Mathematical Logic
暫譯: 數學邏輯

George Tourlakis

Mathematical Logic
  • 出版商: Wiley
  • 出版日期: 2008-08-01
  • 售價: $3,840
  • 貴賓價: 9.5 折 $3,648
  • 語言: 英文
  • 頁數: 320
  • 裝訂: Hardcover
  • ISBN: 0470280743
  • ISBN-13: 9780470280744

    • 海外代購書籍(需單獨結帳)

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商品描述

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.

商品描述(中文翻譯)

一本全面且易於使用的數學推理邏輯指南

《數學邏輯》提供了對形式邏輯方法的全面介紹，以及其作為可靠的推理工具的使用。這本書以其友好的方式，成功地使讀者掌握關鍵概念和方法，以便形成有效的數學論證，這些論證可以用來揭示數學、計算機科學和哲學等多個研究領域的真理。

本書發展了撰寫證明所需的邏輯工具，通過引導讀者了解既定的「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「方程式」風格。章節分為布爾邏輯和謂詞邏輯兩個主題區域。書中還應用了形式邏輯之外的技術，以說明和展示邏輯的力量和局限性，例如：

- 邏輯可以證明真理，且僅能證明真理。
- 邏輯可以證明所有絕對真理（Post 和 Gödel 的完備性定理）。
- 邏輯無法證明所有「條件」真理，例如那些特定於佩亞諾算術的真理。因此，邏輯存在一些嚴重的局限性，這一點通過 Gödel 的不完備性定理得以顯示。

全書提供了大量的例子和習題，進一步促進讀者對邏輯發現數學真理能力的理解。此外，附錄中詳細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，同時還提供了可計算性理論的自成體系的介紹。

憑藉其全面的涵蓋範圍和易於理解的風格，《數學邏輯》是高年級本科生和研究生數學、計算機科學和哲學課程的理想書籍。對於希望學習如何在日常工作中使用邏輯的研究人員和實踐者來說，它也是一本寶貴的參考書。

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