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A formalization of set theory without variables. (English) Zbl 0654.03036
The subject of the book reviewed is clearly indicated by the title. A system \({\mathcal L}^*\) closely related to the equational theory of abstract relation algebras is presented. The compound expressions in binary relations are constructed from two basic symbols (for identity and membership relations) by means of four operations (composition, converse, union and complement). All statements of \({\mathcal L}^*\) are variable-free equations between such expressions. The deductive apparatus of \({\mathcal L}^*\) is based upon logical axiom schemata, nonlogical axioms and one rule of inference.
The book is divided into eight chapters. Chapter 1 opens with a nice description of the formulation of predicate logic, namely the system \({\mathcal L}\) for the language of set theory is presented. In Chapter 2 a conservative extension \({\mathcal L}^+\) of \({\mathcal L}\) is investigated. The notion of equipollence of two systems relative to a common extension is introduced. Chapter 3 deals with the system \({\mathcal L}^*\). Here it is shown that \({\mathcal L}^*\) and \({\mathcal L}^+\) are not equipollent (i.e. \({\mathcal L}^+\) is not a conservative extension of \({\mathcal L}^*\)). The main result of this part is a proof of the equipollence of \({\mathcal L}^*\) with the subsystem \({\mathcal L}_ 3\) of L containing just three variables.
The relative equipollence of \({\mathcal L}\) and \({\mathcal L}^*\) is investigated in Chapter 4. Roughly speaking a set \(\Sigma\) of sentences is constructed in such a way that the modifications of \({\mathcal L}\) and \({\mathcal L}^*\) obtained by adding an \(X\in \Sigma\) between axioms are equipollent. Chapter 5 improves some earlier results. Some consequences of the relative equipollence for axiomatic foundations of set theory are studied in Chapter 6. In Chapter 7 possibilities of extending the results obtained to arbitrary formalism are discussed, e.g. the arithmetic of real numbers is formalized in \({\mathcal L}^*\). Finally, Chapter 8 is devoted to applications to relation algebras.
The well-written book presents results of the research that started in the 19th century (G. Frege, E. Schröder) and finished recently. The text, accompanied by a lot of historical remarks and rather complete bibliography, can be recommended as a good source of information about the subject mentioned in the title.
Reviewer: L.Bukovský

03E30 Axiomatics of classical set theory and its fragments
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations