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THIS DISSERTATION LA THESE A ETE HAS BEEN MICROFILMED MICROFILMEE TELLE QUE EXACTLY AS RECEIVED NOUS L'AVONS REGUE 0 \- I SOME ASPECTS OF BOOLEAN-VALUED MODEL THEORY - F. - ;, 9 Ronald B. Morrow B.A. (Hons), University of Western Ontario, 1972 / J / ' A THESIS SUBMLTTED IN PARTIAL FULFILLMENT OF 9 in the Departmen; ,, . Mathematics @ , RONALDB. MORROW1977 P SIMON FRASER UNIVERSITY July 1977 All rights reserved. This thesis may not be ; reproduced in whole or in part, by photocopy or other means, without permission of the author. d APPROVAL Name : Ronald B. Morrow y, Degree : Master of Science Title of Thesis: Some Aspects of Boolean-valued Model Theory , '\. L Examining Committee: Chairman: Dr. E. Pechlaner - S .K. Thomason, Senior Supervisor . : A.H. ~achlan - -, B. R. Alspach, -- .. , . c .~J~hen External Examiner Date Approved: July 6, 1977 PARTIAL COPYRIGHT LICENSE I hereby grant to Simon Fraser University the right to lend my thesis or dissertation (the title of which is shown below) to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users. 4 further agree that permission f0.r s, mu1 tiple copying of this thesis for scholarly purposes may be granted by me or the Dean of ~raduaee-Studies. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Title of ThesisIDi ssertation : Author: ' .-- , ( signature) 8/77 (date,) ABSTRACT We define the notion of Boolean-valued structure and the semantic tools: morphism, factor structure, product structure, extension;" Boolean power and bounded Boolean power. We investigate , t some of the relationships among these concepts, and prove the , 1 ,Y :I A, - .,- .? 7 I:,"'f 'C 8 completeness theorem for Boolean-valued logic. Fkom this, we prove . i I I ,. the completeness theorem for 2-valued logic using the Boolean power!' - \ We show that the extension relation has,the amalgamati& 3 ir property; that is, for any Boolean-valued structure A. ana.;2ny extensions Al and A2 of A , there is a Booleanvalued 0 5' ' structure A which extends both A1 and A2 . We define stronger 3 - notions of extension, and show that elementary and finitely / contiguous extension have the amalgamation property, whereas contiguous extension does not. By an elementary tower construction we prove an amalgamation result which is equivalent to a generalized version of Robkson's Consistency Lemma. Also, we extend a result of Shelah by showing that elementarily equivalent Boolean-valued structures have isomorphic reduced Boolean powers. (iii) ACKNOWLEDGMENTS 2' I wish to thank my senior supervisor Steve Thomason, also Dolly Rosen who typed the bulk of the thesis, and especially my office mate Robert Woodrow. He was always willing to discuss and and suggesttideas, to.he1p in the other facets of produeg'. the thesis. TABLE OF CONTENTS. C Title Page ........................................................ i d. Approval .......................................................... ii . Abstract ................................L.. ........................ iii 1 -. * --. Acknowledgments .............................................. -/' / iv Introduction ........................................................ 1 . Chapter 1. The Definition of B-structure and the Validity Theorem. 3 Chapter 2. Substructure Morphism Factor' Structure and Product Structure. .............................................../ 13 / 52.4 ........................................................ 22 52.5 ........................................................ 23 . , Chapter 3. The Boolean Power Construction arid the:Cornpleteness Theorem. ................................................... 26 0 Chapter 4. Substructure and the Amalgamation Property ............ 43 Page Chapter 5. ~lementary~ubstructureand the Amalgamation Property ... 49 , , Chapter. 6. Extending a Result of Shelah .......................... & 56.1 ...................................................... Y Chapter 7. Two Other Notions of Substructure and the Amalgamation -2 Property .................................................. Appendix ............................................................- 98 4 Introduction Our interest in Boolean model thepry stems from Alasdair /"\ Urquhart's result [13] that a class, K, of Boolean-valued structures is elementary iff K is closed under product, elementary substructure and 1 elementary morphism. This prompted our reading of two unpublished papers by Urquhart wherein he offers an algebraic version of logic based on Boolean-valued structures. The first two chapters of this thesis attempt -. to lay a firm foundation for Boolean model theory by proving the validity theorem for B-structures, and by investigating some of the relationships among the basic definitions of substructure, morphism and product; conse- quently, these-chapters are somewhat detaile 'In Chapter 3, the canonical B-model \for a eory T is con- strutted; it is especially useful because it..modelsexactly the theorems of T (see also $5.4). A Boolean power of this B-model ppvides an *+' elementary extension which can be factored to produce a of T . Thus, a proof of the 2-valued completeness theorem follows from the Boolean-valued completeness theorem. At the end of his second paper, Urquhart.proves, in part, t& t if A is a B-structure for L = L n L2 and A1 and A2 are 0 0 1 B-structures for L and L2 respectively, such that A and A2 1 , 1 are elementary L -extensions of Ao, then there is a B-structure A3 for 0 F L1 U L which is an elementary L -extension of Al and an elementary - - 2 1 L -extension of This result'is equivalent to a generalized version 2 A2 . of Robinson's Consistency Lemma. Since Urquhart's proof detours through syntax and also depends on the standard proof of the interpolation theorem, ' he comments. t dt it would be desirable to have a direct-model-theoretic L proof. In Chapter 5 we present one which'uses the Boolean power and elementary tower constructions. In Chapter 6'we Gve another proof which uses reduced Boolean powers and extends a result of Shelah [lo]. t The above-mentioned theorem is an amalgamation result. We say b e that a relation R ud the class of B-structures for a language 'L has * the amalgamation property if for any -Ao, A1 and A for 2 L such that A R A and A. R A A for L 0 1 2 i' 3 such that Al R A3 and A2 R A3 . i Having observed that elementary extension has this property, &we naturally asked whether the relation of extension also. enjoys it. In Chapter 4 we answer this affirmatively. In Chapter 7 we define the notion of contiguous extension, finitely