Open access peer-reviewed Edited Volume

# Set Theory

#### Covering

Algebra of classes Family of ClassesNatural numbers Equipotence of sets Operations Ordering Theory of Ordinals and Cardinals Normal form Epsilon numbers Independence results Göedel Theorems Fuzzy sets Rough sets Paraconsistent logic

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Sets theory provides the basis for the developments of all mathematics. In it, several concepts are established and provide the fundamental structure for the creation of a whole construction of mathematical theory. For example, the concept of infinity, which plays a fundamental role, has always been a source of controversy among mathematicians and philosophers. This type of problem, which causes divergences and raises questions, is responsible for the establishment of new theories and sets.

The concepts of sets emerges naturally in the human mind, being an abstract concept created by our human reasoning. An example is the set of natural numbers, which can be seen as an immediate consequence of the notion of sets.

However, when the human mind creates abstract concepts, such as numbers, which are often associated with concrete concepts; However, in some cases, it is impossible to count the number of elements of a set, even knowing that it has a finite amount of elements. The number of residences in the world or stars in the sky is a finite number, no matter how amazingly large. Thus, one of the fundamental questions of the sets theory arises that is "Does the order we use to count things not affect the result?".

There are many questionings of this kind that arise. Despite the protests of several lines of research, such as constructivists and intuitionists, the pillars of the theory of sets developed by Cantor, Zermelo, Frankel, and Von Neumann, served as the basis for the rationale of mathematics. The fundamental idea is to use sets to define all mathematical objects as sets. Everything is set!

This book aims to show new advances and representations in sets theory, asking questions still open and explaining complex axioms. Applications from sets theory to real-world representation problems can also be presented. Philosophical problems and new modeling can also be addressed.

### Publishing process

#### Book initiated and editor appointed

Date completed: November 18th 2019

Applications to edit the book are assessed and a suitable editor is selected, at which point the process begins.

#### Chapter proposals submitted and reviewed

Potential authors submit chapter proposals ready for review by the academic editor and our publishing review team.

#### Approved chapters written in full and submitted

Deadline for full chapters: May 9th 2020

Once approved by the academic editor and publishing review team, chapters are written and submitted according to pre-agreed parameters

#### Full chapters peer reviewed

Review results due: July 28th 2020

Full chapter manuscripts are screened for plagiarism and undergo a Main Editor Peer Review. Results are sent to authors within 30 days of submission, with suggestions for rounds of revisions.

#### Book compiled, published and promoted

Expected publication date: September 26th 2020

All chapters are copy-checked and typesetted before being published. IntechOpen regularly submits its books to major databases for evaluation and coverage, including the Clarivate Analytics Book Citation Index in the Web of ScienceTM Core Collection. Other discipline-specific databases are also targeted, such as Web of Science's BIOSIS Previews.

#### Germano Lambert-Torres

Gnarus Institute

Germano Lambert-Torres received the B.S. and M.Sc. degrees in electrical engineering from the Federal University of Itajuba (UNIFEI), Itajuba, Brazil, the B.S. degree in economics from the South Minas Gerais Economic and Social Sciences Faculty, Itajuba, the B.S. degree in mathematics from the Itajuba Faculty of Sciences and Languages, Itajuba, and the Ph.D. degree in electrical engineering from the École Polytechnique de Montreal, Montreal, QC, Canada, in 1990. From 1983 to 2012, he was a Professor in the Electrical Engineering Department, UNIFEI. From 2000 to 2004, he was the Prorector of Research and Graduate Studies at UNIFEI. He also served a member of two high committees in Brazil, one for Education: Member of National Final Exam for undergraduate electrical engineers (PROVÃO), 1998-2003, and other for Research: Member of National Council for Research (CA-EE CNPq), 2004-2007. From 1995 to 1996, he was a Visiting Professor at the University of Waterloo, Waterloo, ON, Canada. Currently, he is a Director of R&D at PS Solutions, Brazil, and Chair of the Scientific-Technical Council at Gnarus Institute, Brazil. He is a member of the IEEE Fellow Committee in the Industry Application Society (IEEE IAS) and the Education Society (IEEE EdSoc). Also, he is a member of the IEEE Medal in Power Engineering Committee. He also serves as a consultant for many power industries in Brazil and South America, with more than 150 R&D developed projects. He is a member of several committees in Brazilian Governmental and Regulatory Agencies, such as The Ministry of Education (MEC), The Ministry of Science and Technology (MCT), The National Electric Energy Agency (ANEEL), and The National Petroleum Agency (ANP). He has taught numerous IEEE and IFAC tutorials in the U.S., Europe, and Asia. Dr. Lambert-Torres is a member of the International Conference on Intelligent Systems Applications to Power Systems (ISAP) international board. He serves on several committees related to intelligent systems, including IEEE and the International Council on Large Electric Systems (CIGRÉ). He served as the General Chair for ISAP in 1999 and 2009, as well as Vice-General Chair for ISAP in 2001 and the Congress on Logic Applied to Technology in 2003 and 2007. He received several awards, including the 'Technical Committee Working Group Recognition Award on New Technologies and Practical Applications” from the IEEE Power and Energy Society (PES) (2006), 'Outstanding Leadership as Member of the ISAP Board of Directors” from the International Council of ISAP (2007), and 'Technical Committee Working Group Recognition Award on Multi-Agent Systems” from the IEEE-PES (2008). He has completed more than 80 M.Sc. and Ph.D. thesis supervisions and published more than 600 journal and technical conference papers. He is also the author/editor or coauthor of nine books, more than 30 book chapters and 80 transactions papers on intelligent systems and nonclassical logic. Dr. Lambert-Torres is a Fellow of the IEEE, Class 2014, in the Engineer/Scientist category, with the following citation: for contributions to the application of intelligent systems to power systems.

Book chapters authored 6

Books edited 0

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