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Finitely supported mathematics : an introduction /
In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM). It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (...
| Clasificación: | Libro Electrónico |
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| Autor principal: | |
| Otros Autores: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Cham :
Springer,
2016.
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| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- Acknowledgements; 1 Introduction; Abstract; 1.1 Motivation; 1.2 Approaches Related to the Fraenkel-Mostowski Framework; 1.3 Finitely Supported Mathematics; 1.4 Outline; 2 Fraenkel-Mostowski Set Theory: A Framework for Finitely Supported Mathematics; Abstract; 2.1 Axiom of Choice; 2.2 Permutative Renaming; 2.3 Sets with Atoms; 2.4 Constructions of Sets with Atoms; 2.4.1 Powersets; 2.4.2 Cartesian Products; 2.4.3 Disjoint Unions; 2.4.4 Function Spaces; 2.4.5 Categorical Constructions; 2.5 Fraenkel-Mostowski Axioms; 2.6 Logical Notions in the FM Cumulative Universe.
- 2.7 Inconsistency of Choice Principles2.8 Finiteness; 2.9 Freshness; 2.10 Abstraction; 2.10.1 Formal Definition; 2.10.2 Motivation; 2.10.3 Properties; 3 Algebraic Structures in Finitely Supported Mathematics; Abstract; 3.1 Multisets in Finitely Supported Mathematics; 3.1.1 Algebraic Properties of Multisets; 3.1.2 Multisets over Infinite Alphabets; 3.1.3 An Extension of the Framework; 3.2 Generalized Multisets in Finitely Supported Mathematics; 3.2.1 Algebraic Properties of Generalized Multisets; 3.2.1.1 Generalized Multisets as Groups; 3.2.1.2 Orders on Generalized Multisets.
- 3.2.1.3 Generalized Multisets in Reverse Mathematics3.2.2 Generalized Multisets over Infinite Alphabets; 3.3 Order Theory in Finitely Supported Mathematics; 3.3.1 Partially Ordered Sets; 3.3.2 Galois Connections; 3.3.3 Rough Set Approximations; 3.3.4 Abstract Interpretation; 3.3.5 Calculability: Approximations of Fixed Points; 3.3.6 Complete Partially Ordered Sets; 3.3.7 Recursive Equations over CPOs; 3.4 Groups in Finitely Supported Mathematics; 3.4.1 Basic Results; 3.4.2 Isomorphism Theorems; 3.4.3 Embedding Theorems; 3.4.4 Finitely Supported Subgroups; 3.5 General Comments.
- 3.6 Comments on the Methods Used in This Chapter3.7 Conclusion; 4 Extended Fraenkel-Mostowski Set Theory; Abstract; 4.1 Axioms of Extended Fraenkel-Mostowski Set Theory; 4.2 Inconsistency of the Axiom of Choice; 4.3 Algebraic Properties of EFM Sets; 4.4 Topological Properties of EFM Sets; 4.4.1 Subgroup Lattices as Domains; 4.4.2 Scott Topology over the Subgroup Lattice of a Group; 4.4.3 Topological Properties of the Group of Permutations of Atoms in EFM Set Theory; 4.5 Renamings in the Extended Fraenkel-Mostowski Framework; 4.6 Comments; 5 Process Calculi in Finitely Supported Mathematics.