Introduction to Incompleteness: From Gödel's Theorems to Forcing and the Continuum Hypothesis
Batzoglou, Serafim
- 出版商: Birkhauser Boston
- 出版日期: 2024-08-31
- 售價: $6,490
- 貴賓價: 9.5 折 $6,166
- 語言: 英文
- 頁數: 304
- 裝訂: Hardcover - also called cloth, retail trade, or trade
- ISBN: 3031642163
- ISBN-13: 9783031642166
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商品描述
Incompleteness is a fascinating phenomenon at the intersection of mathematical foundations, computer science, and epistemology that places a limit on what is provable. However, despite its importance, it is often overlooked in the mathematics curricula because it is difficult to teach. This book aims to help bridge this pedagogical gap by providing a complete and accessible technical exposition of incompleteness for a wide audience. The author accomplishes this by making conceptually difficult proofs more approachable by providing intuitive explanations of the main ideas. Care is taken to emphasize the different layers of the mathematical argument - the layer within and the metalayer about an axiomatic system.
Structurally, the book efficiently examines key results and arrives at some of the most interesting concepts as quickly as possible. It begins with Gödel's incompleteness theorems before continuing on to challenging concepts in the arithmetized completeness theorem, the Paris-Harrington theorem, and the independence of the continuum hypothesis. Other topics covered include the Lucas-Penrose arguments, ordinals and cardinals, and axiomatic set theory. Additionally, the author's coverage of forcing is a notable addition to the existing literature.
Introduction to Incompleteness will be of interest to researchers, students, and instructors looking for a resource to teach this topic. It may also be suitable for self-study. Knowledge of undergraduate-level theoretical mathematics or computer science is required, as well as a familiarity with abstract proofs.
商品描述(中文翻譯)
不完全性是一個引人入勝的現象，位於數學基礎、計算機科學和認識論的交匯處，對可證明的內容設置了限制。然而，儘管其重要性，這一主題在數學課程中常常被忽視，因為它難以教授。本書旨在幫助彌補這一教學上的空白，為廣泛的讀者提供一個完整且易於理解的不完全性技術性闡述。作者通過提供主要思想的直觀解釋，使概念上困難的證明變得更易接近。特別強調數學論證的不同層次——系統內部的層次和關於公理系統的元層次。
在結構上，本書有效地檢視關鍵結果，並儘快到達一些最有趣的概念。它以哥德爾的不完全性定理開始，然後繼續探討算術完全性定理、巴黎-哈林頓定理以及連續體假設的獨立性等挑戰性概念。其他涵蓋的主題包括盧卡斯-彭羅斯論證、序數和基數，以及公理集合論。此外，作者對強迫法的闡述是對現有文獻的一個顯著補充。
《不完全性導論》將吸引尋找教學資源的研究人員、學生和教師。它也可能適合自學。需要具備本科水平的理論數學或計算機科學知識，以及對抽象證明的熟悉。
作者簡介
Serafim Batzoglou is a computer scientists and geneticist whose work has focused on the application of algorithms and machine learning to the analysis of large-scale biomolecular data. After completing his PhD in Computer Science at MIT, he joined the Stanford AI Laboratory in 2001 where he was professor of Computer Science until 2016, when he transitioned to research leadership positions in industry. He has authored more than 120 peer-reviewed publications primarily in the field of computational biology. In 2003, he was named among the top young technology innovators by Technology Review magazine. He received the inaugural Innovator Award by the International Society of Computational Biology (ISCB) in 2016, and was named an ISCB Fellow in 2020. Serafim is also co-founder of DNAnexus, a precision health data cloud company.
Serafim has been fascinated by logic since his undergraduate years as a math major at MIT. He noticed the dearth of books that cover the topic of incompleteness in an accessible manner. As an educator for 15 years, he had strived to teach technical concepts clearly for a broad audience of engineers and biologists. With this book, he puts his experience to practice in providing a comprehensive and accessible technical exposition of incompleteness for a wide audience.
作者簡介(中文翻譯)
Serafim Batzoglou 是一位計算機科學家和遺傳學家，他的研究專注於將算法和機器學習應用於大規模生物分子數據的分析。完成麻省理工學院的計算機科學博士學位後，他於2001年加入史丹佛大學人工智慧實驗室，並在那裡擔任計算機科學教授直到2016年，之後轉向產業中的研究領導職位。他在計算生物學領域發表了超過120篇經過同行評審的論文。2003年，他被《Technology Review》雜誌評選為頂尖年輕科技創新者之一。2016年，他獲得國際計算生物學學會（ISCB）首屆創新者獎，並於2020年被任命為ISCB Fellow。Serafim 也是 DNAnexus 的共同創辦人，這是一家專注於精準健康數據的雲端公司。
自從在麻省理工學院攻讀數學專業的本科生涯以來，Serafim 就對邏輯產生了濃厚的興趣。他注意到有關不完全性主題的書籍稀缺，且缺乏易於理解的內容。作為一名擁有15年教學經驗的教育者，他一直努力清晰地向廣泛的工程師和生物學家受眾教授技術概念。藉由這本書，他將自己的經驗付諸實踐，為廣泛的讀者提供一個全面且易於理解的不完全性技術闡述。
