Degrees of Unsolvability: Structure and Theory
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759 Richard L. Epstein
Degrees of Unsolvability: Structure and Theory
Springer-Verlag Berlin Heidelberg New York 19 7 9
Author: Richard L. Epstein Department of Mathematics Iowa State University Ames, Iowa 50011 USA
AMS Subject Classifications (1980): primary: 03-02, 03 D30, 03 D35, 03F30 secondary: 01 A65, 03-03, 06 D05 ISBN 3-540-09710-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09710-4 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher © by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Dedicated to URSULA
BRODBECK
'~Like one that stands upon a promontory, And spies a far-off shore where he would tread, Wishing his foot were equal with his eye." Shakespeare, Henry VI, Part 3
ABSTRACT
This book presents the theory of degrees of unsolvability in textbook form. It is accessible to any student with a slight background in logic and recursive function theory. Degrees are defined and their basic properties established, accompanied by a number of exercises. The structure of the degrees is studied and a new proof is given that every countable distributive lattice is isomorphic to an initial segment of degrees. The relationship between these initial segments and the jump operator is studied. The significance of this work for the first-order theory of degrees is analyzed: it is shown that degree theory is equivalent to second-order arithmetic. Sufficient conditions are established for the degrees above a given degree to be not isomorphic to and have different first-order theory than the degrees, with or without jump. The degrees below the halting problem are introduced and surveyed. Priority arguments are presented. The theory of these degrees is shown to be undecidable. The history of the subject is traced in the notes and annotated bibliography.
PREFACE AND A C K N O W L E D G E M E N T S January
1973 I went to Barry Cooper to ask him if anyone had
shown that the degrees
< a is u n d e c i d a b l e
What a good idea for a Ph.D.
thesis!
for every n o n - z e r o
Well,
r.e.
a.
it turned out to be harder
than I imagined. By N o v e m b e r of that year I'd w r i t t e n up the b a c k g r o u n d work on the degrees
< 0' w h i c h was n e c e s s a r y
for
it
(that appeared as
And I had a "plausible" proof that the three element phic to an initial segment of the degrees tial first step towards undecidability. for suffering
through this early
I returned to this project proper proof of that embedding.
chain is isomor-
< a for such I'
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