We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ03-condition. We show that all trees which are not computably categorical have computable dimension ω. Finally, we prove that for every n≥ 1 in ω, there exists a computable tree of finite height which is δ0n+1-categorical but (...) not δ0n-categorical. (shrink)

We study the filter ℒ*(A) of computably enumerable supersets (modulo finite sets) of an r-maximal set A and show that, for some such set A, the property of being cofinite in ℒ*(A) is still Σ0 3-complete. This implies that for this A, there is no uniformly computably enumerable “tower” of sets exhausting exactly the coinfinite sets in ℒ*(A).

We explore the problem of constructing maximal and unbounded filters on computable posets. We obtain both computability results and reverse mathematics results. A maximal filter is one that does not extend to a larger filter. We show that every computable poset has a \Delta^0_2 maximal filter, and there is a computable poset with no \Pi^0_1 or \Sigma^0_1 maximal filter. There is a computable poset on which every maximal filter is Turing complete. We obtain the reverse mathematics result that the principle (...) "every countable poset has a maximal filter" is equivalent to ACA₀ over RCA₀. An unbounded filter is a filter which achieves each of its lower bounds in the poset. We show that every computable poset has a \Sigma^0_1 unbounded filter, and there is a computable poset with no \Pi^0_1 unbounded filter. We show that there is a computable poset on which every unbounded filter is Turing complete, and the principle "every countable poset has an unbounded filter" is equivalent to ACA₀ over RCA₀. We obtain additional reverse mathematics results related to extending arbitrary filters to unbounded filters and forming the upward closures of subsets of computable posets. (shrink)

Zunächst Pfarrer in Berlin-Charlottenburg, dann Professor für Philosophie in Halle, repräsentiert Eberhard die nachwolffianische deutsche Popularphilosophie, der an praktisch und pädagogisch wirksamer Aufklärung im Zusammenspiel mit den fortschrittlichen politischen und religiösen Kräften lag. In diesen Lehrbüchern repräsentiert Eberhard sich als Aufklärer auf dem philosophischen Niveau der ästhetischen, hermeneutischen und historischen Modifikation des Wolffschen Systems und der neuerlichen Rezeption der Philosophie Leibniz’, zumal der Erkenntnistheorie der 1765 erschienenen „Nouveaux essais sur l’entendement humain“. Eberhard wurde aufgrund seiner Theorie des (...) Denkens und Empfindens als Schüler Leibniz’ bezeichnet; und er verfasste denn auch eine Leibniz-Biographie. Er hielt zeitlebens an den wolffianischen Standards von Rationalität fest und widersetzte sich nicht nur dem normativen Rekurs auf „Empfindsamkeit“ in Literatur und Religion, sondern auch der sog. Glaubensphilosophie Friedrich Heinrich Jacobis, erst recht der zeitgenössischen Wiederkehr von Okkultismus und Esoterik. Johann August Eberhard, professor of philosophy in Halle, is one of the foremost exponents of the Popularphilosophie of the German Enlightenment. He was a major contributor to Friedrich Nicolai’s "Allgemeine Deutsche Bibliothek", and advanced unorthodox opinions on religious and political issues. (shrink)

In dieser Untersuchung wird die Instandhaltung technischer Artefakte, neben deren Schaffung und Nutzung die dritte grundlegende Form technischen Handelns, als Teilthema der Technikphilosophie systematisch und umfassend analysiert. Nur durch Instandhaltung können die unermeßlichen und vielfältigen mittels technischer Verfahren erzeugten Werte vor vorzeitigem Verfall und Untergang bewahrt werden. Die Studie erschließt Instandhaltung in phänomenologischer (Grundlagen) und normativer Hinsicht (Instandhaltung und Gesellschaft). Die auf Unumkehrbarkeit des Handelns und ganz allgemein des Geschehens gegründete Endlichkeit seiner Dauer bindet das individuelle technische Artefakt als wesentliches (...) Element in den Verlauf der Allgemein-, Technik-, Kultur-, Wirtschafts- und Sozialgeschichte ein. (shrink)

A trusted guide into the inner realm where our spirits find strength to master life and live for God. It is hard to exaggerate the significance of Innerland, either for Eberhard Arnold or his readers. It absorbed his energies off and on for most of his adult life--from World War I, when he published the first chapter under the title War: A Call to Inwardness, to 1935, the last year of his life. Packed in metal boxes and buried at (...) night for safekeeping from the Nazis, who raided the author_s study a year before his death (and again a year after it), Innerland was not openly critical of Hitler_s regime. Nevertheless, it attacked the spirits that animated German society: its murderous strains of racism and bigotry, its heady nationalistic fervor, its mindless mass hysteria, and its vulgar materialism. In this sense Innerland stands as starkly opposed to the zeitgeist of our own day as to that of the author_s. At a glance, the focus of Innerland seems to be the cultivation of the spiritual life as an end in itself. Nothing could be more misleading. In fact, to Eberhard Arnold the very thought of encouraging the sort of selfish solitude whereby people seek their own private peace by shutting out the noise and rush of public life around them is anathema. He writes in The Inner Life:_These are times of distress. We cannot retreat, willfully blind to the overwhelming urgency of the tasks pressing on society. We cannot look for inner detachment in an inner and outer isolation...The only justification for withdrawing into the inner self to escape today's confusing, hectic whirl would be that fruitfulness is enriched by it. It is a question of gaining within, through unity with eternal powers, a strength of character ready to be tested in the stream of the world._ Innerland, then, calls us not to passivity, but to action. It invites us to discover the abundance of a life lived for God. It opens our eyes to the possibilities of that _inner land of the invisible where our spirit can find the roots of its strength and thus enable us to press on to the mastery of life we are called to by God._ Only there, says Eberhard Arnold, can our life be placed under the illuminating light of the eternal and seen for what it is. Only there will we find the clarity of vision we need to win the daily battle that is life, and the inner anchor without which we will lose our moorings. (shrink)

This paper considers reductions between types of numberings; these reductions preserve the Rogers Semilattice of the numberings reduced and also preserve the number of minimal and positive degrees in their semilattice. It is shown how to use these reductions to simplify some constructions of specific semilattices. Furthermore, it is shown that for the basic types of numberings, one can reduce the left-r.e. numberings to the r.e. numberings and the k-r.e. numberings to the k+1-r.e. numberings; all further reductions are obtained by (...) concatenating these reductions. (shrink)

La cessation hebdomadaire du travail le dimanche est de plus en plus contestée dans notre société. Si les exigences économiques rendent nécessaire le fonctionnement ininterrompu des machines, il faudrait assurer à chaque employé un autre jour de repos. Peut-on alors définir le dimanche comme jour de repos commun à la majorité de la société ? Les chrétiens ont l’habitude de se prononcer en faveur du repos dominical en rappelant le commandement du Décalogue : «Observe le jour du sabbat ». Or (...) cet argument n’est pas pertinent pour des raisons historiques et herméneutiques. Il est toutefois possible de tirer de ce commandement, en particulier de sa version deutéronomique (Dt 5,12-15), des arguments susceptibles d’enrichir les débats d’aujourd’hui. (shrink)

We exhibit a finite lattice without critical triple that cannot be embedded into the enumerable Turing degrees. Our method promises to lead to a full characterization of the finite lattices embeddable into the enumerable Turing degrees.

We characterize the linear order types $\tau $ with the property that given any countable linear order $\mathcal {L}$, $\tau \cdot \mathcal {L}$ is a computable linear order iff $\mathcal {L}$ is a computable linear order, as exactly the finite nonempty order types.

We establish that the isomorphism problem for the classes of computable nilpotent rings, distributive lattices, nilpotent groups, and nilpotent semigroups is \-complete, which is as complicated as possible. The method we use is based on uniform effective interpretations of computable binary relations into computable structures from the corresponding algebraic classes.