Sets for Mathematics 1st Edition by F. William Lawvere (PDF)

Description

Advanced undergraduate or beginning graduate students need a unified foundation for their study of geometry, analysis, and algebra. This book, first published in 2003, uses categorical algebra to build such a foundation, starting from intuitive descriptions of mathematically and physically common phenomena and advancing to a precise specification of the nature of Categories of Sets. Set theory as the algebra of mappings is introduced and developed as a unifying basis for advanced mathematical subjects such as algebra, geometry, analysis, and combinatorics. The formal study evolves from general axioms which express universal properties of sums, products, mapping sets, and natural number recursion. The distinctive features of Cantorian abstract sets, as contrasted with the variable and cohesive sets of geometry and analysis, are made explicit and taken as special axioms. Functor categories are introduced in order to model the variable sets used in geometry, and to illustrate the failure of the axiom of choice. An appendix provides an explicit introduction to necessary concepts from logic, and an extensive glossary provides a window to the mathematical landscape.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This should by no means be thought of as a book on traditional set theory. (For that I would recommend “Classic Set Theory” by Goldrei.) This is a book on category theory showing the majority of its interest in the category of sets.The first few chapters of the book begin to detail a proposed axiomatization of the category of sets, which is finally concluded with the introduction of a natural number object in chapter 9 after being sidetracked for a few chapters by some interesting properties of exponentiation and power sets. This is definitely one of the most interesting mathematics texts I have come across, and I feel like I got a lot out of it.Despite how deceptively simple the first few exercises were, the difficulty level rocketed up fast and I found the book to be extremely challenging overall. The material itself was hard enough, but the sophistication of the authors’ writing only compounded the difficulty. Oftentimes the prose parts of the book felt like something you would see as a “reading comprehension” passage on the GRE, nothing indecipherable, but it certainly took time to process even small bits of content. I quit the book a mere ten pages from the end because I had become completely overwhelmed and was understanding the final material only at a very superficial level.Someone with a better mind than I have might get a lot out of the things I struggled most with. However, the book does have some errors scattered throughout that cause mild confusion. In addition, there were several points in the book where terminology was invoked that I couldn’t recall having read before and couldn’t find in the index of terms. Often I could guess at what the intended meaning was by the context in which they were invoked, but sometimes I simply had to move on without understanding.A word should be made on the appendices. Appendix A.1. may be the most interesting perspective on mathematical logic I have come across. A.2. requires some knowledge of algebra and to be honest I’m not sure why this section was even included in the book – it seems to have no bearing on the rest of the material. Only about sixty or seventy percent of the glossary is really accessible to what I would consider the book’s target audience – for example if you’ve never really encountered adjoint functors before then trying to understand the sections on geometric morphisms or Grothendieck Topoi may be hopeless.While there are parts of the book that invoke knowledge of topology or analysis, these are all brief and easily skipped. Some algebra may help as well, but the only prerequisites that are really important are determination and (as always) mathematical maturity.To summarize: Very difficult, some flaws, but overall a worthwhile read.

⭐There seem to be two types of undergraduate exercises in set theory: the boring ones (e.g. where we are asked to compute the intersection of an indexed family of sets, or draw a diagram of a relational composition) and the exciting ones (where we ask why a set cannot be of the same size as its powerset, or why having two injections f:A->B and g:B->A implies that A and B are isomorphic). The difference is of course that the first kind involves mechanical computation with points, and all data given, and the second kind needs a creative argument in a situation where it sometimes seems that there is not enough data to solve the problem.The book by Lawvere and Rosebrugh made me realise that the exercises I find exciting can be phrased in terms of properties of maps acting on sets, and – yes, indeed – the boring ones can’t.But then the book goes further – it shows that in fact all axioms of sets can be written down in the language of maps. Doesn’t it strike you that as a consequence axiomatic set theory is not boring? :)Some of the discoveries I made during reading are just invaluable. For example, I learned that the *reason* why a set cannot be of the same size as its powerset is that the two-element set have a self map with no fixed point, which is, admittely, the essence of Cantor’s diagonal argument.The Authors say in the Foreword that the book is for students who are beginning the study of algebra, geometry, analysis, combinatorics, … Indeed, being a virtuoso of a particular implementation of set theory such as ZFC does not help much with these subjects. Instead one needs a good knowledge of how sets behave when measured, divided, added, towered, counted – name your favourite operation – and this is precisely the story told in the book.

⭐This book makes the most powerful branch of mathematics accessible to early Mathematics undergraduates. The difficulties people have in comprehending the material are largely due to the abstract nature of the Mathematics itself. I would suggest that a reader takes plenty of time absorbing the material and attempting the exercises. I read Conceptual Mathematics by Dr. Schanuel and Dr. Lawvere before reading this one and I think this helped provide familiarity to the topics discussed. This book contains an ingenious formulation of the familiar Set Theory axioms in the context of the Category of Sets. The Category of Sets here treats all sets with bijections between them as equivalent. A great contribution of one of the authors is the discovery that Cantor adopted this perspective himself. It was Zermelo, who opposed this view, that axiomatized the theory based on the membership relation. This book is a one of a kind read and I would recommend it to ANY sufficiently curious student of Mathematics. I don’t claim mastery of all concepts presented, but I do claim many profound insights and a more modern understanding of mathematics. As an undergraduate myself, I can recommend this to other motivated undergraduates with confidence that it is indeed accessible material. It would not surprise me if the content of this book eventually replaced ZFC in university classrooms.

⭐Je suis seulement un amateur en mathématiques, qui s’intéresse notamment aux fondements des mathématiques et à ses développements les plus récents, comme la théorie des catégories. Donc pour un amateur, ce livre est ardu (il faudrait que je fasse les exercices pour assimiler complètement les notions) mais il est très éclairant. En s’attachant au “comportement” des “ensembles”, en tant que “catégorie”, c’est à dire du point de vue d’une “algèbre abstraite des fonctions sur des ensembles abstraits”, ce livre construit et illustre pas à pas les notions catégoriques.

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