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Foundations without Foundationalism: A Case for Secondorder Logic (Oxford Logic Guides),Used
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The central contention of this book is that secondorder 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 higherorder logic, including a comprehensive discussion of its semantics. He goes on to demonstrate the prevalence of secondorder concepts in mathematics and the extent to which mathematical ideas can be formulated in higherorder logic. He also shows how firstorder languages are often insufficient to codify many concepts in contemporary mathematics, and thus that both first and higherorder 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 LowenheimSkolem theorems. All those concerned with the foundations of mathematics will find this a thoughtprovoking discussion of some of the central issues in the field today.
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