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Mathematical logic : on numbers, sets, structures, and symmetry
Kossak R., Springer International Publishing, New York, NY, 2018. 186 pp. Type: Book (978-3-319972-97-8)
Date Reviewed: Sep 11 2019

The author states his goal in the preface: “to try to explain a certain approach to the theory of mathematical structures.” The book consists of two parts: the first part is about logic, sets, and numbers; and the second part is dedicated to relations, structures, and geometry.

In the exposition, the author deviates from the usual path: instead of formally introducing concepts from the beginning, he starts with examples to illustrate the elements of the language in which mathematical statements can be expressed. The statements are about natural numbers and are used to familiarize the reader with the language of first-order logic. This is what the first chapter is about.

After introducing the syntax of first-order logic, it is time to discuss model-theoretic concepts. Again, the author does not hurry with formal definitions, and instead introduces mathematical structures (sets together with some relations over them) using examples. Chapter 2’s abstract states: “We will be inspecting the structure of the number systems with our logic glasses on, but we need to get used to wearing those glasses.” In the process of “getting used to wearing those glasses,” the reader learns about finite graphs as structures, an important concept of symmetry and its characterization, and is prepared for the next chapters where first-order logic is used to analyze structures made by numbers.

Chapter 3 introduces a set of natural numbers with the relations of addition and multiplication as a mathematical structure. While in chapter 1 the informal notion of natural numbers and their structure are used to introduce first-order logic, here the discussion goes in another way: reconstruct natural numbers based on the introduced logic. This discussion continues in the next chapter (4), where arithmetic structures of the integers and of the rationals are also examined in terms of first-order logic. While some of the details might be tedious, the presented material is intuitively very clear.

However, sometimes the intuition might be misleading, especially if it is about infinity. Then a formal approach becomes very important. This is illustrated in chapter 5, where real numbers are introduced and defined using Dedekind cuts. What follows is a construction of the corresponding number structure, paradoxes, and the relation with another representation via infinite decimals.

With the structure of real numbers, the description of basic number structures is over and the author turns his attention to structures in general. This requires a formal notion of sets, and chapter 6 is dedicated to Zermelo-Fraenkel axiomatic set theory based on the language of first-order logic. The axioms/axiom schemas are accompanied by a discussion of the effects of their introduction.

The first part ends here. The second, more advanced part goes deeper into a model-theoretic discussion. This is not surprising considering model theory is the author’s specialty. In chapter 7, he formally defines relations and starts exploring structures in more detail. The important notion of definability in structures is introduced and a couple of corresponding examples are shown. This discussion continues in the next chapter, where the definability of elements in the field of real numbers is discussed.

In chapter 9, ordered sets of natural numbers, integers, and rational numbers are used as examples of two important classes of structures: minimal and order-minimal. The analysis of definability is relatively easier in these special cases than in general. The author shows how to use the notion of symmetry, introduced in chapter 2 but redefined here for more general structures, for such an analysis.

Chapter 10 considers richer structures: rational numbers with addition and multiplication, and real numbers with addition and multiplication. It introduces the geometry of definable sets and shows the correspondence of its operations with Boolean connectives and quantifiers. One such operation is the projection from higher dimensions to lower ones. The author then shows that a geometric intuition might betray us here: the image of a set under projection from a higher to lower dimension is not necessarily less complex than the set itself. This problem is related to Diophantine equations and Hilbert’s tenth problem, which are discussed at the end of the chapter.

In chapters 11 and 12, the author brings in more advanced materials: axiomatizable theories, models, completeness and compactness theorems, the usage of symmetries to characterize elementary extensions of structures, and Ramsey-type theorems.

Tame or wild? In current model theory jargon, these two terms characterize the complexity of structures; in contrast to wild, tame refers to a structure that is well described and understood. In chapter 13, the author compares two structures, the field of complex numbers (ℂ, +, ⋅) and the standard model of arithmetic (ℕ, +, ⋅), from a tameness/wildness point of view. The former, which looks more complex, turns out to be tame (and also minimal), while the simpler one, the standard model of arithmetic, is rather wild.

Chapter 14 goes beyond first order. It is shown that for some important properties of a structure, for example, finiteness, minimality, and order-minimality, being well ordered cannot be expressed in first-order logic. An example is given, expressing in monadic second-order logic the property of being well ordered.

Chapter 15 gives a brief summary of the structures covered in the book: both the general notion and the concrete examples (over natural numbers, integers, rationals, reals, and complex numbers).

The book ends with a further reading list, proofs of some classical results, and a brief discussion of Hilbert’s program for foundations of mathematics. Carefully chosen exercises at the end of each chapter will facilitate self-study.

The author has made a significant effort to present the (not so easy) material in an understandable way: “I am familiar with not understanding. At the same time, I am also familiar with those extremely satisfying moments when one does finally understand.” I am sure that readers of this well-written book will experience many such satisfying moments.

Reviewer:  Temur Kutsia Review #: CR146688 (1912-0421)
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