Foundations ithout Foundationalism A Case for Second-order Logic [Paperback]

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed description of higher-order logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of second-order concepts in mathematics and the extent to which mathematical ideas can be formulated in higher-order logic. He also shows how first-order languages are often insufficient to codify many concepts in contemporary mathematics, and thus that both first- and higher-order logics are needed to fully reflect current work. Throughout, the emphasis is on discussing the associated philosophical and historical issues and the implications they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic comparable to that provided in a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in the field today.

PART I: Orientation
1. Terms and Questions
2. Foundationalism and Foundations of Mathematics
PART II: Logic and Mathematics
3. Theory
4. Metatheory
5. Second-Order Logic and Mathematics
6. Advanced Metatheory
PART III: History and Philosophy
7. The Historical Triumph of First-Order Languages
8. Second-Order Logic and Rule-Following
9. The Competition

Contains more on second-order logic than is readily available in any other textbook or survey. Philosophically, the book also contains many words of wisdom. --Journal of Symbolic Logic

The most comprehensive all-round account and defense of second-order logic as a vehicle for mathematics known to the reviewer. It is also very compreheló(