The authors show that set theory and number theory can bedeveloped within the framework of a new, different, and simpleequational formalism, closely related to the formalism of thetheory of relation algebras. There are no variables,quantifiers, or sentential connectives. Predicates areconstructed from two atomic binary predicates (which denote therelations of identity and set-theoretic membership) by repeatedapplications of four operators that are analogues of thewell-known operations of relative product, conversion, Booleanaddition, and complementation. All mathematical statements areexpressed as equations between predicates. There are ten logicalaxiom schemata and just one rule of inference: the one ofreplacing equals by equals, familiar from high school algebra.
Though such a simple formalism may appear limited in its powersof expression and proof, this book proves quite the opposite.The authors show that it provides a framework for theformalization 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 todiverse areas of foundational research: propositional logic;semantics; first-order logics with finitely many variables;definability and axiomatizability questions in set theory, Peanoarithmetic, and real number theory; representation and decisionproblems in the theory of relation algebras; and decisionproblems in equational logic.