The first main result isolates some conditions which fail for the class of graphs and hold for the class of Abelian p-groups, the class of Abelian torsion groups, and the special class of "rank-homogeneous" trees. We consider these conditions as a possible definition of what it means for a class of structures to have "Ulm type". The result says that there can be no Turing computable embedding of a class not of Ulm type into one of Ulm type. We apply (...) this result to show that there is no Turing computable embedding of the class of graphs into the class of "rank-homogeneous" trees. The second main result says that there is a Turing computable embedding of the class of rank-homogeneous trees into the class of torsion-free Abelian groups. The third main result says that there is a "rank-preserving" Turing computable embedding of the class of rank-homogeneous trees into the class of Boolean algebras. Using this result, we show that there is a computable Boolean algebra of Scott rank ${\mathrm{\omega }}_{1}^{\mathrm{C}\mathrm{K}}$. (shrink)

This paper calculates, in a precise way, the complexity of the index sets for three classes of computable structures: the class $K_{\omega _{1}^{\mathit{CK}}}$ of structures of Scott rank $\omega _{1}^{\mathit{CK}}$ , the class $K_{\omega _{1}^{\mathit{CK}}+1}$ of structures of Scott rank $\omega _{1}^{\mathit{CK}}+1$ , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete $\Sigma _{1}^{1},\,I(K_{\omega _{1}^{\mathit{CK}}})$ is m-complete $\Pi _{2}^{0}$ relative to Kleen's O, and $I(K_{\omega _{1}^{\mathit{CK}}+1})$ is m-complete $\Sigma _{2}^{0}$ relative to O.

LetEbe a computably enumerable equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized (...) byE. One can also define the following pre-order$ \le _{lo} $on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2 $if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$ \le _{lo} $. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees. (shrink)

Defining the degree of categoricity of a computable structure ${\mathcal{M}}$ to be the least degree d for which ${\mathcal{M}}$ is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d.c.e. in and above 0 (n) can be so realized, as can the degree 0 (ω).

We study the structure of Σ11 equivalence relations on hyperarithmetical subsets of ω under reducibilities given by hyperarithmetical or computable functions, called h-reducibility and FF-reducibility, respectively. We show that the structure is rich even when one fixes the number of properly equation imagei.e., Σ11 but not equation image equivalence classes. We also show the existence of incomparable Σ11 equivalence relations that are complete as subsets of ω × ω with respect to the corresponding reducibility on sets. We study complete Σ11 (...) equivalence relations and show that existence of infinitely many properly Σ11 equivalence classes that are complete as Σ11 sets is necessary but not sufficient for a relation to be complete in the context of Σ11 equivalence relations. (shrink)

We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure A as the family of Turing degrees that compute embeddings between any computable bi-embeddable copies of A; the degree of bi-embeddable categoricity of A is the least degree in this spectrum (if it exists). We extend many known results about categoricity spectra to the case of bi-embeddability. In particular, we exhibit structures without degree of bi-embeddable categoricity, and (...) we show that every degree d.c.e above 0(α) for α a computable successor ordinal and 0(λ) for λ a computable limit ordinal is a degree of bi-embeddable categoricity. We also give examples of families of degrees that are not bi-embeddable categoricity spectra. (shrink)

We study arithmetic and hyperarithmetic degrees of categoricity. We extend a result of E. Fokina, I. Kalimullin, and R. Miller to show that for every computable ordinal $\alpha$, $\mathbf{0}^{}$ is the degree of categoricity of some computable structure $\mathcal{A}$. We show additionally that for $\alpha$ a computable successor ordinal, every degree $2$-c.e. in and above $\mathbf{0}^{}$ is a degree of categoricity. We further prove that every degree of categoricity is hyperarithmetic and show that the index set of structures with (...) degrees of categoricity is $\Pi_{1}^{1}$-complete. (shrink)

A Turing degreedis the degree of categoricity of a computable structure${\cal S}$ifdis the least degree capable of computing isomorphisms among arbitrary computable copies of${\cal S}$. A degreedis the strong degree of categoricity of${\cal S}$ifdis the degree of categoricity of${\cal S}$, and there are computable copies${\cal A}$and${\cal B}$of${\cal S}$such that every isomorphism from${\cal A}$onto${\cal B}$computesd. In this paper, we build a c.e. degreedand a computable rigid structure${\cal M}$such thatdis the degree of categoricity of${\cal M}$, butdis not the strong degree of categoricity (...) of${\cal M}$. This solves the open problem of Fokina, Kalimullin, and Miller [13].For a computable structure${\cal S}$, we introduce the notion of the spectral dimension of${\cal S}$, which gives a quantitative characteristic of the degree of categoricity of${\cal S}$. We prove that for a nonzero natural numberN, there is a computable rigid structure${\cal M}$such that$0\prime$is the degree of categoricity of${\cal M}$, and the spectral dimension of${\cal M}$is equal toN. (shrink)

A critical study of McPeck's recent book, in which he strengthens and develops his arguments against teaching critical thinking (CT). Accepting McPeck's basic claim that there is no unitary skill of reasoning or thinking, I argue that his strictures on CT courses or programs do not follow. I set out what I consider the proper justification that programs in CT have to meet, and argue both that McPeck demands much more than is required, and also that it is plausible that (...) this deflated justification can be met. Specitically, I argue that it is reasonable to expect transfer of learning for basic logical skills. Additional topics covered include: the relation ofliberal education to critical thinking, argument analysis, testing for CT, and the value of conceptual or linguistic analysis. (shrink)

In R. C. Flickinger's (3rd ed. second impression 1929) we read on pp. 171In the iambic trimeters written by Aeschylus a trisyllabic substitution (tribrach, anapaest or dactyl) for the pure disyllabic iambus occurs only once in about twenty-five verses.Tragic Drama of the Greekstrimeters’ lines like.

M. Tullius Cicero was the first among the Roman thinkers, who made the pro_blematic of culture and civilization his serious concern. His investigations led him to the belief that the culture, seen as a whole of traditions, norms and values, is in_separable from artes, i. e. the spiritual phenomena in science and art. From his awarness of the necesssity to create the Roman culture in this sense of inseparability resulted his ambitious philosophical project: the Romans should accept the spiritual experience (...) of the Greeks and develop it in philosophical writings in Latin language. To all doubts and prejudices hostile to this intention he thus responded by philosophical and linguistic writings based on his conviction, that the Latin language is able to express even the most complicated philosophical ideas and concepts and that the human interaction is possible. (shrink)

Se trata de unareflexión interdisciplinar, a partir de W. Benjamin, sobre las relaciones entre experiencia, tiempo y memoriahistórica. La 1. parte analiza el empobrecimiento moderno de la Erfahrung que genera una“nueva barbarie”, a la vez que expenmentación innovadora, y que reclama un concepto más rico de experiencia, ligada ala totalidad concrete de la existencia. La 2. señala algunas paradojas de la memoria, muestra la inconsistencia del tiempo, cristalizado en el mito de Cronos, comoprincipio ordenador, y toma en consideración experiencias relevantes (...) que requieren concebir otro tiempo. En esa dirección, la 3. parte desarrolla la idea de Jetztze¡t, nucleo de la concepción crítica de la historia como tradición de los oprimidos, en cuyos bistantes de peligro relanipagea el angelus novus.Il s’agit d’une reflexion interdisciplinaire sur les liaisons entre expérience, temps et bistoire. La lére partie analyse l’appauvrissement moderne de l’Erfahrung, qui produit, selon W. Benjaniin, experimentation créatice et á la fois “une nouvelle barbarie”, et qui exige la construction d’un concept plus tiche d’expérience, liée á la totalité concréte de l’existence. La 2&me ¡nontre quelques paradoxes de la mémoire, sovligne ]‘inconsistance dv lemps comino principe d’ordre, et rappelle l’existence d’expériences qui demandent un autre concept du temps. La 3éme partie developpe ¡‘idee benjaininienne de Jetztzeit, noyau de sa conception critique de l’histoire conune tradition des opprimés, et analyse particuliérement la figure de l’angelus novus. (shrink)

It has frequently been observed that the Prometheus Vinctus shows certain Sophoclean characteristics of rhythm. In order to vary the rhythm of his iambics and to avoid monotony, Sophocles often knits consecutive trimeters closely together by placing at the end of one line some word which looks forward to the next line, and so precludes the reader from stopping for the natural pause after the sixth foot. Sometimes he uses in this way subordinating words which introduce a dependent clause in (...) the next line ; sometimes proclitics or quasi-proclitics. (shrink)

Ongevraagde ouverture, door F. O. van Gennep.--Ethiek als waagstuk, door F. van Raalten.--Het juridisch aspect van de rabbijnse ethiek, door M. A. Beek.--Is er een eigen christelijke ethiek? Door H. J. Heering.--Het radicalisme in de ethiek, door F. O van Gennep.--Ethiek en praktijk van de geweldloosheid van Dr. Martin Luther King, door J. W. Schulte Nordholt.--Politiek pastoraat, door K. Strijd.--Mythe en ethiek, door F. van Raalten.--Veritas poetica, door C. W. Mönnich.--Ethiek in de counseling, door H. R. Wijngaarden.--Bibliografische gegevens uit het (...) werk van Prof. Dr. E. L. Smelik (p. 141-142). (shrink)