This book offers a systematic map of arithmetical incompleteness and its boundary, one approached from logical, foundational, and philosophical perspectives. Nearly a century after Gödel's discovery, the literature remains organized around the classical theorems themselves, their proofs, their consequences, and their philosophical interpretation, while the limitative side of the phenomenon has been treated piecemeal. Drawing on more than two decades of research on arithmetical incompleteness, the book is not another introduction to the proofs of Gödel's theorems: it asks how far those theorems reach, how the phenomenon they describe can be classified, and where its limits lie. The author's organizing thesis is that arithmetical incompleteness is not a linear property fixed by a single threshold of expressive strength, but a multi-dimensional, abstractly limitless, and intensional phenomenon that is at the same time philosophically rich. The book develops this picture through five interconnected themes: a taxonomy of proofs, from proof-theoretic and recursion-theoretic arguments to concrete-combinatorial ones; a systematic taxonomy of the metamathematical properties that certify incompleteness, the implication relations among which form a partial order with incomparable elements rather than a linear hierarchy; the limits of the first incompleteness theorem, culminating in the result that with respect to interpretability the class of effectively inseparable, equivalently, effectively incomplete theories has no minimal elements, so that effective incompleteness has no logical atom; and the limits of the second incompleteness theorem, including a systematic analysis of its intensionality. The fifth theme is a sustained philosophical reflection on the depth of Gödel's theorems, on mechanism, on arithmetical truth, and on arithmetical pluralism, closing with the proposal that different perspectives on incompleteness lead to different conclusions about its limits, and that, from the metamathematical perspective adopted here, those limits are normative rather than purely logical—shaped by the domain of theories we admit and by the representational demands we impose on consistency statements.

Yong Cheng is currently a professor in the School of Philosophy at Wuhan University, China. He received his PhD in Mathematics (Logic) from the National University of Singapore in 2012, and from 2013 to 2014 was a postdoctoral fellow at the Institute for Mathematical Logic and Foundational Research, University of Münster, Germany. His research centers on logic—particularly the incompleteness phenomenon, foundations of mathematics, and philosophy of mathematics—and extends to the philosophy of science, including the philosophy of logic and the philosophy of computation. His recent work focuses on the logical, foundational, and philosophical study of the incompleteness phenomenon as revealed in Gödel's incompleteness theorems. He is the author of Incompleteness for Higher-Order Arithmetic: An Example Based on Harrington's Principle (Springer, 2019) and of the forthcoming The Landscape of Arithmetical Incompleteness: Gödel's Incompleteness Theorems and Their Limits (Synthese Library, Springer).