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.

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.

Collected Documents on the History of the Pskov Orthodox Mission: A Recent Publication

2018 ◽  
pp. 306-312
Author(s):  
Veniamin F. Zima ◽  
Keyword(s):  
Personal Data ◽  
Prisoners Of War ◽  
Inner Life ◽  
Great Patriotic War ◽  
Occupied Territories ◽  
History Of ◽  
Underground Resistance ◽  
The Baltic ◽  
The Church ◽  
Insight Into

The reviewed work is devoted to a significant, and yet little-studied in both national and foreign scholarship, issue of the clergy interactions with German occupational authorities on the territory of the USSR in the days of the Great Patriotic War. It introduces into scientific use historically significant complex of documents (1941-1945) from the archive of the Office of the Metropolitan Sergius (Voskresensky) of Vilnius and Lithuania, patriarchal exarch in Latvia and Estonia, and also records from the investigatory records on charges against clergy and employees concerned in the activities of the Pskov Orthodox Mission (1944-1990). Documents included in the publication are stored in the archives of Moscow, St. Petersburg, Estonia, Lithuania, Leningrad, Novgorod, and Pskov regions. They allow some insight into nature, forms, and methods of the Nazi occupational regime policies in the conquered territories (including policies towards the Church). The documents capture religious policies of the Nazis and inner life of the exarchate, describe actual situation of population and clergy, management activities and counterinsurgency on the occupied territories. The documents bring to light connections between the exarchate and German counterintelligence and reveal the nature of political police work with informants. They capture the political mood of population and prisoners of war. There is information on participants of partisan movement and underground resistance, on communication net between the patriarchal exarchate in the Baltic states and the German counterintelligence. Reports and dispatches of the clergy in the pay of the Nazis addressed to the Metropolitan Sergius (Voskresensky) contain detailed activity reports. Investigatory records contain important biographical information and personal data on the collaborators. Most of the documents, being classified, have never been published before.

Jan II van Coninxloo en zijn werkzaamheden voor het benedictinessenklooster van Groot-Bijgaarden bij Brussel

1998 ◽  
Vol 112 (2-3) ◽  
pp. 104-126
Author(s):  
Frank Van Der Ploeg
Keyword(s):  
Significant Part ◽  
The Veil ◽  
First Time ◽  
The Relationship ◽  
The Church ◽  
Insight Into

AbstractThis article examines the relationship between the Brussels painter Jan 11 van Coninxloo (ca. 1489-1561 or later) and the Benedictine convent of Groot-Bijgaarden. In earlier publications by J. Maquet-Tombu the link between certain members of the Van Coninxloo family and the Vorst convent have already been pointed out. A new chapter can now be added. In the archive of Groot-Bijgaarden convent are two books in which payments made by the prioresses Françoise and Catherine van Straten for the dccoration of the convent and the church are recorded. The books list a separate item for painting and polychrome work. Here, for the first time, the name Jan van Coninxloo crops up in connection with a sum paid for painting the side panels of the main altar. Van Coninxloo was also paid for painting organ doors, a vaulted ceiling and for 'rough painting'. Four triptychs by Van Coninxloo have also been preserved; they were commissions from women of noble birth who had taken the veil. The names of three of these nuns are known: Anthonine de Locquenghien, Berbel van dcr Noot and Marie Brant. The fourth was called Barbara (Berbel). In view of all this material it may be concluded that Van Coninxloo played a significant part in the decorative appearance of the convent church. He was responsible for triptychs on altars dedicated to St. Anne, St. John and St. Benedict. He also painted the smaller triptych with the Seven Sorrows of the Virgin, the panels of the high altar, doors for an organ and (part of) the ceiling decorations. The article offers a new insight into the context of a group of paintings and adds a number of works to Jan 11 van Coninxloo's oeuvre.

scholarly journals Address of the President Sir Henry Dale, C.B.E., at the Anniversary Meeting, 1 December 1941

1942 ◽  
Vol 179 (978) ◽  
pp. 233-260 ◽  
Keyword(s):  
Technical College ◽  
High Speed ◽  
Large Scale ◽  
Electrical Engineering ◽  
The United States ◽  
The Past ◽  
Pure Mathematics ◽  
Symbolic Structure ◽  
The Church ◽  
Insight Into

This is the first occasion on which I have had the great honour of addressing the Royal Society on this anniversary of its foundation. According to custom, I begin with brief mention of those whom death has taken from our Fellowship during the past year, and whose memories we honour. Alfred Young (1873-1940), distinguished for his contributions to pure mathematics, was half brother to another of our Fellows, Sydney Young, a chemist of eminence. Alfred Young had an insight into the symbolic structure and manipulation of algebra, which gave him a special place among his mathematical contemporaries. After a successful career at Cambridge he entered the Church, and passed his later years in the country rectory of Birdbrook, Essex. His devotion to mathematics continued, however, throughout his life, and he published a steady stream of work in the branch of algebra which he had invented, and named ‘quantitative substitutional analysis’. He lived to see his methods adopted by Weyl in his quantum mechanics and spectroscopy. He was elected to our Fellowship in 1934. With the death of Miles Walker (1868-1941) the Society loses a pioneer in large-scale electrical engineering. Walker was a man of wide interests. He was trained first for the law, and even followed its practice for a period. Later he studied electrical engineering under Sylvanus Thompson at the Finsbury Technical College and became his assistant for several years. Thereafter, encouraged by Thompson, he entered St John’s College, Cambridge, with a scholarship, and graduated with 1st Class Honours in both the Natural Sciences and the Engineering Tripos. Having entered the service of the British Westinghouse Company, he was sent by them to the United States of America to study electrical engineering with the parent company in Pittsburgh. On his return to England he became their leading designer of high-speed electrical generators

Wounded Because of Religion: Identifying the Components of Gender-Specific Religious Persecution of Christians

2021 ◽  
Vol 38 (1) ◽  
pp. 119-141
Author(s):  
Helene Fisher ◽  
Elizabeth Lane Miller ◽  
Christof Sauer
Keyword(s):  
Central African Republic ◽  
Complex Dynamics ◽  
Holistic Approach ◽  
Christian Women ◽  
Religious Persecution ◽  
The World ◽  
Gender Specific ◽  
The Church ◽  
Insight Into

Abstract Emerging understanding of gender-specific religious persecution in some of the world’s most difficult countries for Christians offers timely insight into complex dynamics in which the church and missions have too often been unwittingly complicit due to limited visibility of the components contributing to these wounds. Fresh research into these deeply wounding global phenomena stands as both a warning and a pointer towards an avenue for effective ministrations by churches and Christian ministries that are working in the most severely affected areas of the world. Drawing on the latest trends identified by World Watch Research, outcomes of the Consultation for Christian Women under Pressure for their Faith, a contemporary case study from Central African Republic, and a biblical narrative, we will explore practical opportunities for a holistic approach to bring preparedness, healing, and restoration for communities under severe pressure for their Christian faith.

scholarly journals Verses of Faith and Devotion. Seeing, Reading, and Touching Monumental Crucifixes with Inscriptions (12th–13th century)

2016 ◽  
Vol 6 ◽  
pp. 397-421
Author(s):  
Matko Matija Marušić
Keyword(s):  
Distinctive Feature ◽  
Main Argument ◽  
13Th Century ◽  
The Subject ◽  
The Cross ◽  
The Church ◽  
Insight Into

The paper discusses a group of monumental crucifixes from the 13th-century East Adriatic and Italy, pained or executed in low relief, that display a verse inscriptions on the transverse limb of the cross. The main scope of the paper is to examine the provenance of the text inscribed in order to yield clearer insight into their function, use and original location in the church interiors. The paper specifically aims at analyzing three monumental crucifixes from the East-Adriatic city of Zadar which, although have already been the subject of a respectable number of studies, have not attracted attention as objects of devotion. My interest, therefore, is turned towards verse inscription as their distinctive feature and, as I shall argue, a key aspect in understanding their function. Examining the nature of the text displayed, iconography and materiality of these crucifixes, my main argument is to demonstrate how these objects provoked a multi-faced response from their audience, since were experienced by seeing, hearing and touching respectively.

scholarly journals To Leave Your Kindred and Your Father’s House. Contemporary Dutch Christian Border Narratives

2017 ◽  
Vol 6 ◽  
pp. 174-196
Author(s):  
Sjoerd-Jeroen Moenandar ◽  
Krina Huisman
Keyword(s):  
The Netherlands ◽  
Life Writing ◽  
Narrative Structure ◽  
Christian Faith ◽  
Border Crossings ◽  
Border Narratives ◽  
The Church ◽  
Insight Into ◽  
Dutch People ◽  
Christian Denomination

In this article the authors analyse a collection of essays written by young Dutch people who grew up in the Reformed Liberated Church, a small Christian denomination in the Netherlands. Traditionally, this church is characterised by its inwards nature: members strive to live their lives within the confinements that the church and its institutions stipulate. This has changed over the last few decades and the essays attest to the effects these changes have had on individual lives. We discuss the underlying narrative structure of their accounts and how the authors negotiate different lifestyles and interpretations of the Christian faith on either side of the borders that demarcate the Reformed Liberated tradition. We discuss if – and how – the essays work towards an outcome of ‘discordant concordance’ (Ricœur) where narrative identities remain whole, despite relatively drastic border crossings in the course of the lives that formed them. We address how these stories give insight into how people use the stories they tell to define what needs to be remembered and forgotten when we cross borders. Finally, we discuss the relevance of these essays and our analysis of them for our understanding of today’s globalised and multicultural societies in which many are in a permanent state of transition. This article was submitted to the European Journal of Life Writing on February 17th and published on August 28th 2017.

Eventually infinite time Turing machine degrees: infinite time decidable reals

2000 ◽  
Vol 65 (3) ◽  
pp. 1193-1203 ◽  
Author(s):  
P.D. Welch
Keyword(s):  
Turing Machine ◽  
Initial Segment ◽  
Upper Bounds ◽  
Turing Degrees ◽  
Turing Machines ◽  
Jump Operator ◽  
Infinite Time

AbstractWe characterise explicitly the decidable predicates on integers of Infinite Time Turing machines, in terms of admissibility theory and the constructible hierarchy. We do this by pinning down ζ, the least ordinal not the length of any eventual output of an Infinite Time Turing machine (halting or otherwise); using this the Infinite Time Turing Degrees are considered, and it is shown how the jump operator coincides with the production of mastercodes for the constructible hierarchy; further that the natural ordinals associated with the jump operator satisfy a Spector criterion, and correspond to the Lζ-stables. It also implies that the machines devised are “Σ2 Complete” amongst all such other possible machines. It is shown that least upper bounds of an “eventual jump” hierarchy exist on an initial segment.

Conclusions

2020 ◽  
pp. 240-258
Author(s):  
Mary E. Sommar
Keyword(s):  
Twentieth Century ◽  
Church History ◽  
Historical Background ◽  
General Reader ◽  
History Of ◽  
Specialist Knowledge ◽  
Church Law ◽  
The Church ◽  
Historical Periods ◽  
Insight Into

This is the story of how the church sought to establish norms for slave ownership on the part of ecclesiastical institutions and personnel and for others’ behavior toward such slaves. Chronicles, letters, and other documents from each of the various historical periods, along with an analysis of the various policies and statutes, provide insight into the situations of these unfree ecclesiastical dependents. Although this book is a serious scholarly monograph about the history of church law, it has been written in such a way that no specialist knowledge is required of the reader, whether a scholar in another field or a general reader interested in church history or the history of slavery. Historical background is provided, and there is a short Latin lexicon. This chapter summarizes the conclusions drawn in earlier chapters and provides a brief overview of the question of ecclesiastical servitude up to the twentieth century.

Carolingians and Ecclesiastical Servitude

2020 ◽  
pp. 154-189
Author(s):  
Mary E. Sommar
Keyword(s):  
Church History ◽  
Historical Background ◽  
British Isles ◽  
General Reader ◽  
History Of ◽  
Specialist Knowledge ◽  
Church Law ◽  
The Church ◽  
Historical Periods ◽  
Insight Into

This is the story of how the church sought to establish norms for slave ownership on the part of ecclesiastical institutions and personnel and for others’ behavior toward such slaves. Chronicles, letters, and other documents from each of the various historical periods, along with an analysis of the various policies and statutes, provide insight into the situations of these unfree ecclesiastical dependents. Although this book is a serious scholarly monograph about the history of church law, it has been written in such a way that no specialist knowledge is required of the reader, whether a scholar in another field or a general reader interested in church history or the history of slavery. Historical background is provided, and there is a short Latin lexicon. This chapter discusses slavery in the Carolingian Empire (c. 750–900) and includes an overview of slavery in the British Isles in the first Christian millennium.

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