Sets for Mathematics [Paperback]

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 is a tough read. For starters, there is the subject matter. If you have had a traditional undergraduate training in mathematics, then talking about sets and elements with axioms that deal with functions rather than elements is a stretch. Then, there is the authors' presentation which mixes axiomatic category theory and naive set theory. The approach does motivate category theory with set theory and it does explain how set theory might be replaced with category theory but, as written, it also leaves the reader unsure which mode of thinking is required at any particular moment. This uncertainty is not likely to be fully resolved without a teacher or a second reading. You need one or the other to navigate some early chapters with knowledge gained later.

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.

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