**Colloquium Publications**

Volume: 41;
1987;
318 pp;
Softcover

MSC: Primary 03;

Print ISBN: 978-0-8218-1041-5

Product Code: COLL/41

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-3187-7
Product Code: COLL/41.E**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

# A Formalization of Set Theory without Variables

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*Alfred Tarski; Steven Givant*

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 first-order 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
set-theoretic membership) by repeated applications of four operators
that are analogues of the well-known 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;
first-order 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

# Table of Contents

## A Formalization of Set Theory without Variables

- Cover Cover11
- Title page i2
- Contents iii4
- Explanation of section interdependence diagrams vii8
- Preface xi12
- The formalism ℒof predicate logic 124
- The formalism ℒ⁺, a definitional extension of ℒ 2346
- The formalism ℒ⁺ without variables and the problem of its equipollence with ℒ 4568
- The relative equipollence of ℒ and ℒ⁺, and the formalization of set theory in ℒ^{×} 95118
- Some improvements of the equipollence results 147170
- Implications of the main results for semantic and axiomatic foundations of set theory 169192
- Extension of results to arbitrary formalisms of predicate logic, and applications to the formalization of the arithmetics of natural and real numbers 191214
- Applications to relation algebras and to varieties of algebras 231254
- Bibliography 273296
- Indices 283306
- Back Cover Back Cover1342