Mathematical logic

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Mathematical logic, a branch of mathematics focusing on axiom systems and proofs, also lays the groundwork for information science and technology. The initial five chapters delve into classical mathematical logic, covering essential topics such as the syntax and models of first-order languages, formal inference systems, computability, and Gödel’s theorems. The subsequent chapters extend classical logic by exploring version sequences of formal theories, revision calculus, proschemes (formal proof method descriptions), and inductive inference theory. These concepts contribute to a formal theory of axiomatization, applicable to developing information technology and scientific theories. The book outlines three language environments for theories and the necessary properties of a meta-language environment. It culminates in a workflow for scientific research in the information era, integrating formal methods, interactive software, and human creativity. The second edition features significant revisions on the completeness theorem of the Gentzen system and introduces new content on the logic of scientific discovery, R-calculus without cut, and operational semantics for program debugging. This work serves as a valuable reference for graduate and undergraduate students, as well as researchers in mathematics, information science, and related natural sciences, with the first five chapters suitable for undergraduates and the latter five aim