<b>HOD</b>as a core model

Steel, John R.; Woodin, W. Hugh; Kechris, Alexander S.; Löwe, Benedikt

doi:10.1017/cbo9781139519694.010

Cited by 53 publications

(102 citation statements)

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“…The following lemmas are the analogues of Theorems 3.14 and 3.16 in [StW16]. The proofs are similar to the ones in [StW16] and [SchlTr] and use absoluteness as in the proof of Lemma 5.6; we omit further details. .…”

Section: Moreover We Inductively Definementioning

confidence: 93%

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The Consistency Strength of Long Projective Determinacy

Aguilera¹,

Müller²

2019

J. symb. log.

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We determine the consistency strength of determinacy for projective games of length ω 2 . Our main theorem is that Π 1 n+1 -determinacy for games of length ω 2 implies the existence of a model of set theory with ω + n Woodin cardinals. In a first step, we show that this hypothesis implies that there is a countable set of reals A such that Mn(A), the canonical inner model for n Woodin cardinals constructed over A, satisfies A = R and the Axiom of Determinacy. Then we argue how to obtain a model with ω + n Woodin cardinal from this.We also show how the proof can be adapted to investigate the consistency strength of determinacy for games of length ω 2 with payoff in R Π 1 1 or with σ-projective payoff. The pointclass of all σ-projective sets is the smallest pointclass closed under complements, countable unions, and projections, where countable unions refer to sets which are subsets of the same product space. Moreover, we as usual identify R and ω ω.2 In fact, an argument as in [AMS, Proposition 2.7] with a more careful analysis of the complexity of the payoff sets (using projective determinacy) shows that these two determinacy hypotheses are equivalent. 3 We would like to thank the referee for asking whether (4) and (5) could be added to Corollary 1.2; see also [AgMu].

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Section: Moreover We Inductively Definementioning

confidence: 93%

“…As usual, this notion can easily be generalized to stacks of normal trees, but we omit the technical details. The interested reader can find them in a different setting for example in [StW16], [Sa13], or [MuSa]. The following lemmas are the analogues of Theorems 3.14 and 3.16 in [StW16].…”

Section: Moreover We Inductively Definementioning

confidence: 99%

The Consistency Strength of Long Projective Determinacy

Aguilera¹,

Müller²

2019

J. symb. log.

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“…We start by introducing pre-n-suitable and n-suitable premice. Our definition will generalize the notion of suitability from Definition 3.4 in [StW16] to n > 1. For technical reasons our notion slightly differs from n-suitability as defined in Definition 5.2 in [Sa13].…”

Section: Proving Iterabilitymentioning

confidence: 99%

“…So there are iterates H * of H and N * of N ||ξ such that H * N * . In fact, H N ||ξ N |δ N since the coiteration takes place above α, ρ ω (H ) ≤ α, ρ ω (N ||ξ) ≤ α, and both premice are sound above α. Analogous to Definitions 3.6 and 3.9 in [StW16] we define a notion of short tree iterability for pre-n-suitable premice. Informally a pre-n-suitable premouse is short tree iterable if it is iterable with respect to iteration trees for which there are Q-structures (see Definition 2.4) which are not too complicated.…”

Section: Proving Iterabilitymentioning

confidence: 99%