نظریه مجموعه‌های ساده (متون کارشناسی در ریاضیات)

Every Mathematician Agrees That Every Mathematician Must Know Some Set Theory; The Disagreement Begins In Trying To Decide How Much Is Some. This Book Contains My Answer To That Question. The Purpose Of The Book Is To Tell The Beginning Student Of Advanced Mathematics The Basic Set­ Theoretic Facts Of Life, And To Do So With The Minimum Of Philosophical Discourse And Logical Formalism. The Point Of View Throughout Is That Of A Prospective Mathematician Anxious To Study Groups, Or Integrals, Or Manifolds. From This Point Of View The Concepts And Methods Of This Book Are Merely Some Of The Standard Mathematical Tools; The Expert Specialist Will Find Nothing New Here. Scholarly Bibliographical Credits And References Are Out Of Place In A Purely Expository Book Such As This One. The Student Who Gets Interested In Set Theory For Its Own Sake Should Know, However, That There Is Much More To The Subject Than There Is In This Book. One Of The Most Beautiful Sources Of Set-theoretic Wisdom Is Still Hausdorff's Set Theory. A Recent And Highly Readable Addition To The Literature, With An Extensive And Up-to-date Bibliography, Is Axiomatic Set Theory By Suppes. 1 The Axiom Of Extension -- 2 The Axiom Of Specification -- 3 Unordered Pairs -- 4 Unions And Intersections -- 5 Complements And Powers -- 6 Ordered Pairs -- 7 Relations -- 8 Functions -- 9 Families -- 10 Inverses And Composites -- 11 Numbers -- 12 The Peano Axioms -- 13 Arithmetic -- 14 Order -- 15 The Axiom Of Choice -- 16 Zorn’s Lemma -- 17 Well Ordering -- 18 Transfinite Recursion -- 19 Ordinal Numbers -- 20 Sets Of Ordinal Numbers -- 21 Ordinal Arithmetic -- 22 The Schröder-bernstein Theorem -- 23 Countable Sets -- 24 Cardinal Arithmetic -- 25 Cardinal Numbers. By Paul R. Halmos. This classic by one of the twentieth century's most prominent mathematicians offers a concise introduction to set theory. Suitable for advanced undergraduates and graduate students in mathematics, it employs the language and notation of informal mathematics. There are very few displayed theorems; most of the facts are stated in simple terms, followed by a sketch of the proof. Only a few exercises are designated as such since the book itself is an ongoing series of exercises with hints. The treatment covers the basic concepts of set theory, cardinal numbers, transfinite methods, and a good deal more in 25 brief chapters. "This book is a very specialized but broadly useful introduction to set theory. It is aimed at 'the beginning student of advanced mathematics' ... who wants to understand the set-theoretic underpinnings of the mathematics he already knows or will learn soon. It is also useful to the professional mathematician who knew these underpinnings at one time but has now forgotten exactly how they go. ... A good reference for how set theory is used in other parts of mathematics." — Allen Stenger, The Mathematical Association of America, September 2011. Front Matter....Pages i-vii The Axiom of Extension....Pages 1-3 The Axiom of Specification....Pages 4-7 Unordered Pairs....Pages 8-11 Unions and Intersections....Pages 12-16 Complements and Powers....Pages 17-21 Ordered Pairs....Pages 22-25 Relations....Pages 26-29 Functions....Pages 30-33 Families....Pages 34-37 Inverses and Composites....Pages 38-41 Numbers....Pages 42-45 The Peano Axioms....Pages 46-49 Arithmetic....Pages 50-53 Order....Pages 54-58 The Axiom of Choice....Pages 59-61 Zorn’s Lemma....Pages 62-65 Well Ordering....Pages 66-69 Transfinite Recursion....Pages 70-73 Ordinal Numbers....Pages 74-77 Sets of Ordinal Numbers....Pages 78-80 Ordinal Arithmetic....Pages 81-85 The Schröder-Bernstein Theorem....Pages 86-89 Countable Sets....Pages 90-93 Cardinal Arithmetic....Pages 94-98 Cardinal Numbers....Pages 99-102 Back Matter....Pages 102-104