Starting with D. Scott's work on the mathematical foundations of programming language semantics, interest in topology has grown up in theoretical computer science, under the slogan `open sets are semidecidable properties'. But whereas on effectively given Scott domains all such properties are also open, this is no longer true in general. In this paper a characterization of effectively given topological spaces is presented that says which semidecidable sets are open. This result has important consequences. Not only follows the classical Rice-Shapiro (...) Theorem and its generalization to effectively given Scott domains, but also a recursion theoretic characterization of the canonical topology of effectively given metric spaces. Moreover, it implies some well known theorems on the effective continuity of effective operators such as P. Young and the author's general result which in its turn entails the theorems by Myhill-Shepherdson, Kreisel-Lacombe-Shoenfield and Ceĭtin-Moschovakis, and a result by Eršov and Berger which says that the hereditarily effective operations coincide with the hereditarily effective total continuous functionals on the natural numbers. (shrink)

Effective inseparability of pairs of sets is an important notion in logic and computer science. We study the effective inseparability of sets which appear as index sets of subsets of an effectively given topological T0-space and discuss its consequences. It is shown that for two disjoint subsets X and Y of the space one can effectively find a witness that the index set of X cannot be separated from the index set of Y by a recursively enumerable set, if X (...) intersects the topological closure of an effectively enumerable subset of Y. As a consequence of a more general parametric inseparability result a theorem of Rice-Shapiro type is obtained. Moreover, under some additional requirements it follows that nonopen subsets have productive index sets. This implies a generalized Rice theorem: Connected spaces have only trivial completely recursive subsets. As application some decision problems in computable analysis and domain theory are studied. It follows that the complement of the halting problem can be reduced to the problem to decide of a number whether it is a computable irrational. The same is true for the problems to decide whether two numbers are equal, whether one is not greater than the other, and whether a number is equal to a given number. In the case of an effectively given continuous complete partial order the complexity of the last problem depends on whether the given element is the smallest element, in which case the complement of the halting problem is reducible to it, whether it is a base element and maximal, then the decision problem is recursively isomorphic to the halting problem, or whether it is none of these. In this case, both the halting problem and its complement are reducible to the problem. The same is true in nontrivial cases for the problems whether an element belongs to the basis, whether two elements of the partial order are equal, or whether one approximates the other. In general, for any nonempty proper subset of the partial order either the halting problem or its complement can be reduced to the membership problem of the subset. (shrink)

In examples like the total recursive functions or the computable real numbers the canonical indexings are only partial maps. It is even impossible in these cases to find an equivalent total numbering. We consider effectively given topological T 0 -spaces and study the problem in which cases the canonical numberings of such spaces can be totalized, i.e., have an equivalent total indexing. Moreover, we show under very natural assumptions that such spaces can effectively and effectively homeomorphically be embedded into a (...) totally indexed algebraic partial order that is closed under the operation of taking least upper bounds of enumerable directed subsets. (shrink)

If one wants to compute with infinite objects like real numbers or data streams, continuity is a necessary requirement: better and better (finite) approximations of the input are transformed into better and better (finite) approximations of the output. In case the objects are constructively generated, they can be represented by a finite description of the generating procedure. By effectively transforming such descriptions for the generation of the input (respectively, their codes) into (the code of) a description for the generation of (...) the output another type of computable operation is obtained. Such operations are also called effective. The relationship of both classes of operations has always been a question of great interest. In this paper the setting is extended to the case of multifunctions. Various ways of coding (indexing) sets are discussed and their relationship is investigated. Moreover, effective versions of several continuity notions for multifunctions are introduced. For each of these notions an indexing system for sets is exhibited so that the multifunctions that are effective with respect to this indexing system are exactly the multifunction which are effectively continuous with respect to the continuity notion under consideration. Mostly, in addition to being effective the multifunctions need also possess certain witnessing functions. Important special cases are discussed where such witnessing functions always exist. (shrink)

Published in honor of Victor L. Selivanov, the 17 articles collected in this volume inform on the latest developments in computability theory and its applications in computable analysis; descriptive set theory and topology; and the theory of omega-languages; as well as non-classical logics, such as temporal logic and paraconsistent logic. This volume will be of interest to mathematicians and logicians, as well as theoretical computer scientists.

A strong reducibility relation between partial numberings is introduced which is such that the reduction function transfers exactly the numbers which are indices under the numbering to be reduced into corresponding indices of the other numbering. The degrees of partial numberings of a given set with respect to this relation form an upper semilattice.In addition, Ershov’s completion construction for total numberings is extended to the partial case: every partially numbered set can be embedded in a set which results from the (...) given set by adding one point and which is enumerated by a total and complete numbering. As is shown, the degrees of complete numberings of the extended set also form an upper semilattice. Moreover, both semilattices are isomorphic.This is not so in the case of the usual, weaker reducibility relation for partial numberings which allows the reduction function to transfer arbitrary numbers into indices. (shrink)

The different behaviour of total and partial numberings with respect to the reducibility preorder is investigated. Partial numberings appear quite naturally in computability studies for topological spaces. The degrees of partial numberings form a distributive lattice which in the case of an infinite numbered set is neither complete nor contains a least element. Friedberg numberings are no longer minimal in this situation. Indeed, there is an infinite descending chain of non-equivalent Friedberg numberings below every given numbering, as well as an (...) uncountable antichain. (shrink)

Wer kennt GOTTLOB FREGE (1848-1925)? Nur wenige verbinden mit seinem Namen einen der groaten Logiker aller Zeiten, ohne dessen "Logik von Funktion und Argument" die heutige Computertechnik kaum denkbar ist. Frege fuhlte sich zeitlebens zu seiner Heimat im Norden Deutschlands hingezogen. In Wismar verbrachte er seine Kindheit und in Bad Kleinen, in unmittelbarer Nahe, seinen Lebensabend. Das im Jahre 2000 gegrundete Gottlob-Frege-Zentrum der Hochschule Wismar organisierte ihm zu Ehren im Mai 2013 in seiner Geburtsstadt eine internationale Konferenz, an der Forscher (...) aus der mathematischen Grundlagenforschung und der Analytischen Philosophie teilnahmen. Im Konferenzband erfahren Sie von bekannten Wissenschaftlern u.a., begin{itemize} item was Frege zur Entwicklung der Mathematik, Logik und Philosophie leistete, item welche wissenschaftlichen Auffassungen Frege vertrat und welche er scharf kritisierte, item mit welchen bekannten Wissenschaftlern Frege korrespondierte und wer sein Werk fortsetzte oder in Frage stellte, item was wir heute noch aus Freges Arbeiten lernen konnen. (shrink)

Zu den Grenzfragen des Lebens. Bioethik als Feld intensiver Auseinandersetzung im fachwissenschaftlichen Diskurs und kontroverser Diskussionen in der breiten Öffentlichkeit. Das Handbuch erläutert die Grundlagen und gibt einen Überblick über Bereiche, in denen bioethische Fragen relevant sind, wie Menschenwürde, Leben, Tod und Krankheit. Daneben widmet es sich den speziellen bioethischen Problemen wie etwa Zulässigkeit von Sterbehilfe, Präimplantationsdiagnostik oder embryonaler Stammzellenforschung und zeigt die Schnittstellen der Bioethik zu anderen Disziplinen und gesellschaftlichen Bereichen auf (Bioethik in der Lehre, Biopolitik, Biorecht und Ethikkommissionen).

This book provides a unique survey displaying the power of Riccati equations to describe reversible and irreversible processes in physics and, in particular, quantum physics. Quantum mechanics is supposedly linear, invariant under time-reversal, conserving energy and, in contrast to classical theories, essentially based on the use of complex quantities. However, on a macroscopic level, processes apparently obey nonlinear irreversible evolution equations and dissipate energy. The Riccati equation, a nonlinear equation that can be linearized, has the potential to link these two (...) worlds when applied to complex quantities. The nonlinearity can provide information about the phase-amplitude correlations of the complex quantities that cannot be obtained from the linearized form. As revealed in this wide ranging treatment, Riccati equations can also be found in many diverse fields of physics from Bose-Einstein-condensates to cosmology. The book will appeal to graduate students and theoretical physicists interested in a consistent mathematical description of physical laws. (shrink)

Our starting point is the question ‘What is mass’, what in particular enables mass to constitute duration: so that mass can be regarded as moving as well as at rest (the kinematic principle of relativity). In a thought experiment, this question is attacked here not from the perspective of mass itself, but from that of a standing light wave. In this model, mass-analogous structures can be re- constructed that can be in relative motion to each other. The (empirically known) constancy (...) of the speed of light in every reference system is not as- sumed here. However, it can be shown that the speed of light is identical in all systems constituted by mass-analogous structures that move uniformly relative to each other and that the speed of light moreover is the limiting speed of mo- tions. This indepence of the motion of light from the system of reference, this character of absoluteness, and the relativity of mass motions thus prove to be contrary but inextricably linked moments of the kinematic principle of relativi- ty: an interesting perspective concerning a philosophy of relativity theory. - The mathematical relationships are explained in more detail in the appendix. (shrink)

Ausgangspunkt ist die Frage, was Masse ist, wodurch sie insbesondere befähigt ist, Dauer zu konstituieren und deshalb - im Sinn des kinematischen Relativitätdprinzips - ebenso als bewegt wie als ruhend betrachtet werden kann. In einem Gedankenexperiment wird diese Frage, in Umkehrung der Perspektive, hier nicht von der Masse selbst, sondern von einer stehenden Lichtwelle her angegangen. In diesem Modell lassen sich masse-analoge Strukturen rekonstruieren, die in relativer Bewegung sein können. Die (empirisch bekannte) Unabhängigkeit der Lichtgeschwindigkeit vom Bezugssystem ist dabei nicht (...) vorausgesetzt. Doch kann gezeigt werden, dass die Lichtgeschwindigkeit Grenzgeschwindigkeit solcher relativer Bewegungen ist, dass dies für alle durch masse-analoge Strukturen konstituierte, relativ zueinander gleichförmig bewegten Bezugssysteme gilt und dass die Lichtgeschwindigkeit selbst in allen diesen Systemen gleich ist. Die Absolutheit der Lichtgeschwindigkeit und die Relativität der Massebewegungen erweisen sich so als zwar differente, aber intrinsisch zusammengehörende Momente des kinematischen Relativitätsprinzips: ein Beitrag zu einer Philosophie der Relativitätstheorie. - Die mathematischen Zusammenhänge werden im Anhang ausführlicher dargelegt. (shrink)

Zur Signatur moderner Gesellschaften gehört ihr Pluralismus. Davon ist der Bereich der Ethik nicht ausgenommen. Wofür ein moralischer Pluralismus im Einzelnen steht, das ist allerdings alles andere als eindeutig. Um der Kontroverse über moralische Vielfalt, die in der Gegenwart nicht selten diffus geführt wird, möglichst klare Konturen zu verleihen, empfiehlt es sich, wenigstens drei Grundarten zu unterscheiden: einen deskriptiv-ethischen, einen normativ-ethischen und einen tugendethischen Pluralismus. Innerhalb der jeweiligen Grundart sind, wie in dieser Abhandlung schwerpunktmäßig aufgezeigt wird, etliche Differenzierungen anzubringen. Ohne (...) sich einen Überblick über zentrale Varianten eines moralischen und ethischen Pluralismus zu verschaffen, kann der Diskurs nicht sachgerecht geführt werden. Vorwiegend werden mannigfache Facetten moralischer Diversität eruiert. Einige Hinweise, wie mit dem Pluralismus verantwortlich umgegangen werden kann, werden angefügt. (shrink)

Transzendental wird hier, abweichend von Kant, im Sinn transzendentaler Argumente verstanden. Im Kontext der Letztbegründungsdiskussion sind diese dadurch definiert, dass ihre Verneinung zu einem pragmatischen Widerspruch führt. Nun ist Hegels Philosophie auf die ‚logische Idee‘ gegründet, also auf die in Hegels Verständnis fundamentale dialektische Logik. Sein philosophischer Systementwurf hat dadurch, wie gezeigt wird, transzendentalen Charakter und schließt dementsprechend auch eine transzendentale Naturphilosophie ein. Gezeigt wird weiter, dass Grundbe- stimmungen der Natur, die üblicherweise einfach vorausgesetzt werden, in diesem Rahmen grundsätzlich erklärt (...) werden können: die Existenz der Natur und ihr (räumlich-zeitliches) Auseinandersein, ferner die Erkenn- barkeit der Natur, ihre Gesetzmäßigkeit, Kontingenz und immanente Idea- lisierungstendenz. Dass sich so auch konkrete Naturphänomene erschließen, wird an zwei Exempeln verdeutlicht: in der Begründung dimensionaler Raumstrukturen und in der Begründung des Relativitätsprinzips der Bewe- gung – letzteres mit der Pointe (im Sinn einer Philosophie der (speziellen) Relativitätstheorie), dass die Relativität der Körperbewegung und die Absolutheit der Lichtbewegung so als logisch zusammengehörende Momente kinematischer Relativität verstehbar werden. (shrink)

Was ist Wahrheit?- Vernunft und Offenbarung.- Die Bedeutung von Bekenntnissenin der Gegenwart.

Note critiche alla filosofia sociale di Herbert Marcuse, di D. Ulle.--E' rivoluzionaria la dottrina di Marcuse? di J. Zamoshkin, N. Motroshilova.