Foundations of Finitely Supported Structures: A Set Theoretical Viewpoint

✍ Scribed by Andrei Alexandru, Gabriel Ciobanu

• Publisher: Springer International Publishing;Springer
• Year: 2020
• Tongue: English
• Leaves: 205
• Edition: 1st ed.
• Category: Library

✦ Synopsis

This book presents a set theoretical development for the foundations of the theory of atomic and finitely supported structures. It analyzes whether a classical result can be adequately reformulated by replacing a 'non-atomic structure' with an 'atomic, finitely supported structure’. It also presents many specific properties, such as finiteness, cardinality, connectivity, fixed point, order and uniformity, of finitely supported atomic structures that do not have non-atomic correspondents.

In the framework of finitely supported sets, the authors analyze the consistency of various forms of choice and related results. They introduce and study the notion of 'cardinality' by presenting various order and arithmetic properties. Finitely supported partially ordered sets, chain complete sets, lattices and Galois connections are studied, and new fixed point, calculability and approximation properties are presented. In this framework, the authors study the finitely supported L-fuzzy subsets of a finitely supported set and the finitely supported fuzzy subgroups of a finitely supported group. Several pairwise non-equivalent definitions for the notion of 'infinity' (Dedekind infinity, Mostowski infinity, Kuratowski infinity, Tarski infinity, ascending infinity) are introduced, compared and studied in the new framework. Relevant examples of sets that satisfy some forms of infinity while not satisfying others are provided. Uniformly supported sets are analyzed, and certain surprising properties are presented. Finally, some variations of the finite support requirement are discussed.

The book will be of value to researchers in the foundations of set theory, algebra and logic.

✦ Table of Contents

Front Matter ....Pages i-xi
The World of Structures with Finite Supports (Andrei Alexandru, Gabriel Ciobanu)....Pages 1-14
Finitely Supported Sets: Formal Results (Andrei Alexandru, Gabriel Ciobanu)....Pages 15-37
Choice Principles for Finitely Supported Structures (Andrei Alexandru, Gabriel Ciobanu)....Pages 39-49
Connections with Tarski’s Concept of Logicality (Andrei Alexandru, Gabriel Ciobanu)....Pages 51-54
Partially Ordered Sets in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 55-80
Lattices in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 81-91
Constructions of Lattices in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 93-109
Galois Connections in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 111-116
Several Forms of Infinity for Finitely Supported Structures (Andrei Alexandru, Gabriel Ciobanu)....Pages 117-156
Properties of Atoms in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 157-165
Freshness in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 167-169
Abstraction in Finitely Supported Mathematics (Andrei Alexandru, Gabriel Ciobanu)....Pages 171-177
Relaxing the Finite Support Requirement (Andrei Alexandru, Gabriel Ciobanu)....Pages 179-192
Conclusion (Andrei Alexandru, Gabriel Ciobanu)....Pages 193-197
Back Matter ....Pages 199-204

✦ Subjects

Computer Science; Theory of Computation; Mathematics of Computing; Mathematical Logic and Foundations