We work in set theory ZF without axiom of choice. Though the Hahn-Banach theorem cannot be proved in ZF, we prove that every Gateaux-differentiable uniformly convex Banach space E satisfies the following continuous Hahn-Banach property: if p is a continuous sublinear functional on E, if F is a subspace of E, and if f: F → ℝ is a linear functional such that f ≤ p|F then there exists a linear functional g : E → (...) ℝ such that g extends f and g ≤ p. We also prove that the continuous Hahn-Banach property on a topological vector space E is equivalent to the classical geometrical forms of the Hahn-Banach theorem on E. We then prove that the axiom of Dependent choices DC is equivalent to Ekeland's variational principle, and that it implies the continuous Hahn-Banach property on Gateaux-differentiable Banach spaces. Finally, we prove that, though separable normed spaces satisfy the continuous Hahn-Banach property, they do not satisfy the whole Hahn-Banach property in ZF+DC. (shrink)

In set theory without the axiom of choice math formula, three-space type results for the Hahn-Banach property are provided. We deduce that for every Hausdorff compact scattered space K, the Banach space C of real continuous functions on K satisfies the continuous Hahn-Banach property in math formula. We also prove in math formula Rudin's theorem: “Radon measures on Hausdorff compact scattered spaces are discrete”.

We determine the computational complexity of the Hahn-Banach Extension Theorem. To do so, we investigate some basic connections between reverse mathematics and computable analysis. In particular, we use Weak König's Lemma within the framework of computable analysis to classify incomputable functions of low complexity. By defining the multivalued function Sep and a natural notion of reducibility for multivalued functions, we obtain a computational counterpart of the subsystem of second-order arithmetic WKL0. We study analogies and differences between WKL0 and (...) the class of Sep-computable multivalued functions. Extending work of Brattka, we show that a natural multivalued function associated with the Hahn-Banach Extension Theorem is Sep-complete. (shrink)

The classical Hahn–Banach Theorem states that any linear bounded functional defined on a linear subspace of a normed space admits a norm-preserving linear bounded extension to the whole space. The constructive and computational content of this theorem has been studied by Bishop, Bridges, Metakides, Nerode, Shore, Kalantari Downey, Ishihara and others and it is known that the theorem does not admit a general computable version. We prove a new computable version of this theorem without unrolling the classical proof (...) of the theorem itself. More precisely, we study computability properties of the uniform extension operator which maps each functional and subspace to the set of corresponding extensions. It turns out that this operator is upper semi-computable in a well-defined sense. By applying a computable version of the Banach–Alaoglu Theorem we can show that computing a Hahn–Banach extension cannot be harder than finding a zero in a compact metric space. This allows us to conclude that the Hahn–Banach extension operator is ${\bf {\Sigma^{0}_{2}}}$ -computable while it is easy to see that it is not lower semi-computable in general. Moreover, we can derive computable versions of the Hahn–Banach Theorem for those functionals and subspaces which admit unique extensions. (shrink)

We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to ; “ can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality ”, thus the latter statement is true in every Fraenkel‐Mostowski model of ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality ” is not provable in (...) ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality ” is provable in ; (the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X, and, then there is a unit vector such that for all and all scalars α”) is provable in ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ball is compact” is provable in ; (Principle of Dependent Choices) + “ can be well‐ordered” does not imply the Hahn‐Banach Theorem () in ; and “no infinite‐dimensional Banach space has a Hamel basis of cardinality ” are independent from each other in ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between (the Axiom of Countable Choice) and ; implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of, but not a theorem of ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ” implies “every Dedekind‐finite set is finite”. (shrink)

Sur la décomposition des ensembles de points en parties respectivement congruentes.

In this analysis of one of the most difficult and neglected topics in Hegelian studies, Songsuk Susan Hahn tackles the status of contradiction in Hegel's ...

Karl Hahn, Jahrgang 1937, war bis zu seiner Emeritierung im Jahr 2003 Professor für Politische Theorie und Ideengeschichte an der Westfälischen Wilhelms-Universität Münster. Hierbei hat er die Klassiker der abendländischen Geistesgeschichte nicht in einem kanonisch-schematischen Profil gelehrt, sondern die jeweiligen Themen und Theoreme mit lebensfüllender Leidenschaft vermittelt. Seine Vorlesungen und Seminare an der Universität zeugen, wie auch seine Vorträge und Tagungsbeiträge, von einem streitbaren Geist, der stets auch um die aktualisierende Bezugnahme bemüht war. Stets hat er pointiert Stellung bezogen, (...) auch gegen den Mainstream des Faches, das er kritisch hinterfragt hat. Ob Platon, Fichte oder auch Simone Weil: Bei Karl Hahn stehen philosophische Überlegungen für eine Suche nach der Vollendung des Daseins in seinen unendlich vielen und offenen Aussagemöglichkeiten. Hahns großes Lebensthema war und ist hierbei das tragische Element des menschlichen Wesens, dessen Kleinheit und Begrenztheit angesichts der großen Fragen der menschlichen Existenz. Das hat er seinen Studierenden mit auf den Weg gegeben. Und auch den Kolleginnen und Kollegen, die bereit waren, zuzuhören. Mit dem vorliegenden Band wird der Wissenschaftler zum Anlass seines 80. Geburtstages geehrt, der seine Verdienste nicht in der flüchtigen Wertschätzung des Feuilletons, sondern grundsätzlich im Diskurs der Seminarräume und Kongressveranstaltungen erworben hat. (shrink)

Metaphysics, geometry, and the problems with diagrams -- The Pythagorean theorem: Euclid I.47 and VI.31 -- Thales and geometry: Egypt, Miletus, and beyond -- Pythagoras and the famous theorems -- From the Pythagorean theorem to the construction of the cosmos out of right triangles.

Yoga teacher, podcaster, and blogger Kori Hahn presents a new-age guide for harnessing the principles of yoga to manifest a better life. The book teaches readers how to develop simple, personalized rituals using techniques such as affirmations, breath work, meditation, journaling, and visualization.

Auch die subtilsten geistigen Kompetenzen haben eine organische Basis. Mag es auch zunächst überraschen, so besteht daher doch eine entscheidende Verbindung zwischen Gedächtnis und Körper. Es geht also darum, Gedächtnis und Erinnerung nicht als bloßes Wissen misszuverstehen. Vor allem als soziale Tatsache bedarf Erinnerung stets wiederholter Beschwörungen, um nicht zu verschwinden. Spricht man vom individuellen Gedächtnis und der persönlichen Erinnerung, denkt man im Allgemeinen an bewusste Prozesse, dabei ist es häufig viel eher so, dass bestimmte Lernerfahrungen gar nicht mehr bewusst, (...) sondern gleichsam zum weitestgehend unbewussten Moment geistiger und vor allem auch körperlicher Prägungen geworden sind. Diese Verbindung zwischen Körper und Gedächtnis faltet der vorliegende Band systematisch aus. (shrink)

This collection of essays discusses the work of Jurgen Habermas - the philosopher and exponent of the tradition known as Critical Theory. His works defend the Enlightenment ideas of rationality, humanism, and the possibilities of discourse.

Auf der Suche nach objektiver Wahrheit wurde in den Wissenschaften das Prinzip der Relationalitat popular: Gegenstandsbereiche gleich welcher Art werden als reine Beziehungsgeflechte beschrieben, nach dem Vorbild der Mathematik (diese Notwendigkeit entstand nicht zuletzt durch die Relativitatstheorie). Um Reales theoretisch zu erfassen, etwa um Dinge zahlen zu konnen, braucht man ein Identitatsprinzip fur Gegenstande. Das formal beste und einzig universelle ist die Identitat der Ununterscheidbaren. In Verbindung mit dem modernen Strukturalismus fuhrt dies aber zum geisterhaften Verschwinden realer Dinge in der (...) Beschreibung, wenn beziehungsmassig Symmetrie vorliegt. Objektive Beschreibungen erfassen also nur asymmetrische Strukturen korrekt. Manches Reale wird somit nicht objektiv erfasst.". (shrink)

The Chairman of Volkswagen's Board of Management made the following presentation in London last November at a Conference on‘Business and Moral Standards in Post‐Communist Europe’, held under the auspices of the Cardinal Archbishop of Westminster and sponsored by the Sedgwick Group and KPMG Peat Marwick. Dr Hahn's lecture is reproduced with permission.

In der Einleitung in den Schwerpunkt „Ethik der Globalisierung und kollektive Verantwortung“ machen wir uns für ein Verständnis von globaler Ethik als einer neuen Disziplin innerhalb der angewandten Ethik stark. Darin geht es wesentlich um Probleme, die sich aus der globalen Kooperation und Konkurrenz in wirtschaftlicher, politischer, sozialer und kultureller Hinsicht ergeben. Nach dieser Auffassung hat es globale Ethik grundsätzlich mit der Bestimmung globaler kollektiver Verantwortlichkeiten zu tun, da sich das Handeln individueller und kollektiver Akteure heute vor dem Hintergrund einer (...) globalisierten Struktur von Handlungspfaden, Anreizen und Regeln vollzieht. In allen Bereichsethiken der globalen Ethik ist daher die Frage, wie sich kollektive Verantwortung unstrukturierten Gruppen zuschreiben lässt, zentral geworden. Worin besteht die kollektive Verantwortung von Unternehmen und Konsument*innen für globale Ausbeutung, Weltarmut oder die Folgen der Klimawandels? Die Antworten der fünf in dieser Einleitung vorgestellten Beiträge greifen dazu in erhellender Weise auf die handlungstheoretischen Debatten zu kollektiver Verantwortung zurück. Es zeigt sich, dass das Verantwortungsprinzip durch seinen prospektiven, Kollektive in den Blick nehmenden, die moralische, politische und juridische Dimension verbindenden sowie auf Strukturen scharf stellenden Charakter besonders geeignet ist, diese Probleme zu behandeln. (shrink)

The debate over environmental, health, and safety regulation has reached a new crescendo in the 104th Congress. So impassioned is the debate on occasion, and so high the feelings, that even the tools of regulatory analysis have become part of the combat.To some, the term cost-benefit analysis, for example, is virtually a swearword, a nefarious tool used by big business to undermine regulations aimed at benefiting the people at large. To others, it is the mechanism for achieving more effective regulation (...) at less cost.This new book on the subject of reforming regulation is aimed at increasing the light and turning down the heat. It consists of an introduction and nine chapters written by scientists, public policy analysts, and economists on various aspects of regulation and regulatory analysis.The book advances the latest knowledge and evidence on risk analysis, cost-benefit analysis, and related techniques for improving regulation--all based on experience and data accumulated over the quarter century since regulation in the areas of environment, health, and safety became a major player on the governmental stage. The strengths and limitations of those analytical tools are an important part of the portrait that is painted, but a common thesis is that, properly used, the tools can be helpful in getting better regulatory results. (shrink)

It is virtuous for individual and collective agents to be tolerant. However, toleration is difficult, both in practice and in conceptualization. Firstly, given that toleration can be understood in various ways (Walzer 1997, Forst 2007), it seems that to determine what is the proper conception of toleration would be controversially difficult. Here I shall suggest one particular conception of toleration is more suitable than others. This conception allows, as I shall explain, us to better understandthe difficulties of toleration. Thus, this (...) particular conception of toleration should lead us to see what is more adequate for dealing with the difficulties of toleration. To be more precise, I shall argue for a political conception of toleration, which different from the attitudinal conception of toleration as being indifferent, or the ethical conception of toleration as respect. There is the suggestion of toleration as recognition (Galeotti 2002). These alternative understandings of toleration do not provide better diagnoses of the difficulties of toleration. The political conception of toleration is intended to be grounded on some moral considerations, notpragmatic purpose. It is political in that it recognizes the fact that toleration is essentially practiced to deal with a power relationship among the parties of toleration. Where these is no such power relationship, as I shall argue, there is no issue of toleration. Secondly, this proposed conception of toleration is political in the sense that it shall not deal with differences coming from, to use John Rawls’s phrase, the fact of pluralism by adopting any comprehensive doctrine such as an ethics of respect or recognition. (shrink)

An alternative proof is given that to each LLM proof net there corresponds at least one LLM sequent proof. The construction is inspired by the sweepline technique from computational geometry and includes a treatment of the multiplicative constants and of proof boxes.