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].

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 |