HOD L(ℝ) is a Core Model Below Θ

Bulletin of Symbolic Logic 1 (1):75-84 (1995)
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Abstract

In this paper we shall answer some questions in the set theory of L, the universe of all sets constructible from the reals. In order to do so, we shall assume ADL, the hypothesis that all 2-person games of perfect information on ω whose payoff set is in L are determined. This is by now standard practice. ZFC itself decides few questions in the set theory of L, and for reasons we cannot discuss here, ZFC + ADL yields the most interesting “completion” of the ZFC-theory of L.ADL implies that L satisfies “every wellordered set of reals is countable”, so that the axiom of choice fails in L. Nevertheless, there is a natural inner model of L, namely HODL, which satisfies ZFC.. The superscript “L” indicates, here and below, that the notion in question is to be interpreted in L.) HODL is reasonably close to the full L, in ways we shall make precise in § 1. The most important of the questions we shall answer concern HODL: what is its first order theory, and in particular, does it satisfy GCH?These questions first drew attention in the 70's and early 80's.

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Citations of this work

The comparison lemma.John R. Steel - forthcoming - Annals of Pure and Applied Logic.
In inner models with Woodin cardinals.Sandra Müller & Grigor Sargsyan - 2021 - Journal of Symbolic Logic 86 (3):871-896.

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References found in this work

Set Theory.Thomas Jech - 1999 - Studia Logica 63 (2):300-300.
Inner models with many Woodin cardinals.J. R. Steel - 1993 - Annals of Pure and Applied Logic 65 (2):185-209.
Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.
Thin collections of sets of projective ordinals and analogs of L.Howard Becker - 1980 - Annals of Mathematical Logic 19 (3):205-241.

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