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Full Description
Naïve Set Theory: A Rigorous Approach aims to provide a complete and unitary presentation of naive set theory as the foundation of the whole mathematics. Suitable for undergraduate students, this book focuses on the main foundational issues, trying to clarify explicitly what is (necessarily) assumed as intuitively known and what is rigorously founded on more elementary concepts. It analyses in detail the nature of natural numbers, both as metatheoretical objects and through their set-theoretical model. The author also pays particular attention to some topics that are not usually covered by the literature on naive set theory, like the universal properties of Cartesian product and disjoint union, that lead to a precise formulation of their basic features (associativity, commutativity, and distributivity). The exposition is organized coherently from the initial meta-theoretical notions to the construction of the universe of well-founded sets.
Features:
Replete with exercises, partially spread within the text and partially listed at the end of each chapter with a solutions manual available on www.Routledge.com/9781032933047.
Covers foundational topics that are not usually discussed in the literature on naive set theory, such as universal properties of Cartesian product and disjoint union, through which one can properly state associativity, commutativity, and distributivity of these operations.
Contents
1. Preliminaries. 2. Hereditary Sets and Basic Operations. 3. Relations and Functions. 4. Sets and Indexed Sets. 5. Universal Properties. 6. Sets and Classes. 7. Embedding Natural Numbers in Set Theory. 8. Well-Orders. 9. Axioms of Choice. 10. Cardinality. 11. Ordinals. 12. Cardinals. 13. Well-Founded Sets.
