Mathematical Logic

A comprehensive and user-friendly guide to the use of logic in mathematical reasoning

Mathematical Logic presents a comprehensive introduction to formal methods of logic and their use as a reliable tool for deductive reasoning. With its user-friendly approach, this book successfully equips readers with the key concepts and methods for formulating valid mathematical arguments that can be used to uncover truths across diverse areas of study such as mathematics, computer science, and philosophy.

The book develops the logical tools for writing proofs by guiding readers through both the established "Hilbert" style of proof writing, as well as the "equational" style that is emerging in computer science and engineering applications. Chapters have been organized into the two topical areas of Boolean logic and predicate logic. Techniques situated outside formal logic are applied to illustrate and demonstrate significant facts regarding the power and limitations of logic, such as:

• Logic can certify truths and only truths.
• Logic can certify all absolute truths (completeness theorems of Post and Gödel).
• Logic cannot certify all "conditional" truths, such as those that are specific to the Peano arithmetic. Therefore, logic has some serious limitations, as shown through Gödel's incompleteness theorem.

Numerous examples and problem sets are provided throughout the text, further facilitating readers' understanding of the capabilities of logic to discover mathematical truths. In addition, an extensive appendix introduces Tarski semantics and proceeds with detailed proofs of completeness and first incompleteness theorems, while also providing a self-contained introduction to the theory of computability.

With its thorough scope of coverage and accessible style, Mathematical Logic is an ideal book for courses in mathematics, computer science, and philosophy at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners who wish to learn how to use logic in their everyday work.

《數學邏輯》是一本全面且易於使用的指南，介紹了邏輯在數學推理中的應用。本書以親切易懂的方式，成功地為讀者提供了關鍵概念和方法，用於構建有效的數學論證，並可應用於數學、計算機科學和哲學等多個研究領域，揭示真理。

本書通過引導讀者進行已確立的「希爾伯特」風格的證明寫作，以及在計算機科學和工程應用中出現的「等式」風格，來發展寫作證明的邏輯工具。書中的章節分為布林邏輯和謂詞邏輯兩個主題領域。此外，還運用在形式邏輯之外的技巧，以說明和展示關於邏輯的能力和限制的重要事實，例如：

- 邏輯可以證明真理，且僅限於真理。
- 邏輯可以證明所有絕對真理（Post和哥德爾的完備性定理）。
- 邏輯無法證明所有「條件性」真理，例如那些特定於皮亞諾算術的真理。因此，邏輯有一些嚴重的限制，這在哥德爾的不完備性定理中得到了證明。

書中提供了大量的例子和問題集，進一步幫助讀者理解邏輯發現數學真理的能力。此外，詳細的附錄介紹了塔斯基語義學，並提供了完備性和第一不完備性定理的詳細證明，同時還提供了一個獨立的可計算性理論入門。

《數學邏輯》的廣泛範圍和易於理解的風格，使其成為高年級本科和研究生課程中的理想教材。對於希望在日常工作中學習如何使用邏輯的研究人員和從業人員來說，它也是一本有價值的參考書。