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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].

##### MSC:
 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