Table Of ContentHANDBOOK OF
RECURSIVE MATHEMATICS
Volume 2:
Recursive Algebra, Analysis and Combinatorics
STUDIES IN LOGIC
AND
THE FOUNDATIONS OF MATHEMATICS
VOLUME 139
Honorary Editor:
P. SUPPES
Editors:
S. ABRAMSKY, London
S. ARTEMOV, Moscow
R.A. SHORE, Ithaca
A.S. TROELSTRA, Amsterdam
ELSEVIER
AMSTERDAM (cid:12)9L AUSANNE (cid:12)9N EW YORK (cid:12)9O XFORD (cid:12)9S HANNON (cid:12)9S INGAPORE (cid:12)9T OKYO
HANDBOOK OF
RECURSIVE MATHEMATICS
Volume 2"
Recursive Algebra, Analysis and Combinatorics
Edited by
Yu. L. ERSHOV A. NERODE
Sobolev Institute of Mathematics Department of Mathematics
pr. Akademika Koptuga, 4 Cornell University
Novosibirsk, 630090, Russia Ithaca, NY 14853- 7901, U.S.A.
S.S. GONCHAROV J.B. REMMEL
Sobolev Institute of Mathematics Department of Mathematics
pr. Akademika Koptuga, 4 University of California at San Diego
Novosibirsk, 630090, Russia La Jolla, CA 92093, U.S.A.
Associate Editor
V.W. MAREK
Department of Computer Science
University of Kentucky
Lexington, KY 40506, U.S.A.
1998
ELSEVIER
AMSTERDAM (cid:12)9 LAUSANNE (cid:12)9 NEW YORK (cid:12)9 OXFORD (cid:12)9 SHANNON (cid:12)9 SINGAPORE (cid:12)9 TOKYO
ELSEVIER SCIENCE B.V.
Sara Burgerhartstraat 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Library of Congress Cataloging in Publication Data
A catalog record from the Library of Congress has been applied for.
ISBN: 0 444 50106 1 volume 2 (SiL vol. 139)
ISBN: 0 444 50003 0 volume 1 (SiL vol. 138)
ISBN: 0 444 50107 X (set: vols. 1 & 2)
(cid:14)9 1998 Elsevier Science B.V. All rights reserved.
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Printed in The Netherlands.
Contents
Introduction to the Handbook of Recursive Mathematics vii
Yu. L. Ershov, S. S. Goncharov, A. Nerode and J. B. Remmel
Part One Recursive Model Theory
1 Pure Computable Model Theory 3
V. S. Harizanov
2 Elementary Theories and their Constructive Models 115
Yu. L. Ershov and S. S. Goncharov
3 Isomorphic Recursive Structures 167
C. J. Ash
4 Computable Classes of Constructive Models 183
V. P. Dobritsa
5 E-Definability of Algebraic Structures 235
Yu. L. Ershov
6 Autostable Models and Algorithmic Dimensions 261
S. S. Goncharov
7 Degrees of Models 289
J. F. Knight
8 Groups of Computable Automorphisms 311
A. S. Morozov
9 Constructive Models of Finitely Axiomatizable Theories 347
M. G. Peretyat 'kin
10 Complexity Theoretic Model Theory and Algebra 381
D. Cenzer and J. B. Rernmel
11 A Bibliography of Recursive Algebra and Recursive Model Theory 515
I. Kalantari
12 A Bibliography of Recursive Analysis and Recursive Topology 583
V. Brattka and I. Kalantari
Part Two Recursive Algebra, Analysis,
and Combinatorics
13 II ~ Classes in Mathematics 623
D. Cenzer and J. B. Remmel
14 Computability Theory and Linear Orderings 823
R. G. Downey
15 Computable Algebras and Closure Systems: Coding Properties 977
R. G. Downey and J. B. Remmel
16 A Survey of Recursive Combinatorics 1041
W. Gasarch
17 Constructive Abelian Groups 1177
N. G. Khisamiev
18 Recursive and On-Line Graph Coloring 1233
H. A. Kierstead
19 Polynomial-Time Computability in Analysis 1271
K. Ko
20 Generally Constructive Boolean Algebras 1319
S. P. Odintsov
21 Reverse Algebra 1355
S. G. Simpson and J. Rao
Introduction to the Handbook of
Recursive Mathematics
Y. L. Ershov, S. S. Goncharov,
A. Nerode, and J. B. Remmel
Preface
Recursive or Computable Mathematics is the study of the effective or com-
putable content of the techniques and theorems of Mathematics. Recursive
Mathematics has been an active area of research for the last 25 years. Its
tools are the techniques of modern Computability Theory. These tools have
been applied to analyze the effective content of results in a wide variety of
mathematical fields including Algebra, Analysis, Topology, Combinatorics,
Logic, Model Theory, Algebraic Topology, and Mathematical Physics.
This volume consists of surveys and new results from many of the leading
researchers in the field. Almost all of the major subfields of Recursive Mathe-
matics are represented in the volume. There are contributions which analyze
the effective content of results from Algebra, Analysis, Combinatorics, Or-
derings, Logic, Non-monotonic Logics, Topology and Model Theory.
In this introduction, we will describe the historical roots of Recursive
Mathematics as well as summarize some of the major themes that have arisen
in the study of Recursive Mathematics and that are covered by the articles
in this volume. We also provide a brief description of the articles of this
volume. Finally we will end with what the editors believe are promising
areas for further research.
HANDBOOK OF RECURSIVE MATHEMATICS vii
Edited by Yu. L. Ershov, S. S. Goncharov,
A. Nerode, and J. B. Remmel
{~) 1998 Elsevier Science B.V. All rights reserved.
viii Y. L. Ershov, S. S. Goncharov, A. Nerode, and, J. B. Remmel
Historical roots of recursive and
constructive mathematics
Mathematics consists of constructions of new mathematical objects from old
and of proofs that these objects have certain properties and lack other prop-
erties. Cantor's work on set theory in the late nineteenth century presented
to the mathematical world what Hilbert called the paradise of set theory.
Although Cantor's work was controversial at that time, the set theoretic
point of view is at present widely accepted. Thus in the twentieth century,
most working mathematicians have accepted Zermelo-Fraenkel set theory as
the foundation of mathematics within which their work takes place. That is,
almost all working mathematicians generally accept that Zermelo-Fraenkel
set theory provides a foundation for mathematics with the caveat that occa-
sionally one might have to extend ZF or ZFC by new axioms to accommodate
certain proofs and constructions.
Within Zermelo-Fraenkel set theory, sets are built up by operations which
construct sets from sets such as union, intersection, and power set. In addi-
tion, one employs comprehension-like schemes which assert that a set exists
consisting of all elements having a property expressed in the language of set
theory. The expression of the property in the language of set theory may it-
self involve sets as is the case with the axiom of replacement. Moreover, one
assumes that the basic objects, sets, are extensional. That is, if two different
properties are satisfied by the same elements, and one property defines a set,
then the other property defines the same set.
However there are much narrower conceptions of mathematical objects,
constructions, and proofs proposed by finitists and constructivists. In one
way or another, they can be more or less adequately described within the
classical Zermelo-Fraenkel set theory, but some of them are better viewed as
alternate foundations of mathematics.
The earliest instance of such constructivist tendencies is the classical
Greek fascination with what can be constructed by limited means. For in-
stance consider the concept of a ruler and compass construction. Here one
is naturally led to ask questions like: "What are the ruler and compass con-
structible regular polygons?" or "Is a segment the length of a circumference
of circle ruler and compass constructible from a segment the length of a
diameter?".
Introduction to the Handbook of Recursive Mathematics ix
Algebra as developed by the Greek Diophantus and by the Persian al-
Khwgrismi and his successors was wholeheartedly constructive. They wanted
to find formulas for the solution of algebraic equations. In fact, it is thought
by some that the word "algorithm" comes from the name al-Khwgrismi.
Descartes' seventeenth century coordinatization of Geometry was intended
as a reduction of geometric construction problems to the problem of solving
algebraic equations. This endeavor was highly constructive. In fact, the
explicit formula-based approach to Algebra continued into the nineteenth
century to the time of Galois and Abel. By then negative results could be
obtained since, through Lagrange, the notion of a solution by radicals of the
fifth degree equation became precise and Galois' theory showed there were
no such solutions for the general fifth degree equation.
Of the two strongest algebraists of the middle of the nineteenth century,
Kronecker and Dedekind, Kronecker was the one to develop explicit formula
and algorithms for all of his complex and Diophantine Algebraic Geometry.
Kronecker's philosophy was that the true subject matter of mathematics
is concretely representable structures whose operations are combinatorial.
Dedekind preferred a set theoretic approach, which was not algorithmic. One
need merely contrast Dedekind's set-theoretic definition of prime ideals and
primary ideal factorizations with Kronecker's wholly constructive description
via polynomials found by factoring algorithms. Dedekind explicitly said that
he wanted set-theoretic definitions which are independent of the algorithm
used. Thus Dedekind distinguishes himself from Kronecker and started to
lead us away" from algorithmic mathematics and toward Cantor's absolutist
set theory.
In Analysis, Newton and Leibniz put great emphasis on solving geometric
and physical problems by translating them to purely symbolic forms. Newton
especially emphasized power series. For physicists ever since, publishing
proofs without algorithms is just not satisfactory. Almost all of Analysis
was highly constructive till the Weierstrassian revolution in the 1850's when
exact definitions and proofs were developed for real n-dimensional Analysis
and pure existence proofs using compactness arguments started to become
popular. After the time of Dedekind in Algebra and Weierstrass in Analysis,
general non-algorithmic set-theoretic methods took hold.
However, very early on, mathematicians, including Cantor himself,
realized that there were paradoxes within naive set theory. The response
to these paradoxes led to an extensive interest in the foundation of mathe-
matics by many of the leading mathematicians including the likes of Hilbert
