The Cantor–Bernstein theorem: how many proofs?

2019 ◽  
Vol 377 (2140) ◽  
pp. 20180031 ◽  
Author(s):  
Wilfried Sieg
Keyword(s):  
Argument Structure ◽  
Proof Theory ◽  
Careful Analysis ◽  
Methodological Framework ◽  
Theme Issue ◽  
Bernstein Theorem ◽  
Computational Tools ◽  
Chain Theory ◽  
The Many

Dedekind's proof of the Cantor–Bernstein theorem is based on his chain theory, not on Cantor's well-ordering principle. A careful analysis of the proof extracts an argument structure that can be seen in the many other proofs that have been given since. I contend there is essentially one proof that comes in two variants due to Dedekind and Zermelo , respectively. This paper is a case study in analysing proofs of a single theorem within a given methodological framework, here Zermelo–Fraenkel set theory (ZF). It uses tools from proof theory, but focuses on heuristic ideas that shape proofs and on logical strategies that help to construct them. It is rooted in a perspective on Beweistheorie that predates its close connection and almost exclusive attention to the goals of Hilbert's finitist consistency programme. This earlier perspective can be brought to life (only) with the support of powerful computational tools. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

THE IMPACT OF MESSAGE-BUFFER ALIGNMENT ON COMMUNICATION PERFORMANCE

2005 ◽  
Vol 15 (01n02) ◽  
pp. 49-65 ◽  
Author(s):  
LEON ARBER ◽  
SCOTT PAKIN
Keyword(s):  
Network Performance ◽  
Performance Test ◽  
Careful Analysis ◽  
Communication Performance ◽  
Accepted Practice ◽  
The Many ◽  
Performance Discrepancy ◽  
Relationship Of ◽  
The Impact

Of the many factors that contribute to communication performance, perhaps one of the least investigated is that of message-buffer alignment. Although the generally accepted practice is to page-align buffer memory for best performance, our studies show that the actual relationship of buffer alignment to communication performance cannot be expressed with such a simple formula. This paper presents a case study in which porting a simple network performance test from one language to another resulted in a large performance discrepancy even though both versions of the code consist primarily of calls to messaging-layer functions. Careful analysis of the two code versions revealed that the discrepancy relates to the alignment in memory of the message buffers. Further investigation revealed some surprising results about the impact of message-buffer alignment on communication performance: (1) different networks and node architectures prefer different buffer alignments; (2) page-aligned memory does not always give the best possible performance, and, in some cases, actually yields the worst possible performance; and, (3) on some systems, the most significant factor affecting network performance is the relative alignment of send and receive buffers with respect to each other.

Coroboration of magnetotelluric investigations with other geophysical anomalies for a case study located in North-West to central part of Romania

2020 ◽  
Author(s):  
Natalia-Silvia Asimopolos ◽  
Laurentiu Asimopolos
Keyword(s):  
Transition Zone ◽  
Sedimentary Cover ◽  
Careful Analysis ◽  
Geological Structure ◽  
North West ◽  
The North ◽  
The Many ◽  
Geophysical Anomalies

<p>The results of the magnetoteluric investigations carried out along the profiles are presented in the form of sections in which the variations of the different parameters, the 2-D modeling, as well as the inversions.</p><p>The results of the geophysical researches (magnetotellurics, gravity, geomagnetics) obtained were aimed at obtaining a unitary image on the deep geological structure in the investigated area. A number of information was obtained regarding such as:</p><ul><li>Determination of the thickness of the package of formations belonging to the post-tectogenetic sedimentary cover of the Transylvanian Depression; sedimentary sedimentary cover, conductive, with a maximum thickness of approximately 4000 m in the Pannonian Depression;</li> <li>Contouring of the Tethysian Major Suture (near the town of Alba Iulia in the Transylvanian Basin), represented by the Transylvanian nappes system (ofiolitic complex and sedimentary formations), with resistivities of about 500 Ohm*m, which separates two blocks with continental crust of different thicknesses (22- 27 km for Internal Dacids and 32-36 km for Median Dacids);</li> <li>Highlighting the change of nappes systems belonging to the Transylvanians, with a wide development both to the north (Căpâlnas-Techereu nappes and the nappes of Groşi and Criş), as well as to the east (the ophiolite complex and sedimentary cover), over the Biharia nappes system, respectively Central-Eastern Carpathian nappes; extension of the Codru and Biharia - Arieşeni nappes, the last with higher resistivities (200 Ohm*m);</li> <li>Highlighting the transcrustal fault that marks the contact between the Inner Dacides and the Median ones;</li> <li>Individualization at the level of the lower crust of a transition zone; significant decrease of resistivity, as a consequence of the presence of the fluids in the transition zone, from the pressure in the pores from lithostatic type to the hydrostatic type (occurs at depths of 22 - 30 km and at temperatures of 350º - 400º C).</li> </ul><p>The electromagnetic data reflect the anomalies of electrical conductivity in a sensitive way, but due to the many causes that can generate them, a careful analysis of the particularities existing for each case, especially the superficial ones, was necessary.</p><p>The correlation of the all the information provided in sections (resistivity, phases, density, magnetic susceptibility), inversions, modeling, lead to the validation of the model.</p>

NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

2019 ◽  
pp. 1-35
Author(s):  
WILFRIED SIEG ◽  
PATRICK WALSH
Keyword(s):  
Proof Theory ◽  
Mathematical Proof ◽  
Formal Proofs ◽  
Bernstein Theorem ◽  
Search System ◽  
Inference Mechanism ◽  
Mathematical Proofs ◽  
Computational Work ◽  
Definitional Extension

Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

The Nonconformist Inheritance

2018 ◽  
Author(s):  
Isabel Rivers
Keyword(s):  
Eighteenth Century ◽  
Seventeenth Century ◽  
Church Of England ◽  
John Bunyan ◽  
John Wesley ◽  
Religious Attitudes ◽  
Pilgrim's Progress ◽  
The Many ◽  
John Owen

This chapter analyses the editions, abridgements, and recommendations of texts by seventeenth-century nonconformists that were made by eighteenth-century dissenters, Methodists, and Church of England evangelicals. The nonconformist writers they chose include Joseph Alleine, Richard Baxter, John Flavel, John Owen, and John Bunyan. The editors and recommenders include Philip Doddridge, John Wesley, Edward Williams, Benjamin Fawcett, George Burder, John Newton, William Mason, and Thomas Scott. Detailed accounts are provided of the large number of Baxter’s works that were edited, notably A Call to the Unconverted and The Saints Everlasting Rest, and a case study is devoted to the many annotated editions of Bunyan’s Pilgrim’s Progress and the ways in which they were used. The editors took into account length, intelligibility, religious attitudes, and cost, and sometimes criticized their rivals’ versions on theological grounds.

scholarly journals Morphological Change of Dhaka City Over a Period of 55 Years: A Case Study of Two Wards

1970 ◽  
Vol 2 ◽  
pp. 30-38 ◽  
Author(s):  
Bayes Ahmed ◽  
Muhammad Rakibul Hasan Raj ◽  
KM Maniruzzaman
Keyword(s):  
Urban Form ◽  
Spatial Dynamics ◽  
Morphological Structure ◽  
General Interest ◽  
Morphologic Change ◽  
Dhaka City ◽  
Territorial Expansion ◽  
Physical Form ◽  
The Many

Dhaka City has undergone radical changes in its physical form, not only by territorial expansion, but also through internal physical transformations over the last decades. These have created entirely new kinds of fabric. With these changes, the elements of urban form have changed. Plots and open spaces have been transformed into building areas, open squares into car parks, low land and water bodies into reclaimed built-up lands etc. This research has its general interest in the morphologic change of Dhaka City. It focuses on the spatial dynamics of urban growth of Dhaka over the last 55 years from 1952-2007. In the research, the transformation of urban form has been examined through space syntax. The aim behind using this technique is to describe aspects of relationships between the morphological structure of man-made environments and social structures and events. To conduct this research, Wards 49 and 72 of Dhaka City Corporation were selected as the study areas, of which Ward 72 is an indigenous and Ward 49 is a planned type of settlement. Being a planned residential area, the syntactic measures from this morphological analysis are showing quite unchanged and high values in all phases for Ward 49 and the physical characteristics of Ward 72 (Old Dhaka) still represent the past. The syntactic values are found to be higher for Ward 72 and than Ward 49. Higher values indicate that the street network is highly connective among each other. Time affects differently the layout of cities and the architecture of buildings. Of the many human creations, street systems are among the most resistant to change. This has been emphasized in this study, thereby facilitating the comparison of urban layouts across space and time. The interpretation of history in the light of quantitative accounts, as demonstrated in this study, will be of value to urban planners and urban designers for the future planning of modern Dhaka City.DOI: http://dx.doi.org/10.3329/jbip.v2i0.9554  Journal of Bangladesh Institute of Planners Vol. 2, December 2009, pp. 30-38

Logistics management in China: A case study of Haier

2004 ◽  
Vol 23 (1) ◽  
pp. 15-27
Author(s):  
Jason C.H. Chen ◽  
Binshan Lin ◽  
Lingli Li ◽  
Patty S. Chen
Keyword(s):  
Communication Networks ◽  
Global Economy ◽  
Collaborative Advantage ◽  
Logistics Systems ◽  
Business Partners ◽  
Business Operations ◽  
Global Brand ◽  
The Many ◽  
Foreign Competitors

Chinese businesses began with a weak foundation in the intense world trade environment, similar to the many other companies that grew from developing countries. How were these Chinese businesses able to compete with foreign competitors armed with strong capital structures and efficient communication networks? Haier is an excellent example of how Chinese companies have successfully adapted to and prospered in the global economy, using information technology as a strategic weapon to improve its competitive advantage and further to create collaborative advantage. Haier's growth is miraculous: in less than two decades, it grew from a state-owned refrigerator factory into an innovative international giant. The company has become China's first global brand and the fifth largest appliance seller in the world. What are the secrets of Haier's success? Many researchers have conducted extensive studies on Haier's management and found the key is Management Information Systems such as e-Commerce and logistics systems that improve business operations between its suppliers, customers, and business partners. This article recounts the journey of Haier's achievements to excellence through its MIS, and provides analyses of the company's business model, the market chain management model.

scholarly journals William James and the Sustainable Transformation of Values, with a Case Study for Rethinking the American Dream

Human Arenas ◽  
2021 ◽  
Author(s):  
Paul Croce
Keyword(s):  
Economic Growth ◽  
Social Transformation ◽  
American Dream ◽  
William James ◽  
Social Issues ◽  
Virus Spread ◽  
Global Pandemic ◽  
The Face ◽  
The Many

AbstractThis article addresses the call of the Psychology of Global Crises conference for linkage of academic work with social issues in three parts: First, examples from conference participants with their mix of bold calls for social transformation and realization of limits, a combination that generated few clear paths to achieving them. Second, presentation of Jamesian practical idealism with psychological insights for moving past impediments blocking implementation of ideals. And third, a case study of impacts from the most recent prominent crisis, the global pandemic of 2020, which threatens to exacerbate the many crises that had already been plaguing recent history. The tentacles of COVID’s impact into so many problems, starting with economic impacts from virus spread, present an opportunity to rethink the hope for constant economic growth, often expressed as the American Dream, an outlook that has driven so many of the problems surging toward crises. Jamesian awareness of the construction of ideological differences and encouragement of listening to those in disagreement provide not political solutions, but psychological preludes toward improvements in the face of crises.

Jeremiah and Inner Biblical Exegesis

2021 ◽  
pp. 281-308
Author(s):  
Dalit Rom-Shiloni
Keyword(s):  
Biblical Exegesis ◽  
Separate Spheres ◽  
Prophetic Literature ◽  
Literary Evolution ◽  
The Many

Bringing inner biblical exegesis as a methodology to the study of Jeremiah has opened venues to discuss Jeremiah in two quite separate spheres: The book and its literary evolution, as well as the prophetic activity in its early oral-written stages. This chapter is aimed at presenting the great benefits and the many pitfalls that these cross-lines (of methodology and Jeremiah) provide for the study of the prophetic book, and not least, for the basic methodological presumptions of inner biblical exegesis as part of the study of intertextuality in prophetic literature. Focusing on interpretive (i.e., adaptation/actualization) techniques within the plethora of intertextual relationships, this chapter takes the utilization of pentateuchal traditions (rarely, texts) in Jeremiah as a case study, and calls to question some of the basic scholarly assumptions concerning Jeremiah: the differences of style (poetry and prose) and the options to differentiate the prophet from his followers/tradents/editors.

scholarly journals The local church as a non-governmental organisation in the fight against poverty: A historical overview of Bethulie 1933–1935

2014 ◽  
Vol 70 (1) ◽  
Author(s):  
Johan Van der Merwe
Keyword(s):  
South Africa ◽  
Local Church ◽  
Dutch Reformed ◽  
Historical Overview ◽  
Reformed Church ◽  
Main Challenge ◽  
Organisation And Administration ◽  
The Many ◽  
The Church

Poverty is one of the greatest threats to society. In South Africa it is also one of the biggest challenges. This article starts with the challenges put to society by Mr Trevor Manuel at the Carnegie 3 conference. It then explores the possibility of if and how the church can act as a non-governmental organisation in the fight against poverty. A historical overview of the actions of Rev. E.P. Groenewald, during the drought of 1933–1934 in the Dutch Reformed Church Bethulie, serves as a case study of how the church can make a difference. It, however, also illustrates the many pitfalls on this challenging road. The article comes to the conclusion that the main challenge of the church in the fight against poverty is to act as a non-governmental organisation, which transforms values and assists society with good organisation and administration.

scholarly journals Kilburn High Road Revisited

2016 ◽  
Vol 1 (2) ◽  
pp. 128-140 ◽  
Author(s):  
Cristina Capineri
Keyword(s):  
Sense Of Place ◽  
Service Management ◽  
Street Life ◽  
The Many ◽  
High Road ◽  
Simple Narrative ◽  
Unique Relationship ◽  
Over Time ◽  
Selection Of

Drawing on John Agnew’s (1987) theoretical framework for the analysis of place (location, locale and sense of place) and on Doreen Massey’s (1991) interpretation of Kilburn High Road (London), the contribution develops an analysis of the notion of place in the case study of Kilburn High Road by comparing the semantics emerging from Doreen Massey’s interpretation of Kilburn High Road in the late Nineties with those from a selection of noisy and unstructured volunteered geographic information collected from Flickr photos and Tweets harvested in 2014–2015. The comparison shows how sense of place is dynamic and changing over time and explores Kilburn High Road through the categories of location, locale and sense of place derived from the qualitative analysis of VGI content and annotations. The contribution shows how VGI can contribute to discovering the unique relationship between people and place which takes the form given by Doreen Massey to Kilburn High Road and then moves on to the many forms given by people experiencing Kilburn High Road through a photo, a Tweet or a simple narrative. Finally, the paper suggests that the analysis of VGI content can contribute to detect the relevant features of street life, from infrastructure to citizens’ perceptions, which should be taken into account for a more human-centered approach in planning or service management.

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