Contributions to the Founding of the Theory of Transfinite Numbers

Georg Cantor

This timeless mathematical classic introduced a groundbreaking field of mathematics that continues to have a profound impact on various areas such as topology, number theory, analysis, logic, and more. What sets this work apart is its ability to explain complex ideas in a clear and straightforward manner, making it accessible to anyone with a solid understanding of college-level mathematics.

In this book, Cantor starts by establishing the basic definitions and operations of cardinal and ordinal numbers. He delves into concepts like “cardinality” and “ordinality,” exploring topics such as the addition, multiplication, and exponentiation of cardinal numbers. He also examines the smallest transfinite cardinal number, the ordinal types of ordered sets, operations on ordinal types, and the ordinal type of the linear continuum, among others. Additionally, Cantor presents a theory of well-ordered sets and explores the ordinal numbers of these sets, as well as the properties and extent of transfinite ordinal numbers.

To provide context, Philip E. B. Jourdain, a renowned mathematical historian, offers an 82-page introduction. He discusses the contributions of Cantor’s predecessors such as Veierstrass, Cauchy, Dedekind, Dirichlet, Riemann, Fourier, and Hankel, and summarizes and analyzes Cantor’s earlier work. The book also includes a bibliographical note that references further investigations into the theory of transfinite numbers by influential mathematicians like Frege, Peano, Whitehead, Russell, and others.

This book is an excellent choice for students looking to explore this exciting branch of mathematics. It serves as a comprehensive resource and introduction to the theory of transfinite numbers, making it a valuable addition to any math enthusiast’s library.