Sanchis Luis E. (Syracuse Univ.,Syracuse,NY)
Set theory - an operational approach.
Gordon and Breach Science Publishers,Inc., Newark, NJ.
Among mathematical theories, set theory has a special place, because it is formulated in formal logic, but it usually provides the semantics for formal logic. In practice, set theory is used as if it described one particular (mathematical) reality, the sets we intuitively feel are unique, although it is known that the axioms of set theory can have many different models.
In this book, Sanchis attempts to bring the theory of sets closer to practice by limiting all set references to sets that can be constructed from other sets, and all set properties to properties that can be constructed from other set properties. In particular, this means that in the definitions of sets and properties, all quantification is restricted to members of specified sets. Unrestricted quantification is not allowed because it is over the whole universe of sets, an assumed entity that is not well defined.
A side effect of Sanchis's operational approach to set theory is that it is not necessary to prove that sets exist before talking about them; existence comes automatically with the fact that all introduced sets are constructed from other sets. Of particular interest to mathematicians is the way that Sanchis makes induction and recursion primitive rules in his theory, rather than the derived rules that they are in other formulations of set theory. Given two previously defined set properties and a set operation, the set induction rule introduces a new set property by means of four axioms. The first axiom defines the new property to hold on a set Z whenever the first given property holds on Z (the base case) or the second property holds on Z and the new property holds on every member of the set obtained from Z by application of the given set operation. The other axioms introduce a related set operation and require a foundation condition: whenever a set contains a set on which the new property holds, there is a least member of the set on which the new property holds. The set recursion rule introduces a set operation that is defined in terms of a set property that has been previously introduced by the set induction rule.
The book is written in the style of an advanced mathematics textbook, with many theorems and their proofs interspersed with remarks, notes, and exercises. It covers most of the topics typically covered in an introduction to set theory, including ordinals, the set of natural numbers, the power set, and cardinals. In the last two chapters, Sanchis presents a formal logic, based on Gentzen's calculus of sequents, for formally proving the theorems in his set theory. This logic is like classical logic when quantification is restricted members of specific sets, but it is like intuitionistic logic when quantification is unrestricted. In the appendix, Sanchis considers the consequences of replacing his ordinal permutation rule by a weaker ordinal enumeration rule.
The book is aimed at professionals and graduate students in mathematics and philosophy who are interested in the role of set theory in the foundations of mathematics. It will also be of interest to people who want to apply set theory in their research without having to worry about the existence of the sets they define. To fully understand the book, however, the interested reader should be already familiar with set theory and formal logic.
D.L. Chester, DE.