The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. At the same time, it is often viewed as a foundation of mathematics so that in the most prevalent, current mathematical practice "to make a notion precise" simply means "to define it in set theory." This book tries to do justice to both aspects of the subject: it gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets (including the basic results that have applications to computer science), but it also attempts to explain precisely how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author added solutions to the exercises, and rearranged and reworked the text in several places to improve the presentation.
The book is aimed at advanced undergraduate or beginning graduate mathematics students and at mathematically minded graduate students of computer science and philosophy.

lgli/D:/!genesis/library.nu/33/_99775.33880338389021202fac382360436026.pdf

nexusstc/Notes on Set Theory/33880338389021202fac382360436026.pdf

scihub/10.1007/0-387-31609-4.pdf

Yiannis N. Moschovakis

Copernicus

Telos

Undergraduate Texts in Mathematics, Second edition, New York, NY, 2006

Undergraduate texts in mathematics, 2nd ed, New York, ©2006

Undergraduate Texts in Mathematics, 2nd, 2005

United States, United States of America

December 21, 2005

December 8, 2005

2nd, 2005-12-08

2nd, US, 2005

до 2011-01

lg550241

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Whatthisbookisabout. Thetheoryofsetsisavibrant,excitingmathematical Theory, With Its Own Basic Notions, Fundamental Results And Deep Open Pr- Lems,andwithsigni?cantapplicationstoothermathematicaltheories. Atthe Sametime,axiomaticsettheoryisoftenviewedasafoundationofmathematics: It Is Allegedthat All Mathematical Objectsare Sets, And Theirpropertiescan Be Derived From The Relatively Few And Elegant Axioms About Sets. Nothing So Simple-minded Can Be Quite True, But There Is Little Doubt That In Standard, Current Mathematical Practice, “making A Notion Precise” Is Essentially S- Onymouswith“de?ningitinsettheory”. Settheoryistheo?ciallanguageof Mathematics,just Asmathematicsisthe O?ciallanguageof Science. Like Most Authors Of Elementary, Introductory Books About Sets, I Have Triedtodojusticetobothaspectsofthesubject. From Straight Set Theory, These Notes Cover The Basic Facts About “abstract Sets”, Includingthe Axiom Of Choice, Trans?nite Recursion, And Cardinal And Ordinal Numbers. Somewhat Less Common Is The Inclusion Of A Chapter On “pointsets” Which Focuses On Results Of Interest To Analysts And Introduces The Reader To The Continuum Problem, Central To Set Theory From The Very Beginning. There Is Also Some Novelty In The Approach To Cardinal Numbers, Whichare Brought In Very Early (following Cantor, But Somewhatdeviously), So That The Basic Formulas Of Cardinal Arithmetic Can Be Taught As Quickly As Possible. Appendixagivesamoredetailed“construction”oftherealnumbers Thaniscommonnowadays,whichinadditionclaimssomenoveltyofapproach And Detail. Appendix B Is A Somewhat Eccentric, Mathematical Introduction To The Study Of Natural Models Of Various Set Theoretic Principles, Including Aczel’s Antifoundation. It Assumes No Knowledge Of Logic, But Should Drive Theseriousreaderto Studyit. About Set Theory As A Foundation Of Mathematics, There Are Two Aspects Of These Notes Which Are Somewhat Uncommon. Equinumerosity -- Paradoxes And Axioms -- Are Sets All There Is? -- The Natural Numbers -- Fixed Points -- Well Ordered Sets -- Choices -- Choice’s Consequences -- Baire Space -- Replacement And Other Axioms -- Ordinal Numbers. Yiannis Moschovakis. Includes Index.

The axiomatic theory of sets is a vibrant part of pure mathematics, with its own basic notions, fundamental results, and deep open problems. It is also viewed as a foundation of mathematics so that "to make a notion precise" simply means "to define it in set theory." This book gives a solid introduction to "pure set theory" through transfinite recursion and the construction of the cumulative hierarchy of sets, and also attempts to explain how mathematical objects can be faithfully modeled within the universe of sets. In this new edition the author has added solutions to the exercises, and rearranged and reworked the text to improve the presentation.

"The book is aimed at advanced undergraduate or beginning graduate mathematics students and at mathematically minded graduate students of computer science and philosophy." "Important changes in the Second Edition include the re-writing of many parts to make the book easier to study and to teach from, the addition of material about Baire space and ordinal numbers (the arithmetic of ordinals), the addition of solutions to the exercises (but not the more challenging problems), and the correction of many typographical errors from the First Edition."--Jacket

2011-06-04