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Vincent Mazzo
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What are the best books on Set Theory?? I mean a book with all axioms and theory and dispenses other books.
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micromass said:"Set theory" by Jech is my favorite, but it's only good if you already know a bit set theory.
dustbin said:How much would you consider adequate? I am familiar with sets from Apostol, Principles of Mathematics, and a couple of proof-intro books.
Set theory is the foundation of mathematics and provides a framework for understanding the relationships between different mathematical objects. It also has applications in other fields such as computer science and philosophy.
The axioms of set theory are the fundamental principles that govern the behavior of sets. They include the axiom of extension, which states that two sets are equal if and only if they have the same elements, and the axiom of regularity, which ensures that sets do not contain themselves as elements.
The Zermelo-Fraenkel set theory, also known as ZF set theory, is a commonly used system of axioms for set theory. It includes the axioms of extension, regularity, and several others that provide a basis for building the rest of mathematics.
The difference between ZF set theory and ZFC set theory is that ZFC includes an additional axiom, the axiom of choice, which states that for any collection of non-empty sets, there exists a set containing exactly one element from each set in the collection. This axiom has important implications in certain areas of mathematics.
Set theory provides a framework for understanding the relationships between different mathematical objects, which can help in understanding and proving theorems in other branches of mathematics. It also introduces concepts such as cardinality and ordinals that have applications in other areas of math, such as topology and algebra.