The End of Infinity

The way we think about infinity shapes the way we think about other subjects. The author says, "My heart fluttered the first time I heard the paradoxical claim of a 19th-century mathematician who said infinity was real, that there was more than one type of infinity, and that not all infinities were the same size! It was either one of the most absurd or one of the most important theories in intellectual history." The chapters that follow investigate the issue from a variety of perspectives. This book focuses mainly on the philosophy of infinity; it does not require an extensive understanding of mathematics, although relevant terms and concepts are introduced and explained. From the late 19th to the early 20th century, great thinkers like Cantor, Frege, and Russell attempted to provide a new foundation for mathematics by replacing the intuitions of Kant with logic, rigorous definitions, and axiomatic set theory. One of the results of their work was a new conception of infinity, which included the surprising claims that infinite sets exist, that there are different types of infinite sets, and that not all infinite sets are the same size. However, the great logician Gödel proved that all axiomatic systems, like set theory, are necessarily incomplete and therefore cannot not provide a solid foundation for mathematics, evidenced by the fact that there are known mathematical truths that cannot be proven or disproven with set theory. Despite this setback, set theory continues to dominate modern mathematics, with limited attempts to reconsider its validity - until now.