The Logic of Infinity
Few mathematical results capture the imagination like Georg Cantor's groundbreaking work on infinity in the late nineteenth century. This opened the door to an intricate axiomatic theory of sets which was born in the decades that followed. Written for the motivated novice, this book provides an overview of key ideas in set theory, bridging the gap between technical accounts of mathematical foundations and popular accounts of logic. Readers will learn of the formal construction of the classical number systems, from the natural numbers to the real numbers and beyond, and see how set theory has evolved to analyse such deep questions as the status of the continuum hypothesis and the axiom of choice. Remarks and digressions introduce the reader to some of the philosophical aspects of the subject and to adjacent mathematical topics. The rich, annotated bibliography encourages the dedicated reader to delve into what is now a vast literature.
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Avoiding Russells paradox
Relations and order
Ordinal numbers and the Axiom of Infinity
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abstract algebraic alternative arbitrary assume Axiom of Choice axiomatic axioms of ZF bijection Cantor cardinal collection complete complex numbers construction continued fraction Continuum Hypothesis contradiction converges countable defined definition described element empty set equipollent equivalence classes example exists finite sets first-order theory formal foundational function f geometry Gödel Gödel number Hilbert idea Incompleteness Theorem induction infinite sets injection integers intuitive isomorphic limit ordinal logic Math mathematicians mathematics model of ZF multiplication natural numbers negation non-empty notion objects operations order isomorphic order type ordered pairs ordinal numbers paradox Peano Arithmetic polynomial predicate prime primitive principle proof provable prove rational numbers real numbers recursively relation Remarks result satisfies second-order sentence sequence set theory simply smallest standard transitive model statement subset successor ordinals symbols transfinite true variables well-formed formula well-ordering Well-Ordering Theorem Zermelo–Fraenkel set theory