Models and Sets: Proceedings of the Logic Colloquium held in Aachen, July 18–23, 1983 Part I

489 Pages·1984·5.729 MB·English

by  Jos Baeten (auth.), Gert H. Müller, Michael M. Richter (eds.)

Lecture Notes ni
Mathematics
Edited yb .A Dold dna .B Eckmann
3011
Models dna Sets
Proceedings of the Logic Colloquium
held ni Aachen, July 18-23, 1983
Patti
Edited yb .G .H Meller and M.M. Richter
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
Editors
Gert .H rellUM
Heidelberg Universit~t Institut, Mathematisches
mI Feld Neuenheimer 294, 6900 Heidelberg, laredeF Republic of Germany
Michael .M Richter
Lehrgebiet ehcsitamehtam Grundlagen der ,kitamrofnI Aachen RWTH
nebargrelpmeT ,46 Aachen, 5100 laredeF Republic of Germany
AMS Subject Classification :)0891( 03 ,C 03 ,E 03 ,G 03 H
ISBN 1-00931-045-3 galreV-regnirpS York New Heidelberg Berlin oykoT
ISBN 1-00931-783-0 galreV-regnirpS Berlin Heidelberg York New oykoT
This work is subject to copyright. All rights are reserved, whether the whole or part of the material
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© by Springer-Verlag Berlin Heidelberg 1984
Printed in Germany
Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
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VORWORT
Dieser Band enth~it einen Teil der Proceedings des Logic Colloquiums
'83, welches vom 18. - 23. Juli 1983 in Aachen stattfand; es war
dies gleichzeitig der Europiische SommerkongreB der Association for
Symbolic Logic. Ein weiterer Band der Proceedings erscheint unter
dem Titel "Computation and Proof Theory" ebenfalls in den Lecture
Notes in Mathematics des Springer-Verlages.
Insgesamt hatte das Logic Colloquium '83 189 Teilnehmer aus 26
L~ndern. Zusitzlich zu den eingeladenen Hauptvortr~gen wurden
siebzig angemeldete Vortr~ge gehalten. Ein Tell davon fand in
"Special Sessions" statt: Boole'sche Algebren (organisiert von
S.Koppelberg), Topologische Modelltheorie (J.Flum), Nonstandard
Analysis (K.H.Diener), Logic versus Computer Science (E.B6rger).
Abstracts aller angemeldeten Vortrige sowie eine Vollstindige Liste
aller eingeladenen Vortr~ge werden im Bericht der Veranstalter im
Journal of Symbolic Logic ver~ffentlicht.
Das Logic Colloquium '83 wurde erm~glicht durch groBz~gige
finanzielle UnterstOtzung der Deutschen Forschungsgemeinschaft, des
Landes Nordrhein-Westfalen, der Division of Logic, Methodology and
Philosophy of Science, dem Deutschen Akademischen Austauschdienst,
der Deutschen Stiftung fur Internationale Entwicklung, der
Maas-Rhein-Euregio, der Stadt Aachen, der RWTH Aachen und nicht
zuletzt der deutschen Industrie. Ihnen allen sei herzlich gedankt!
Die alte Kaiserstadt Aachen gab einen w~rdigen Rahmen ab.
Oberstadtdirektor Dr. H. Berger erbffnete den KongreB als Schirmherr
im Namen der Maas-Rhein-Euregio. Die erste BUrgermeisterin Frau
Prof. Dr. W. Kruse lud ein zu einem Empfang im Kr~nungssaal des
Rathauses, der fur die Teilnehmer zu einem bleibenden Erlebnis
wurde. Auch hierfUr ein herzliches "Danke"!
Die Herausgeber
CONTENTS
VORWORT III
J. Baeten I
Filters and Ultrafilters over Definable Subsets
of Admissible Ordinals
B. Benninghofen
Superinfinitesimals and the Calculus of the
Generalized Riemann Integral
A.J. Berner, I. Juhasz 53
Point-Picking Games and HFD's
R. Bonnet 67
On Homomorphism Types of Superatomic Interval
Boolean Algebras
G.L. Cherlin* 83
Decidable Theories of Pseudo-Algebraically
Closed Fields
G.L. Cherlin* 102
Definability in Power Series Rings of Npnzero
Characteristics
G.L. Cherlin, H. Volger 113
Convexity Properties and Algebraic Closure Operators
J. Czelakowski 147
Remarks on Finitely Based Logics
J.M. Font 169
Monadicity in Topological Pseudo-Boolean Algebras
.W Hodges* 193
Finite Extensions of Finite Groups
G.F. van der Hoeven, I. Moerdijk* 207
Constructing Choice Sequences from Lawless Sequences
of Neighbourhood Functions
E. Kranakis, I. Phillips 235
Partitions and Homogeneous Sets for Admissible Ordinals
.W Lenski 261
Elimination of Quantifiers for the Theory of Archimedean
Ordered Divisible Groups in a Logic with Ramsey Quantifiers
S.C. Liu 281
A Proof-Theoretic Approach to Non Standard Analysis
(Continued)
K.L. Manders 297
Interpretations and the Model Theory of the Classical
Geometries
A. Marcja, C. Toffalori 331
On Cantor-Bendixson Spectra Containing (I ,I) - I (°)
Vl
D. Mundici 351
Abstract Model-Theory and Nets of C*-Algebras:
Noncommutative Interpolation and Preservation
Properties
R. Murawski 379
A Contribution to Nonstandard Teratology
P.H. Schmitt 389
Model- and Substructure Complete Theories of Ordered
Abelian Groups
V. Weispfenning* 419
Quantifier Elimination and Decision Procedures for
Valued Fields
Ph. Welch I 473
On 2 Z
* Invited Lecture
TABLE FO STNETNOC - PART 11
(published in L~I vol. )4011
VORWORT
.K AMBOS-SPIES *
Contiguous R.E. Degrees
H.-G. CARSTENS, P. PNPPINGHAUS * 39
Abstract Construction of Counterexamples in Recursive
Graph Theory
C.T. CHONG, C.G. JOCKUSCH 63
Minimal Degrees and 1-generic Sets below 0'
J.N. CROSSLEY, J.B. REMMEL * 79
Undecidability and Recursive Equivalence II
L. DENENBERG, H.R. LEWIS lol
Logical Syntax and Computational Complexity
E.C. DENNIS-JONES, S.S. WAINER 117
Subrecursive Hierarchies via Direct Limits
E.J. FARKAS, M.E. SZABO * 129
A Star-Finite Relational Semantics for Paral]el Programs
S. FEFERMAN 143
Between Constructive and Classical Mathematics
.G GERMANO, S. MAZZANTI * 163
Partial Closures and Semantics of While: Towards
an Iteration-Based Theory of Data Types
Y. GUREVICH 175
Toward Logic Tailored for Computational Complexity
.P HAJEK 217
On a New Notion of Partial Conservativity
P.G. HINMAN 233
Finitely Approximable Sets
Ch. KREITZ, .K WEIMRAUCH* 259
A Unified Approach to Constructive and Recursive
Analysis
.P LINDSTRUM* 279
On Faithful Interpretability
Y.N. MOSCHOVAKIS 289
Abstract Recursion as a Foundation for the Theory
of Algorithms
D. RUDDING t 363
Some Logical Problems Connected with a Modular Decomposition
Theory of Automata
VIII
U.R. SCHMERL * 389
Diophantine Equations in a Fragment of Number Theory
P. SCHROEDER-HEISTER ~ 399
Generalized Rules for Quantifiers and the Completeness
of the Intuitionistic Operators &, v, D, ~, V, 3
P.H. SLESSENGER * 427
On Subsets of the Skolem Class of Exponential Polynomials
D. SPREEN, .P YOUNG * 437
Effective Operators in a Topological Setting
T. UESU ~ 453
An Axiomatization of the Apartness Fragment of the
Theory DLO + of Dense Linear Order
t Professor Dr. D. R6dding died on June ,~4 1984
* An asterisk indicates a contributed paper.
FILTERS AND ULTRAFILTERS OVER DEFINABLE SUBSETS
OF ADMISSIBLE ORDINALS
by Jos Baeten, Technische Hogeschool Delft, The Netherlands.
This article is part of the author's Ph.D. thesis, to appear
under the same title at the University of Minnesota.
0. Abstract
The search for a recursive analogue of a measurable cardinal
leads to a study of filters and ultrafilters over certain definable
subsets of an admissible ordinal, using the hierarchy of
constructible sets.
Connections with admissibility are explored in sections 2 and 3,
and we find that the existence of a normal filter is stronger than
the existence of the same type of filter (section 3). We look at the
analoghes of certain classical filters, namely the co-finite filter
in section 2 and the normal filter of closed unbounded sets in
section 3.
In section 4, we find that any filter (resp. normal filter) of a
certain type can, on a countable ordinal, be extended to an
ultrafilter (resp. normal ultrafilter) of the same type.
2 J.Baeten
I. Preliminaries and notation
Lower case Greek letters represent ordinals, and lower case Latin
letters represent non-negative integers. We work in the constructible
hierarchy L = U{L~ :~eOrd}. The L4vy hierarchy of ~-m' ][m and A_ m
formulas is defined as usual. A relation R on K L is ~L K (where ~ is a
set of e- formulas), if it is definable by a {-formula with
parameters from L~. An ordinal M iS~n-admissible if for all ~ e ~L~
and all ~<~ LK ~ ~/B<~ ~ ) ~ ~Vg<~<~ ~(~,~).
Pow(~) is the powerset of ~, /k{X~ : 8<M} = {~< ~: ~B<~ ~ eXs},
X\Y = {xex : x~Y}.
Note: to avoid unnecessary complications we will always assume ~ is
an ordinal with ~K=~ and n>0.
2. Filters
2.1 Definition: Let F~Pow(~) be a nonprincipal filter on M, i.e.
i. X, YeF >- X~Y e F
ii. X e F & X~ Y >- Y e F
iii. ~ ~ F
iv. ~;4~<K ~&\{B} ~ F.
Let ~ be a sat of e-formulas (usually Zn, ~n or /k s).
a. F is a ~-filter if
~Q<K ~/<X B : 8<~> e ~L~]~F 6]{X~ : ~<~} e F.
b. F is a ~-normal filter if
V<x~ : 8<K> e ~L~/~F /k{X 8 : 8<~(} e F.
c. F is ~-ultra if X/~ e ~L M X e F or &<\X e F.
d. F is a ~-ultrafilter if a. and c. hold;
F is a ~-normal ultrafilter if b. and c. hold.
2.2 Note: a /kn-filter is often considered to be defined just on the
Boolean algebra of KLn___~ sets; a ~n- or ~n-filter on the Boolean
algebra generated by the ~nL ~ sets (the so-called BnL ~ sets).
2.3 Definition: H = {X~ : ~\X is bounded in ~}.
Note that if F is a /kl-filter, then H~F.
For the next theorem we need a lemma from Kranakis [1982a]:
2.4 Lemma: The following are equivalent:
i. K is ~n+l-admissible
ii. For all ~<K and all _A~nL f : ~--> p there is a ~<~ such that
f-l({8}) is cofinal in M.
2.5 Theorem: The following are equivalent:
i. ~W is ~n+l-admissible
ii. there is a /kn-filter on
iii. there is a ~n-filter on
proof : iii >- ii : immediate.
ii >- i: we use 2.4. Let F be a /~n-filter on .,V Suppose ~<I(,
f :~--> ~ is /knL ~ but for each 8<~ f-l({fi}) is bounded in .~I Then
-i
for each ~<~ we have that ~\f ({~}) e H~.F (by 2.3), so
= t-%{v,\f-l({8}) : ~<~} e F, a contradiction.
i ~- iii: we show H is a ~n-filter on .(~ It is easy to see that
H is a nonprincipal proper filter on ~. Thus let ~<H and
<X~ : ~<~> e ~nL~;q~H. Take ~ e ~nL~ such that
e X~ <~ L~ ~(~,~) (for ~<~). Then ~)L ~ ~/B<~ (0~ ~>j~ ~(~,~).
Since ,~ iS~n+l-admissible , there is a ~<~ such that
~ ~ ~<~ 3,x<,~ ~.>~V Tc~,~), o~ L~ ~ ~.>~V c~<~ ~c~,~)).or
/~{X s : S<~} e H.