BOUNDS ON WEAK SCATTERING
GERALD E. SACKS In Memory of Jon Barwise
Contents 1. Introduction 2. Scott Analysis and Rank 3. Small ZF Sets 0 4. Enumeration of Models for Scattered Theories 5. Absoluteness of Vaught' Conjecture s 6. Bounds on Scattered Theories 7. Iterated Classical Bounding 8. Enumeration of Models under Weak Scattering 9. Bounds on Weakly Scattered Theories 10. Further Results and Open Questions References 1 4 6 8 13 15 17 19 24 28 28
1. Introduction This paper has two themes less disparate than they seem at …rst reading: Extending classical descriptive set theoretic results that impose bounds on suitably de…ned functions from ! ! into ! 1 : Extending and clarifying some early results on Scott ranks of countable structures sketched in [12]1. Let F be a function, possibly partial, from ! ! into ! 1 : A typical classical bounding theorem says the range of F is bounded by a countable ordinal if the graph of F has a suitable de…nition. For example, the graph of F is boldface 1 with real parameter p; in this formulation the graph of F 1 is viewed as a subset of ! ! ! 1 by requiring each value of F to be a well ordering of !: Let F (X) ambiguously denote the well ordering and also the
Date: December 9, 2004. 1991 Mathematics Subject Classi…cation. 03C70, 03D60. Key words and phrases. weakly scattered theories, bounds on Scott rank. Many thanks to Julia Knight for her patience and encouragement. 1 [12] was a hasty writeup of a talk given at the 1971 meeting of the International Congress of Logic, Methodology and Philosophy of Science. Some details absent from [12] but needed here are presented below..
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GERALD E. SACKS
ordinal represented by the well ordering. For each X, F (X) is the unique solution of a 1 formula with parameters p; X: Consequently F (X) (the well 1 ordering) is hyperarithmetic in p; X; and so F (X) < ! p;X ; 1 (1.1) the least ordinal not recursive in p; X: The e¤ective version of the theorem says that the bound on the range of F is an ordinal less than ! p : 1 A recursion-theoretic approach to the e¤ective bound originated by Kleene is as follows. (See Sacks[13] for details.) Suppose (8 ]: 1

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