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Scales in \({\mathbf K}(\mathbb R)\). (English) Zbl 1167.03032
Kechris, Alexander S. (ed.) et al., Games, scales, and Suslin cardinals. The Cabal Seminar, Vol. I. Reprints of papers and new material based on the Los Angeles Caltech-UCLA Logic Cabal Seminar 1976–1985. Cambridge: Cambridge University Press; Urbana, IL: Association for Symbolic Logic (ASL) (ISBN 978-0-521-89951-2/hbk). Lecture Notes in Logic 31, 176-208 (2008).
From the introduction: In this paper, we extend the fine-structural analysis of scales in \(\mathbf{L}(\mathbb R)\) [J. R. Steel, “Scales in \({\mathbf L}(\mathbb R)\)”, ibid. 130–175 (2008; Zbl 1159.03323), reprint from Lect. Notes Math. 1019, 107–156 (1983; Zbl 0529.03028)] and \({\mathbf L}(\mu,\mathbb R)\) [D. Cunningham, The real core model. PhD thesis, UCLA (1990)] to models of the form \({\mathbf L}(\vec E,\mathbb R)\), constructed over the reals from a coherent sequence \(\vec E\) of extenders. We show that in the natural hierarchy in an iterable model of the form \({\mathbf L}(\vec E,\mathbb R)\) satisfying AD, the appearance of scales on sets of reals not previously admitting a scale is tied to the verification of new \(\Sigma_1\) statements about \(\vec E\) and individual reals in exactly the same way as it is in the special case \(\vec E=\emptyset\) of [Steel, loc. cit.]. For example, we show:
Theorem 1.1. Let \({\mathcal M}\) be a passive, countably iterable premouse over \(\mathbb R\), and suppose \({\mathcal M}\models \text{AD}\); then the pointclass consisting of all \(\Sigma^{\mathcal M}_1\) sets of reals has the scale property.
Section 2 is devoted to preliminaries. In Section 3 we show that for any \(\mathbb R\)-mouse \(\mathcal M\) satisfying “\(\Theta\) exists”, \(\mathbf{HOD}^{\mathcal M}\) is a \(T\)-mouse, for some \(T\subseteq\Theta^{\mathcal M}\). We use this representation of \(\mathbf{HOD}^{\mathcal M}\) in the proof of Theorem 1.1, which is given in Section 4. There we also extend the proof of Theorem 1.1 so as to obtain a complete description of those pointclasses which have the scale property and are definable over initial segments of \({\mathbf K}(\mathbb R)\) satisfying AD.
For the entire collection see [Zbl 1149.03002].

03E15 Descriptive set theory
03E45 Inner models, including constructibility, ordinal definability, and core models
03E60 Determinacy principles
91A44 Games involving topology, set theory, or logic