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The Number Systems: Foundations of Algebra and Analysis

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There are many books that present the construction of the number systems from the ground up. You can read the beautifully written little book of Thurston. Or you can (probably must) read the greatest classic of them all, Landau's Foundations of Analysis. I believe that each of them has its own virtues. So what makes this book by Solomon Feferman, a very known logician, different from the others, and worth to read? I believe that there are two main reasons. Firstly, it gives a lot of sidelights on topics rarely touched upon in other works. For instance, it discusses how matemathical entities can be differently defined: either directly, either with a finite computation scheme, either with an infinite computation scheme or either without any way to compute. And those differences, very important to know and be aware even if you are not a construtivist and don't worry about such discussions, are passed over in most treatments. His discussion of sets is pleasant and not boring as usually is, because he discusses many subtle points, informally of course (it is not a course in set theory). Secondly, it is more biased towards algebra, as the author itself points in the preface. This is good because usually the subject of number systems is a collateral topic in analysis courses. But the own subject matter of analysis gives you practice in working the concepts of analysis that are introduced to explain number systems. So, besides giving a sound foundation of the edifice, little is gained and used thereafter. But for algebra the methods give a concrete application of many concepts that are given in a very abstract setting in algebra courses. So it makes those concepts much clearer. An example is Feferman's treatment of polynomials and number fields, that stops short of Galois' theory. I learnt Galois' theory firstly in a very abstract setting, and its most beautiful application of insolvability of quintic equations was just touched upon, to my dismay. This book allows you to follow the reverse path: understand concretely what are the objects that mathematics studies, at least one detailed example of them, and of course makes the abstract setting much clearer and better motivated.