Sacks forcing does not always produce a minimal upper bound

AbstractTheorem. There is a countable admissible set, Ol, with ordinal ωCK1 such that if S is Sacks generic over Ol then ω1S > ωCK1 and S is a nonminimal upper bound for the hyperdegrees in Ol. (The same holds over Ol for any upper bound produced by any forcing which can be construed so that the forcing relation for Σ1 formulas is Σ1.) A notion of forcing, the “delayed collapse” of ωCK1, is defined. The construction hinges upon the symmetries inherent in how this forcing interacts with Σ1 formulas. It also uses Steel trees to make a certain part of the generic object Σ1 over the final inner model, Ol, and, indeed, over many generic extensions of Ol

Models Without Indiscernibles Author

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Speech Communication

Contains table of contents for Part IV, table of contents for Section 1 and reports on five research projects.Apple Computer, Inc.C.J. Lebel FellowshipNational Institutes of Health (Grant T32-NS07040)National Institutes of Health (Grant R01-NS04332)National Institutes of Health (Grant R01-NS21183)National Institutes of Health (Grant P01-NS23734)U.S. Navy / Naval Electronic Systems Command (Contract N00039-85-C-0254)U.S. Navy - Office of Naval Research (Contract N00014-82-K-0727

Interpolation theorems for program schemata

A schema is thought of as determining a class of structures—namely, those interpretations for the schema for which a convergent computation occurs. (This view is analogous to the ordinary use of first-order formulas to determine a class of structures.) Model theoretic questions, such as compactness, have been explored in this context. This paper answers the question of whether or not the Craig Interpolation Theorem holds for program schemata, and for several extensions of program schemata. It is shown that, in a certain setting, there is a trade-off between implicit versus explicit definition and bounded versus unbounded storage requirements