Colloquium Publications 1987; 318 pp; softcover Volume: 41 Reprint/Revision History: reprinted 1988 ISBN10: 0821810413 ISBN13: 9780821810415 List Price: US$90 Member Price: US$72 Order Code: COLL/41
 Completed in 1983, this work culminates nearly half a century of the late Alfred Tarski's foundational studies in logic, mathematics, and the philosophy of science. Written in collaboration with Steven Givant, the book appeals to a very broad audience, and requires only a familiarity with firstorder logic. It is of great interest to logicians and mathematicians interested in the foundations of mathematics, but also to philosophers interested in logic, semantics, algebraic logic, or the methodology of the deductive sciences, and to computer scientists interested in developing very simple computer languages rich enough for mathematical and scientific applications. The authors show that set theory and number theory can be developed within the framework of a new, different, and simple equational formalism, closely related to the formalism of the theory of relation algebras. There are no variables, quantifiers, or sentential connectives. Predicates are constructed from two atomic binary predicates (which denote the relations of identity and settheoretic membership) by repeated applications of four operators that are analogues of the wellknown operations of relative product, conversion, Boolean addition, and complementation. All mathematical statements are expressed as equations between predicates. There are ten logical axiom schemata and just one rule of inference: the one of replacing equals by equals, familiar from high school algebra. Though such a simple formalism may appear limited in its powers of expression and proof, this book proves quite the opposite. The authors show that it provides a framework for the formalization of practically all known systems of set theory, and hence for the development of all classical mathematics. The book contains numerous applications of the main results to diverse areas of foundational research: propositional logic; semantics; firstorder logics with finitely many variables; definability and axiomatizability questions in set theory, Peano arithmetic, and real number theory; representation and decision problems in the theory of relation algebras; and decision problems in equational logic. Table of Contents  The formalism \(\mathcal L\)of predicate logic
 The formalism \(\mathcal L^+\), a definitional extension of \(\mathcal L\)
 The formalism \(\mathcal L^+\) without variables and the problem of its equipollence with \(\mathcal L\)
 The relative equipollence of \(\mathcal L\) and \(\mathcal L^+\), and the formalization of set theory in \(\mathcal L^\times\)
 Some improvements of the equipollence results
 Implications of the main results for semantic and axiomatic foundations of set theory
 Extension of results to arbitrary formalisms of predicate logic, and applications to the formalization of the arithmetics of natural and real numbers
 Applications to relation algebras and to varieties of algebras
 Bibliography
 Indices
