On the notational independence of various hierarchies of degrees of unsolvability

1965 ◽  
Vol 30 (1) ◽  
pp. 69-86
Author(s):  
Gustav Hensel ◽  
Hilary Putnam
Keyword(s):  
Natural Number ◽  
Point Of View ◽  
Arithmetic Hierarchy ◽  
The Church ◽  
Degrees Of Unsolvability ◽  
Number Class ◽  
Natural Way

From the work of Kleene, Post and Davis it is well-known that the arithmetic sets can be characterized as those sets recursive in ∅(n) for some natural number n, where ∅(0) = ∅ and . Actually the arithmetic sets which can be expressed in prenex form with n alternating quantifiers (applied to recursive predicates) are recursive in ∅(n). Hence, starting with ∅, the “jump operation”, which takes a set A into the set , serves to increase the “complexity” of sets in a uniform way.As far as extending the ω-sequence of degrees of unsolvability (i.e. the degrees represented by the ∅(n)) into the 2nd number class, there is an immediate problem. One knows from Spector [17], corollary 2, p. 585, that there is no l.u.b. for the ω-sequence. So, at ω one must pick a degree in some other “natural” way. Unfortunately, what has seemed “natural” to some mathematicians has not seemed natural to others. Kleene and Davis (cf. [10] and [3] respectively) extended this arithmetic hierarchy of degrees of unsolvability by making use of natural number notations for a certain segment of ordinals of the 1st and 2nd number classes, the “constructive” ordinals. Using the Church-Kleene system S3, one can “sum up” previously obtained sets at the limit notations in a way that is certainly natural from a notational point of view.

Systems of notations and the ramified analytical hierarchy

1974 ◽  
Vol 39 (2) ◽  
pp. 243-253 ◽  
Author(s):  
Joan D. Lukas ◽  
Hilary Putnam
Keyword(s):  
Initial Segment ◽  
Turing Degrees ◽  
Power Set ◽  
The Church ◽  
Insight Into ◽  
Degrees Of Unsolvability ◽  
Number Class

The purpose of this paper is to show that arithmetically minimal systems of notations can be constructed which provide notations for all ramified analytical ordinals (all the ordinals in the minimum β-model for analysis). This is a much larger section of the second number class than the Church-Kleene constructive ordinals (although still only an initial segment of the ordinals). Arithmetic minimality means that if H is an “H-set” associated with an ordinal α in our system and H′ is an H-set associated with the same ordinal α in an arbitrary system of notations S, then H is arithmetical in H′. Thus the arithmetical degrees associated with ordinals in our system are as low as possible.In order to clarify the structure of degrees of unsolvability and, more generally, to gain a deeper insight into the power set of the integers, coarser but neater classifications than the structure of Turing degrees have been sought. Several hierarchies of sets of integers have been studied, each of which organizes a certain class of sets (or their degrees of unsolvability) into a well-ordering of levels with increasing complexity of nonrecursiveness appearing at each new level. The best known of these hierarchies is the Kleene hierarchy of arithmetical sets.

Canons to the Left of Him: Brevard Childs, his Critics, and the Future of Old Testament Theology

1997 ◽  
Vol 50 (1) ◽  
pp. 1-38 ◽  
Author(s):  
Iain Provan
Keyword(s):  
Old Testament ◽  
18Th Century ◽  
Biblical Theology ◽  
Point Of View ◽  
The Bible ◽  
The Difference ◽  
Probable Origin ◽  
The Individual ◽  
The Relationship ◽  
The Church

It is well known that the seeds from which the modern discipline of OT theology grew are already found in 17th and 18th century discussion of the relationship between Bible and Church, which tended to drive a wedge between the two, regarding canon in historical rather than theological terms; stressing the difference between what is transient and particular in the Bible and what is universal and of abiding significance; and placing the task of deciding which is which upon the shoulders of the individual reader rather than upon the church. Free investigation of the Bible, unfettered by church tradition and theology, was to be the way ahead. OT theology finds its roots more particularly in the 18th century discussion of the nature of and the relationship between Biblical Theology and Dogmatic Theology, and in particular in Gabler's classic theoreticalstatementof their nature and relationship. The first book which may strictly be called an OT theology appeared in 1796: an historical discussion of the ideas to be found in the OT, with an emphasis on their probable origin and the stages through which Hebrew religious thought had passed, compared and contrasted with the beliefs of other ancient peoples, and evaluated from the point of view of rationalistic religion. Here we find the unreserved acceptance of Gabler's principle that OT theology must in the first instance be a descriptive and historical discipline, freed from dogmatic constraints and resistant to the premature merging of OT and NT — a principle which in the succeeding century was accepted by writers across the whole theological spectrum, including those of orthodox and conservative inclination.

ON ROBERTS’ PROOF OF THE TURAEV-WALKER THEOREM

1996 ◽  
Vol 05 (04) ◽  
pp. 427-439 ◽  
Author(s):  
RICCARDO BENEDETTI ◽  
CARLO PETRONIO
Keyword(s):  
Euler Characteristic ◽  
A Priori ◽  
Algebraic Approach ◽  
Fusion Rule ◽  
Point Of View ◽  
Witten Invariant ◽  
Manifolds With Boundary ◽  
3 Dimensional ◽  
Skein Theory ◽  
Natural Way

In this paper we discuss the beautiful idea of Justin Roberts [7] (see also [8]) to re-obtain the Turaev-Viro invariants [11] via skein theory, and re-prove elementarily the Turaev-Walker theorem [9], [10], [13]. We do this by exploiting the presentation of 3-manifolds introduced in [1], [4]. Our presentation supports in a very natural way a formal implementation of Roberts’ idea. More specifically, what we show is how to explicitly extract from an o-graph (the object by which we represent a manifold, see below), one of the framed links in S3 which Roberts uses in the construction of his invariant, and a planar diagrammatic representation of such a link. This implies that the proofs of invariance and equality with the Turaev-Viro invariant can be carried out in a completely “algebraic” way, in terms of a planar diagrammatic calculus which does not require any interpretation of 3-dimensional figures. In particular, when proving the “term-by-term” equality of the expansion of the Roberts invariant with the state sum which gives the Turaev-Viro invariant, we simultaneously apply several times the “fusion rule” (which is formally defined, strictly speaking, only in diagrammatic terms), showing that the “braiding and twisting” which a priori may exist on tetrahedra is globally dispensable. In our point of view the success of this formal “algebraic” approach witnesses a certain efficiency of our presentation of 3-manifolds via o-graphs. In this work we will widely use recoupling theory which was very clearly exposed in [2], and therefore we will avoid recalling notations. Actually, for the purpose of stating and proving our results we will need to slightly extend the class of trivalent ribbon diagrams on which the bracket can be computed. We also address the reader to the references quoted in [2], in particular for the fundamental contributions of Lickorish to this area. In our approach it is more natural to consider invariants of compact 3-manifolds with non-empty boundary. The case of closed 3-manifolds is included by introducing a correction factor corresponding to boundary spheres, as explained in §2. Our main result is actually an extension to manifolds with boundary of the Turaev-Walker theorem: we show that the Turaev-Viro invariant of such a manifold coincides (up to a factor which depends on the Euler characteristic) with the Reshetikhin-Turaev-Witten invariant of the manifold mirrored in its boundary.

scholarly journals Control Systems and Number Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-28
Author(s):  
Fuhuo Li
Keyword(s):  
Zeta Functions ◽  
Feedback System ◽  
State Equation ◽  
The State ◽  
Point Of View ◽  
Pid Controllers ◽  
Primary Concern ◽  
Feedback Connection ◽  
Proper Understanding ◽  
Natural Way

We try to pave a smooth road to a proper understanding of control problems in terms of mathematical disciplines, and partially show how to number-theorize some practical problems. Our primary concern is linear systems from the point of view of our principle of visualization of the state, an interface between the past and the present. We view all the systems as embedded in the state equation, thus visualizing the state. Then we go on to treat the chain-scattering representation of the plant of Kimura 1997, which includes the feedback connection in a natural way, and we consider theH∞-control problem in this framework. We may view in particular the unit feedback system as accommodated in the chain-scattering representation, giving a better insight into the structure of the system. Its homographic transformation works as the action of the symplectic group on the Siegel upper half-space in the case of constant matrices. Both ofH∞- and PID-controllers are applied successfully in the EV control by J.-Y. Cao and B.-G. Cao 2006 and Cao et al. 2007, which we may unify in our framework. Finally, we mention some similarities between control theory and zeta-functions.

scholarly journals The hermit сharacter as a new type of character in realistic prose by Maxim Gorky

2021 ◽  
Vol 9 (SPE2) ◽  
Author(s):  
Svetlana Strokina ◽  
Lenie Taymazova ◽  
Elvina Useinova ◽  
Ruslan Adonin
Keyword(s):  
Point Of View ◽  
Art Form ◽  
New Type ◽  
Maxim Gorky ◽  
First Time ◽  
The Church

1922-1924 was a fruitful period of Maxim Gorky's literature work. It is related to searching a new art form. The cycle “Stories of 1922-1924” is an expressive example of “new prose”. For the first time, the hermit character appeared in the cycle “Stories of 1922-1924”. From the point of view of generally recognized morality and the Church, the new type of character is ambiguous. It is characterized by both sinfulness and holiness.

scholarly journals The view of the Ecumenical Synods on the state

1972 ◽  
Vol 6 ◽  
pp. 121-127
Author(s):  
Metropolitan of Helsinki Johannes
Keyword(s):  
The State ◽  
Point Of View ◽  
Basic Principles ◽  
The Mind ◽  
The Church

When one wants to learn to know the basic principles of the Ancient, undivided Church concerning the State, as authoritatively and validly as possible, one has to turn to the teachings of the so-called Ecumenical Synods, which is the term used of those great synods of the bishops which were recognised as representing the mind of the Church and whose declarations and rulings thus were—and are, from the Orthodox point of view—binding on the whole Church. The period, during which the said synods of the ancient Church took place, falls between the 4th and the 8th centuries, the latest of them being held in 787.

scholarly journals The School Culture Heritage of the Reformation

2017 ◽  
Vol 7 (1) ◽  
pp. 116-121
Author(s):  
László Trencsényi
Keyword(s):  
Point Of View ◽  
16Th Century ◽  
Church Reform ◽  
College Culture ◽  
Educational Innovations ◽  
The Reformation ◽  
Culture Heritage ◽  
History Of ◽  
Contemporary Education ◽  
The Church

Abstract On the occasion of the 500th anniversary of the Reformation, this essay analyses those educational innovations in the history of central European education that were introduced by the Church reform in the 16th century, following these modernizations and their further developments through the spreading of the universal school systems in the late 18th and early 19th centuries. Drawing examples from the innovations in the college culture of the period, the author emphasises that those pedagogical values established in the 16th century are not only valid today, but are exemplary from the point of view of contemporary education. From these the author highlights: pupils’ autonomy (in the form of various communities), cooperation with the teachers and school management and the relative pluralism of values.

scholarly journals Liturgical renovation of modern churches in Rome (Italy)

2019 ◽  
Vol 2 (3) ◽  
pp. 39
Author(s):  
Daniela Concas
Keyword(s):  
Twentieth Century ◽  
Roman Catholic ◽  
Point Of View ◽  
Second Vatican Council ◽  
Direct Reading ◽  
Liturgical Movement ◽  
Vatican Council ◽  
Architectural Language ◽  
Historical Moment ◽  
The Church

At the beginning of the first half of the twentieth century the bond between ars-venustas and cultus-pietas has produced many churches of Roman Catholic cult.It’s between the 20s and 60s of the twentieth century that the experiments of the Liturgical Movement in Germany lead to the evolution of the liturgical space, which, even today, we see engraving in modern churches in Rome (Italy).The Council of Trent (1545-1563) constitutes the precedent historical moment, in which the Church recognised the need for major liturgical renovation of its churches. In comparison with this, the Second Vatican Council (1959-65) introduced some radical changes within the church architectural spaces.The observations come from the direct reading of the present architectural space and the interventions already realised in modern churches in Rome. The most significant churches from an historical-artistic point of view were selected (1924-1965). Significantly, although every single architecture is unique for dimensions, architectural language and used materials, a comparison, in order to gather the discovered characteristics and to compare the restrictions regarding the different operations, would extremely effective, as demonstrated below.Since the matter is considerably vast, in this work, only some brief notes regarding the liturgical renovation of the Presbytery area will be outlined.

scholarly journals A pastoral investigation into some of the challenges associated with aging and retirement in the South African context

2015 ◽  
Vol 49 (3) ◽  
Author(s):  
Magezi E. Baloyi
Keyword(s):  
South African ◽  
Elderly People ◽  
Family Relationships ◽  
Point Of View ◽  
South Africans ◽  
Black South Africans ◽  
African Context ◽  
Long Life ◽  
The Church

This article is a contribution to a project that congratulates from the work of George Lotter, a pastor, pastoral counsellor and academic who wrote much on matters relating to pastoral care and counselling. Elderliness and retirement can be understood as a period in the lives of elderly people that allows them to rest after a long life of activity and service. From another point of view, old age is also a time that offers pastoral caregivers an opportunity to care for people who have contributed to their families and society. Pastoral caregivers have an important role to play in the lives of elderly people. This applies particularly to elderly black South Africans, who often find themselves confronted by poverty and other related problems. This article investigates the challenges and problems affecting retired and elderly black South Africans with particular focus on the economic impact of aging and its influence on family relationships in the lives of elderly people. To conclude: it is the duty of pastoral caregivers to search for and establish guidelines for the roles the church can play in improving elderly people’s quality of life.’n Pastorale ondersoek na enkele van die uitdagings ten opsigte van veroudering en aftredein die Suid-Afrikaanse konteks. Hierdie artikel is ’n bydrae tot ’n projek wat voorspruit uit die werk van George Lotter, ’n pastor, pastorale berader en akademikus wat baie geskryf het oor kwessies wat met pastorale sorg en berading verband hou. Bejaardheid en aftrede kan verstaan word as ’n tyd in in ouer persone se lewe waartydens hulle kan rus ná ’n lang aktiewe en diensbare lewe. Bejaardheid kan ook gesien word as ’n tyd wat aan pastorale versorgers die geleentheid gee om na die mense wat bygedra het tot hulle families en die gemeenskap se versorging, om te sien. Pastorale versorgers speel ’n belangrike rol in die lewens van bejaardes. Dit is spesifiek van toepassing op bejaarde swart Suid-Afrikaners wat dikwels gekonfronteer word met armoede en aanverwante probleme. Hierdie artikel ondersoek die uitdagings en probleme waarmee bejaarde swart Suid-Afrikaanse afgetredenes te kampe het. Dit fokus spesifiek op die ekonomiese uitwerking van veroudering en die invloed wat dit op familieverhoudinge in die lewens van bejaardes het. Die slotsom word gemaak dat dit die plig van pastorale versorgers is om riglyne te soek en te bied vir die rol wat die kerk kan speel ter verbetering van die lewensomstandighede van die bejaardes.

Thomas Fuller as Royalist Country Parson During the Interregnum

1979 ◽  
Vol 16 ◽  
pp. 301-314
Author(s):  
W. B. Patterson
Keyword(s):  
Civil War ◽  
Late Summer ◽  
Political Conflict ◽  
Point Of View ◽  
First Year ◽  
The City ◽  
The Church

Although Thomas Fuller, the church historian, spent the first year of the civil war in London, where he articulated from the pulpit a political point of view consonant with that of the parliamentary peace party, there can be little doubt that his allegiance was with the king in that struggle. In the late summer of 1643 Fuller left London for the royalist capital at Oxford and before the end of the year entered the service of Lord Hopton as a chaplain in the royal army. During the latter stages of the civil war he resided in Exeter, where he served as chaplain to the infant princess Henrietta Anne, and where he enjoyed close relations with members of the court circle there. Fuller left the royalist community in Exeter only when the city itself surrendered in 1646, two months before the fall of Oxford. Because of these activities and because of a series of plainspoken books and pamphlets during the years of religious and political conflict, Fuller was widely known as an adherent of the royalist cause, albeit never as militant or as uncritical a partisan as many others in the king’s camp.

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