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Mathematical Logic

Roman Kossak
Publication Date: 
Number of Pages: 
Springer Undergraduate Texts in Philosophy
[Reviewed by
Dennis W. Gordon
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“My first encounters with mathematical logic were traumatic…. I really know what it is not to understand.” – Roman Kossak

Such modesty and humility. Wow. Here is an outstanding book. In the beginning, we learn of the difficulties the author encountered as a student while learning some of the very topics he writes about in this book. So successfully has the author conquered his youthful difficulties that model theory is now his research specialty and is also an important component of this book. As defined by William Weiss and Cherie D'Mello in their Fundamentals of Model Theory, “Model theory is the part of mathematics which shows how to apply logic to the study of structures in pure mathematics. On the one hand, it is the ultimate abstraction; on the other, it has immediate applications to every-day mathematics.” And, like other topics covered which are based on first-order logic – the unifying concept of the book - this underlying principle is given considerable attention in these pages. This logic is very important since “…a large portion of mathematics can be formalized in first-order logic.” Second and third order logics are briefly discussed. For individuals with little or no familiarity with these concepts they might sound rather advanced yet they are all developed from basic principles with no prerequisites necessary “… other than some genuine curiosity about the subject.”

The book is written in two parts with Part I entitled Logic, Sets, and Numbers, and it here that this delightful passage appears: The German mathematician Leopold Kronecker famously said that ‘God made the integers, all else is the work of man.’ We outlined that work in great detail. We built everything from the natural numbers. But we did not assume that God gave them to us. We built them from scratch ourselves.” The more advanced Part II - Relations, Structures, Geometry – is certainly more advanced than the first part, but Part I offers excellent preparation.

In addition to the two main parts of the book, there are also two appendices, the first one including several important proofs while the second appendix covers Hilbert’s program for the foundations of mathematics. Following these appendices is a bibliography listing both printed and internet sources. As for choices of format there are three to choose from – hardcover, softcover, and Kindle – all at reasonable prices.

In spite of having studied chemistry (Wayne State University and The University of Kansas) and enjoying a professional career in both academic and industrial research, Dennis’ greatest personal interest in science is mathematics. Now retired, he is a voracious reader, and with his wife Sally, they enjoy traveling in their sports car, bluegrass music, and the wonders of Wisconsin. Dennis may be contacted at


See the table of contents in the publisher's webpage.