In the light of logic - Feferman, Solomon (Oxford University Press, 1998)
Includes bibliographical references (pages 309-330) and index
In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support
I. Foundational Problems - Deciding the undecidable: Wrestling with Hilbert's problems - Infinity in mathematics: Is Cantor necessary? -- - II. Foundational Ways - The logic of mathematical discovery versus the logical structure of mathematics - Foundational ways - Working foundations -- - III. Godel - Godel's life and work - Kurt Godel: Conviction and caution - Introductory note to Godel's 1933 lecture -- - IV. Proof Theory - What does logic have to tell us about mathematical proofs? - What rests on what? The proof-theoretic analysis of mathematics - Godel's Dialectica interpretation and its two-way stretch -- - V. Countably Reducible Mathematics - Infinity in mathematics: Is Cantor necessary? (Conclusion)