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Download Subsystems of Second Order Arithmetic (Perspectives in Logic) epub

by Stephen G. Simpson

Foundations of mathematics is the study of the most basic concepts and logical structure of mathematics, with an eye to the unity of human knowledge. Almost all of the problems studied in this book are motivated by an overriding foundational question: What are the appropriate axioms for mathematics? Through a series of case studies, these axioms are examined to prove particular theorems in core mathematical areas such as algebra, analysis, and topology, focusing on the language of second-order arithmetic, the weakest language rich enough to express and develop the bulk of mathematics. In many cases, if a mathematical theorem is proved from appropriately weak set existence axioms, then the axioms will be logically equivalent to the theorem. Furthermore, only a few specific set existence axioms arise repeatedly in this context, which in turn correspond to classical foundational programs. This is the theme of reverse mathematics, which dominates the first half of the book. The second part focuses on models of these and other subsystems of second-order arithmetic. Additional results are presented in an appendix.
Download Subsystems of Second Order Arithmetic (Perspectives in Logic) epub
ISBN: 0521150140
ISBN13: 978-0521150149
Category: Science
Subcategory: Mathematics
Author: Stephen G. Simpson
Language: English
Publisher: Cambridge University Press; 2 edition (February 18, 2010)
Pages: 464 pages
ePUB size: 1447 kb
FB2 size: 1232 kb
Rating: 4.4
Votes: 421
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Extremely useful for understanding the concepts behind reverse mathematics.
I received a new copy of the softcover version of the book from Amazon yesterday and the print quality of this book is quite bad. The block of printed texts on the pages are very slanted so that the side margin is smaller at the top and (about twice as) larger at the bottom. These types of print issues one often sees in cheap books printed in India. Now the Cambridge University Press seems to be adopting them. I am considering returning the book.
Very inspiring and informative. Lot of examples from different mathematical areas, are logically structured inside several formal systems (and language of second-order arithmetics). First chapter is easy and attractive reading, other ones are really mathematical enjoy.
In my opinion this is one of the best books about one of the most prominent active fields in logic
Simpson's "Subsystems of Second Order Arithmetic" is the essential textbook on reverse mathematics and models of subsystems of second-order arithmetic. The first part of the book deals with reverse mathematics for five essential subsystems of second-order arithmetic, and shows how many usual mathematical theorems are equivalent, over the base system, to one of these five systems, thus convincingly establishing that they are fairly natural. The second part focuses on models of the same five systems (omega-models, then beta-models, and more general models in the last chapter).

The content is fairly self-contained, requires minimal foreknowledge of logic, and covers a good territory. Proofs are concise but readable. The discussion gives the reader a good grasp of "what is going on" at an intuitive level. The general plan is clear (although a table of implication of all subsystems introduced would have been a welcome addition).

This book should be of interest to logicians and non-logician mathematicians. Even philosophers with an interest in mathematical logic should find it interesting (at least for the first part).

The main question of "which set existence axioms are needed to prove the known theorems of ordinary mathematics" is successfully answered, although the author is more convincing with regards to the three weakest theories as with regards to the two strongest ones, whose equivalent theorems are mostly about descriptive set theory.

My main regret is that this "second edition" is basically just a reprint of the first: only typographical mistakes have been corrected, and the bibliography has been updated, whereas the author could have used this occasion to describe some new results. This is not a substantial criticism, however: even if it is no longer completely up-to-date, this book remains an essential reference on the subject.