Logic Year 1979–80: The University of Connecticut, USA

333 Pages·7.097 MB·English

by  John T. Baldwin (auth.), Manuel Lerman, James H. Schmerl, Robert I. Soare (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
859
I I
Logic Year 1979-80
The University of Connecticut, USA
Edited by .M Lerman, .J .H Schmerl, and .R .i Soare
galreV-regnirpS
Berlin Heidelberg New York 1891
srotidE
Manuel Lerman
James H. Schmerl
Department of Mathematics, The University of Connecticut
Storrs, CT 06268, USA
Robert I. Soare
Department of Mathematics, The University of Chicago
Chicago, IL 60637, USA
AMS Subject snoitacifissalC :)0891( 03-06, 03C30, 03C45, 03C60,
03C65, 03C75, 03D25, 03D30, 03D55, 03D60, 03 D65, 03D80,
03F30, 03G30
ISBN 3-540-10708-8 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-10708-8 Springer-Verlag NewYork Heidelberg Berlin
yrarbiL of Cataloging Congress in year title: Logic under entry Main Data Publication
1979-80, University the of notes (Lecture Connecticut. in ;scitamehtam 859)
Bibliography: .p index. Includes I. Symbolic Logic, .sessergnoC:-lacitamehtam dna
.I ,namreL .M ,)leunaM( .-3491 .II ,lremhcS .J .H semaJ( ,)yrneH .-0491 III. ,eraoS .R .I treboR(
.-0491 ,)gnivrI .VI in notes Lecture Series: scitamehtam ;)galreV-regnirpS( 859. QA3.L28
510s 859 [QA9.AI] vol. ]3.115[ AACR2 81-5628
0-387-10708-8 ISBN ).S.U(
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PREFACE
Each year the Mathematics Department of the University of Connecticut
sponsors a special year which is an intense concentration in a specific area
of Mathematics. The year ]979-80 was devoted to Mathematical Logic, with
special emphasis on recursion theory and model theory. Visitin~ scholars
from other institutions, either for the whole year or for one of the two
semesters, formed the core of this successful year. Stephen Simpson (Pennsylvania
State University) and David Kueker (University of Maryland) were visitors for
the entire year; Richard Shore (Cornell University) and Robert Soare (University
of Chicago) visited just for the fall semester; and Michael lforley (Cornell
University) and Joram Hirschfeld (Tel--Aviv University) visited just for the
spring semester. Visiting graduate students included: Klaus Ambos, Stephen
~rackin, Steven Buechler, David Cholst, Peter Fejer, David ~'iller, Charles
Steinhorn, and Galen Weitkamp.
The highlight of the year was the Conference on Mathematical Logic, which
took place November 11-13, 1979, at Storrs. There were 80 logicians in
attendance° Included on the program vere ten invited hour addresses, twenty
contributed fifteen minute talks, ar~l two papers presented by title.
~his volume represents the proceedinF.s both of the Logic Year and also
of the Confe:'er~e. ~imost all of the papers include~! herein have been based
eithe1" on talks presented at the Conference or on presentations made to one
of the various seminars, includin~ the joint University of Connec£icut -
Yale - ~Tesleyan logic seminars, that were repularly helN during the course
of the year.
~e Logic Year and the Conference could not have been so successful without
the ~reatly appreciated assistance and cooperation of many organizations and
individuals. We thank the National Science Foundation for financial support
under Frant MCS 79-03308; we thank the Research Foundation for additional
financial assistance; we t~ank the University of Connecticut Office of
Conferences, Institutes and Administrative Services for their able har~!ling
of the organization of the Conference; we thank our consulting editors
Steve Simpson, Richard Shore and David F~aeker for their expertise; and finally
we thank all of those individuals who by attending the Conference contributed
to makin~ it an outstanding event.
M. Lerman
.J Schmer i
R. Soare
CONFERENCE PROGRAM
I. Invited Addresses
i. Herr ing ton ~ Leo: Building Arithmetical Models of Peano Arithmetic.
2. Jockusch, Carl: Some Easy Constructions of r. e. Sets.
3. Macintyre, Angus: Primes in Non-standard Models of Arithmetic.
4. Hakkai, Michael: The Category of Models of a Theory.
5. Millar, Terrence: Topics in Recursive Model Theory.
6. Morley Michael: Theories with few Models.
7. Moschovakis, Yiannis: Ordinal Games and Recursion Theory.
8. Nerode, Anil: Recursive Model Theory and Constructive Algebra.
9. Sacks, Gerald: The Limits of Recursive Enumerability.
i0. Vaught, Robert: Infinitary Languages and Topology.
II. Contributed Papers
i. Baldwin, John: Why Superstable Theories are Super.
.2 Byerly, Robert E.: An Invariance Notion in Recursion Theory.
3. Cherlin, Gregory: Real Closed Rings.
4. DiPaola, Robert: The Theory of Partial ~-Recursive Operators,
Effective Operations, and Completely Recursively Enumerable Classes.
5. Epstein, Richard, Hass, Richard, and Kramer, Richard: A Hierarchy
of Sets and Degrees Below 0'.
6. Fejer, Peter A.: Structure of r.e. Degrees.
7. Glass, Andrew: On Elementary Types of Automorphism Groups of
Linearly Ordered Sets.
8. Kaufmann, Matt: On Existep~e of E End Extensions.
n
9. Kierstead, Henry and Rermmel, Jeffrey B.: On the De~rees of
Indiscernibles in Decidable Models.
i0. Kolaitis, Phokion G.: Spector-Gandy Theorem for Recursion in E and
Normal Functiona .si
Kranakis, Evangelos: On E Partition Relations.
n
12. Maass, Wolfgang: R.E. Generic Sets.
13. Manaster, Alfred: Recursively Categorical, Topologically Dense,
Decidable Two Dimensional Partial Orderings.
14. Odifreddi, Piergiorgio: Strong Recudibilities.
15. Posner, David: The Upper Semilattice of Degrees < O' is Complemented.
16. Slaman, Theodore A. and Sacks, Gerald E.: Inadmissible Forcing.
17. Smith, Kay Ellen: Boolean-Valued Models and Galois Theory for
Comrmatative Regular Rings.
18. Smith, Rick: A Survey of Effectiveness in Field Theory.
Vl
CONFERENCE PROGRAM (CONT.)
II. Contributed Papers (Cont.)
19. Srebrny, Marian: Measurable Uniformization.
20. Watnick, Richard: Recursive and Constructive Linear Orders,
III. Papers Presented by Title
i. Calude, Christian: Category Methods in Computational Complexity.
.2 Miller, David: The Degree of Semirecursive ~-hyperhypersimple sets.
TABLE OF CONTENTS
Baldwin, John T.
Definability and the Hierarchy of Stable Theories . . . . . . . . . . . 1
Berline, C., and Cherlin, G.
QE Rings in Characteristic p . . . . . . . . . . . . . . . . . . . . . . 16
Epstein, Richard L., Haas, Richard and Kramer, Richard L.
Hierarchies of Sets and Degrees Below 0' . . . . . . . . . . . . . . . . 32
Pejer, Peter A. and Soare, Robert I.
The Plus-cupping Theorem for the Recursively Enumerable Degrees .... 49
Friedman, Sy D.
Natural s-RE Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Glass, A.M.W., Gurevich, Yuri, Holland, Charles, W. and
Jamh~-Giraudet, Mich~le
Elementary Theory of Automorphism Groups of Doubly Homogeneous
Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Jockusch, Carl G. Jr.
Three Easy Constructions of Recursively Enumerable Sets ........ 83
l.~aufmann, ~at t
On Existence of Z End Extensions . . . . . . . . . . . . . . . . . . . 92
n
Kolaitis, Phokion G.
Hodel ~neoretic Characterizations in Generalized Recursion Theory . . . 104
Kueker, David
L~I ~elementarily Equivalent Models of Power ~0 I . . . . . . . . . . 120
Lerman, Manuel
On 2ecursive Linear Orderings . . . . . . . . . . . . . . . . . . . . . 132
}~a¢intyr e, Angus
The Complexity of Types in Field Theory . . . . . . . . . . . . . . . . 143
Makkai, M.
The Topos of Types . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Manaster, A. B. and Remmel, J. B.
Some Decisio1~ Problems for Subtheories of Two-dimensional Partial
Ord erings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Millar, Terr ence
Counter-examples via Model Completions . . . . . . . . . . . . . . . . . 215
Miller, David P.
High Recursively Enumerable Degrees and the Anti-cupping Property . . . 230
Moschovakis, Yiannis N.
On the Grilliot-Harrington-MacQueen Theorem . . . . . . . . . . . . . . 246
Schmerl, James H.
Recursively Saturated, Rather Classless Models of Peano Arithmetic . . . 268
Shore, Richard A°
The Degrees of Unsolvability: Global Results . . . . . . . . . . . . . . 283
Smith, Rick L.
Two Theorems on Autostability in p-Groups . . . . . . . . . . . . . . . 302
Watnick, Richard
Constructive and Recursive Scattered Order Types . . . . . . . . . . . . 312
CONFERENCE PARTICIPANTS
Ambos, Klaus Lin, Charlotte
Baldwin, John .T Maass, Wolfgang
Barnes, Robert Macintyre, Angus
Bohorquez, Jaime Makkai, Mihaly
Brackin, Stephen .H Manaster, Alfred
Brady, Stephen Mansfield, R. .B
Bruce, Kim Marker, Dave
Buechler, Steven A. Mate~ Attila
Byer ,yl Robert McKenna, Kenneth
Cherlin, Greg Millar, Terrenc e
DiPaola, Robert Miller, David .P
Cholst, David Morley, Michael
Dorer, David Moschovakis, Y. .N
Dougher ,yt Dan Nerode, Anil
Epstein, Richard L. Odell, David A.
Fejer, Peter ifreddi, Od George
Fisher, Edward R. Posner, David .B
Friedman, Sy D. Poweil, William .C
Glass, .A .M .W Sacks, Gerald
Gold, Bonnie Schmerl, James
Grif for, Edward Scowcroft, Philip
Harrington, Leo Shamash, Josephine
Hay, Louise ield, enf Sho Joseph
do!l ,se Harold Shore, Richard .A
~omer, Steven Simpson, Stephen G.
Hoover, D. N. Slaman, Theodore
Hrbacek, Karel Smith, Carl
Jockusch, Carl, Jr. Smith, Kay
Joseph, Debra Smith, Rick
F~namo ri, A. Smith, Stuart .T
mann, Kauf Matt Soare, Robert
Keisler, .H Jerome ebrny, Sr Marian
Kierstead, H~ary .A Stob, Michael
Kolaitis, Phokion G. Van den Dries, Lou
Kramer, Richard .L Vaught, Robert
Kranakis, Evangelo s Weiss, Michael
Krause, Ralph M. Weitkamp, Galen
Kueker, David ,kcintaT~ Richard
Landrait ,si Charles Weaver, George
Lerman, Manuel Welaish, Jeffrey
Wood, Carol
DEFINABILITY AND THE HIERARCHY OF STABLE THEORIES
John .T Baldwin
tI is well known that a theory T is stable fi and only fi for every A contained ni a
model of T and every type p in S(A), p is definable over A in the following sense:
The type p in S(A) si definable over B by the map d fi for each formula ~(x;y)
there si a formula d~(y) with parameters from B such that for each sequence a in
:A ~(x;a) si in p if and only if d~(a) holds.
In fact, ni ]2[ we proposed that a slight variant of this property be taken as the
definition of a stable theory. There si a natural objection to this proposal; the
usual definition of stable, superstable, and totally transcendental theories in terms
of the cardinality of the space of types yields immediately the hierarchy: totally
transcendental implies superstable implies stable. Is there a similar hierarchy of
definability which defines totally transcendental and superstable ni terms of
"definability of types"? In this paper we provide such a hierarchy. Namely, we will
show the following results.
Let S )T( denote the collection of n-types over the empty set. ~e say T is a small
n
theory if for each n, ISn(T)I~IT .I
THEOREM .I The countable small theory T si totally transcendental fi and only if for
every A contained ni a model of T and every p in S(A), there is a finite subset B of
A such that p is definable over B.
We will define below the concept "p is definable almost over B".
THEOREM .2 The countable theory T si superstable fi and only if for every A
contained in a model of T and every p in S(A), there si a finite B contained in A
such that p is definable almost over .B
Most of the results in this paper are easy corollaries to theorems in [7]. The main
claim to novelty lies in the recognition that a nice hierarchy can be defined in
terms of definability. However, our viewpoint is much different from Shelah's.
Several notions of rank are central to his development. His results and even some of
his definitions depend upon properties of these ranks. In contrast, our development
depends only upon the basic properties of forking as developed either along Shelah's
line or along that of Lascar-Poizat. With one exception which we will discuss later,
the results in this paper hold for uncountable languages with essentially the same
proofs. For simplicity of notation, we concentrate on the countable case. The paper
is designed to be read by anyone who has read III.1 and III.2 of [7] or [5] or[3].
We follow various notational conventions common in this subject which are explained
in these sources. For example, all our constructions take place within a very
saturated "monster model". Since it is not usually important to know the length of a
finite sequence of variables or elements we write x or a omitting the usual
overscore. When the length is important, it is given explicitly.
Section I. The notion of forking (or more precisely non-forking) provides an
explication in a general model theoretic context of the idea of algebraic
independence. In particular, if A C B and t(c;B) does not fork (d.n.f.) over A
(t(c;B) denotes the type of c over )B then, intuitively, "c obeys no more relations
over B than it does over A". More detailed explanations occur in the three
references cited above. More formally, we adopt the following definition.
DEFINITION 1.1. Let A C B and let c be an arbitrary element. Then, t(c;B) forks
over A if there is a formula ~(x;y) a sequence b from B and sequences i b for i <
such that:
)i t(bi;A) = t(b;A) for all i.
ii) ~(x;b) e t(c;B).
iii) The set {~(x;bi): i < w} si n-inconsistent for some .n (That is, no
more than n of these formulas can be simultaneously satisfied.)
This definition is slightly simpler than the one given in [7] but is equivalent to
that definition for stable theories. In fact, the precise definition of forking used
is of little importance for this paper. After the next technical lemma where we rely
on the definition, we will list the principal properties of forking. In the
remainder of the paper (except for 3.4) we will rely not on the definition of forking
but only on the properties listed here.
1.2 LEMMA. Let .a for i ni I be a sequence of n element sequences such that
I
Pi=t(ai,B) d.n.f, over A, where p=piIA. fI D is a ultrafilter on I and a denotes the
ultraproduct of the oa with respect to ,D then a=t(a,B) d.n.f, over .A
i
PROOF. fI ~(x;b) e t(a;B) then for almost all (with respect to )D ,i ~(x;b) e Pi"
But then, since iP d.n.f, over ,A the formula ~(x;b) does not cause t(a;B) to fork
over .A Since this holds for each formula ~(x;b), t(a;B) d.n.f, over .A
1.3 THEOREM. fI T si a stable theory then:
)i fI p e S(A) then p does not fork over .A
)ii If A C B C C and p e S(C) then
)a If p does not fork over A then p d.n.f, over B and plB d.n.f.
over A.
b) If p d.n.f, over B dna pIB d.n.f, over A then p d.n.f, over A.
iii) If A B C C C dna p ~ S(B) d.n.f, over A then there exists na
extension p' of P in S(C) which d.n.f, over .A
iv) If b is in B dna t(B;C) d.n.f, over A then t(b;C) d.n.f, over .A
v) If ~ A B, p e S(A) dna a is na extension of p in S(B) which does
not fork over A dna if p is not algebraic over A, then q is not algebraic
over .B