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Introduction to incompleteness = from Gödel's theorems to forcing and the continuum hypothesis /

紀錄類型:	書目-語言資料,印刷品 : Monograph/item
正題名/作者:	Introduction to incompleteness/ by Serafim Batzoglou.
其他題名:	from Gödel's theorems to forcing and the continuum hypothesis /
作者:	Batzoglou, Serafim.
出版者:	Cham :Springer Nature Switzerland : : 2024.,
面頁冊數:	xi, 302 p. :ill., digital ; : 24 cm.;
Contained By:	Springer Nature eBook
標題:	Gödel's theorem. -
電子資源:	https://doi.org/10.1007/978-3-031-64217-3
ISBN:	9783031642173

Introduction to incompleteness = from Gödel's theorems to forcing and the continuum hypothesis /
Batzoglou, Serafim.

Introduction to incompletenessfrom Gödel's theorems to forcing and the continuum hypothesis /[electronic resource] :by Serafim Batzoglou. - Cham :Springer Nature Switzerland :2024. - xi, 302 p. :ill., digital ;24 cm.

Part 1: Godel's Theorems -- Formal Axiomatic Systems -- Peano Arithmetic and Recursive Functions -- Godel's Incompleteness Theorems -- Structures, Models, and Satisfaction -- Completeness and Compactness Theorems -- Completeness and Peano Arithmetic -- The Lucas-Penrose Arguments -- Part II: Incompleteness in arithmetic and set theory -- Incompleteness in Finite Combinatorics -- Consistency of PA and E0 Induction -- Set Theory -- Independence of CH--forcing -- Independence of CH--forcing CH and -CH.

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.

ISBN: 9783031642173

Standard No.: 10.1007/978-3-031-64217-3doiSubjects--Topical Terms:

833019
Gödel's theorem.

LC Class. No.: QA9.65

Dewey Class. No.: 511.3

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520 $a 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.
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