primitivo recursivo. En este artículo tratamos de mostrar que su tesis "Las funciones finitistas son precisamente las funciones primitivas recursivas" es discutible y que otra, también defendida por él, resulta insostenible. La segunda tesis es que los teoremas finitistas son precisamente las clausuras universales de las ecuaciones que pueden demostrarse en PRA. (shrink)

In his paper `Finitism', W.W.~Tait maintained that the chief difficulty for everyone who wishes to understand Hilbert's conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argued that all finitist reasoning is essentially primitive recursive. In our paper, we attempt to show that his thesis ``The finitist functions are precisely the primitive recursive functions'' is disputable and that another, likewise defended by him, is (...) untenable. The second thesis is that the finitist theorems are precisely those $\Pi^0_1$-sentences that can be proved in. (shrink)

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work is in the "public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Die jungsten Finanzkrisen haben die Diskussion uber die Moral der Markte wiederbelebt: Welche Verantwortung haben Unternehmen, Banken, Politik und letztlich jeder Einzelne in unserer Gesellschaft? Droht die Okonomisierung aller Lebensbereiche, der Ausverkauf des Menschen? Ist der Markt moralfrei? Brauchen wir neue Perspektiven und Werte? Diese drangenden Fragen werden im vorliegenden Buch von namhaften Autorinnen und Autoren wie Franz Muntefering und Rita Sussmuth beantwortet: Der Mensch muss wieder ins Zentrum rucken, Verantwortung und Moral sollten grundlegender Massstab auch des wirtschaftlichen Handelns sein. (...) Mit diesen Forderungen appellieren die BeitragerInnen an einen echten Dialog zwischen Gesellschaft, Wirtschaft und Ethik. Mit Beitragen von Bernhard Emunds, Jurgen Hardt, Linus Hauser, Esther Junger, Josef Kaiser, Siegfried Karl, Andreas Lenz, Uta Meier-Grawe, Franz Muntefering, Thomas Petersen, Marcus K. Reif, Rainer Schwarz, Rita Sussmuth und Joachim Wiemeyer". (shrink)

1. Frühe Schriften.--2. Janaer Schriften, 1801-1807.--3. Phänomenologie des Geistes.--4. Nürnberger und Heidelberger Schriften, 1808-1817.--5-6. Wissenschaft der Logik.--7. Grundlinien der Philosophie des Rechts.--8-10. Enzyklopädie der philosophischen Wissenschaften im Grundrisse (1830).--11. Berliner Schriften, 1818-1831.--12. Vorlesungen über die Philosophie der Geschichte.--13-15. Vorlesungen über die Ästhetik.--16-17. Vorlesungen über die Philosophie der Religion.--18-20. Vorlesungen über die Geschichte der Philosophie.

We study the idea of implantation of Piron's and Bell's geometrical lemmas for proving some results concerning measures on finite as well as infinite-dimensional Hilbert spaces, including also measures with infinite values. In addition, we present parabola based proofs of weak Piron's geometrical and Bell's lemmas. These approaches will not used directly Gleason's theorem, which is a highly non-trivial result.

Representative works are interpreted in light of the two great political movements of the nineteenth century: the abolition of slavery and the women's rights movement.

Zu Hegels verschiedenen Vorlesungen über die «Philosophie des Rechts» gab es bisher jeweils nur eine Mit- oder Nachschrift seiner Schüler. Deshalb ließ sich schwer einschätzen, inwieweit diese die sich ändernden Vorstellungen Hegels über Recht und Staatsverfassung authentisch wiedergaben. Die neu aufgetauchte und erstmals kritisch edierte Nachschrift K. W. L. Heyses stammt aus dem Wintersemester 1822/23, aus dem es bisher die Mitschrift H. G. Hothos gab. Beide ergänzen sich gegenseitig und zeigen, daß Hotho und Heyse Hegels Darlegungen genau verfolgt und in (...) verkürzter Form notiert haben. Damit ist für Hegels Haltung wenige Jahre nach den Karlsbader Beschlüssen eine sicherere Textgrundlage als bisher gegeben. (shrink)

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be (...) preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Especially where a patient's ability to communicate is limited, the receptive use of art visually offers an entry point into contemplation and insight. Here, receptive work with visual art can help to dissolve limitations in verbal communication. In receptive art therapy, an independent therapeutic resource is attributed to the artistic work and its symbolic pictorial content, which can initiate a healing process. A form of indirect communication is stimulated, in which not only a change on the symbolic but also on (...) the emotional level takes place through the emotionally highly charged pictorial-symbolic expression in the confrontation. Different approaches and basic concepts of the method are presented. This conference volume is addressed to the patients, therapists and clinical as well as inclusive companions in these processes. With contributions by Dr. Alexandra Daszkowski, Prof. Dr. Georg Franzen, Dr. Barbara Laimböck, Prof. Andreas Mayer-Brennenstuhl, Prof. Dr. Karl-Heinz Menzen, Karolina Sarbia, Prof. Dr. Martin Schuster and Ass.-Prof. Dr. Karla Villavicencio. (shrink)