arXiv subjects

Sandra Müller

Publications and source records attributed to Sandra Müller.

28 records · Page 2Linked to original sources

arXiv2021

Projective Games on the Reals

Let $M^\sharp_n(\mathbb{R})$ denote the minimal active iterable extender model which has $n$ Woodin cardinals and contains all reals, if it exists, in which case we denote by $M_n(\mathbb{R})$ the class-sized model obtained by iterating the topmost measure of $M_n(\mathbb{R})$ class-many times. We characterize the sets of reals which are $Σ_1$-definable from $\mathbb{R}$ over $M_n(\mathbb{R})$, under the assumption that projective games on reals are determined: (1) for even $n$, $Σ_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}Π^1_{n+1}$; (2) for odd $n$, $Σ_1^{M_n(\mathbb{R})} = \Game^\mathbb{R}Σ^1_{n+1}$. This generalizes a theorem of Martin and Steel for $L(\mathbb{R})$, i.e., the case $n=0$. As consequences of the proof, we see that determinacy of all projective games with moves in $\mathbb{R}$ is equivalent to the statement that $M^\sharp_n(\mathbb{R})$ exists for all $n\in\mathbb{N}$, and that determinacy of all projective games of length $ω^2$ with moves in $\mathbb{N}$ is equivalent to the statement that $M^\sharp_n(\mathbb{R})$ exists and satisfies $\mathsf{AD}$ for all $n\in\mathbb{N}$.

math.LO↗

arXiv2021

Infinite decreasing chains in the Mitchell order

It is known that the behavior of the Mitchell order substantially changes at the level of rank-to-rank extenders, as it ceases to be well-founded. While the possible partial order structure of the Mitchell order below rank-to-rank extenders is considered to be well understood, little is known about the structure in the ill-founded case. The purpose of the paper is to make a first step in understanding this case, by studying the extent to which the Mitchell order can be ill-founded. Our main results are (i) in the presence of a rank-to-rank extender there is a transitive Mitchell order decreasing sequence of extenders of any countable length, and (ii) there is no such sequence of length $ω_1$.

math.LO↗

arXiv2021

HOD in inner models with Woodin cardinals

We analyze the hereditarily ordinal definable sets $\operatorname{HOD}$ in $M_n(x)[g]$ for a Turing cone of reals $x$, where $M_n(x)$ is the canonical inner model with $n$ Woodin cardinals build over $x$ and $g$ is generic over $M_n(x)$ for the Lévy collapse up to its bottom inaccessible cardinal. We prove that assuming $\boldsymbolΠ^1_{n+2}$-determinacy, for a Turing cone of reals $x$, $\operatorname{HOD}^{M_n(x)[g]} = M_n(\mathcal{M}_{\infty} | κ_\infty, Λ),$ where $\mathcal{M}_\infty$ is a direct limit of iterates of $M_{n+1}$, $δ_\infty$ is the least Woodin cardinal in $\mathcal{M}_\infty$, $κ_\infty$ is the least inaccessible cardinal in $\mathcal{M}_\infty$ above $δ_\infty$, and $Λ$ is a partial iteration strategy for $\mathcal{M}_{\infty}$. It will also be shown that under the same hypothesis $\operatorname{HOD}^{M_n(x)[g]}$ satisfies $\operatorname{GCH}$.

math.LO↗

arXiv2021

Long Games and $σ$-Projective Sets

We prove a number of results on the determinacy of $σ$-projective sets of reals, i.e., those belonging to the smallest pointclass containing the open sets and closed under complements, countable unions, and projections. We first prove the equivalence between $σ$-projective determinacy and the determinacy of certain classes of games of variable length ${<}ω^2$ (Theorem 2.4). We then give an elementary proof of the determinacy of $σ$-projective sets from optimal large-cardinal hypotheses (Theorem 4.4). Finally, we show how to generalize the proof to obtain proofs of the determinacy of $σ$-projective games of a given countable length and of games with payoff in the smallest $σ$-algebra containing the projective sets, from corresponding assumptions (Theorems 5.1 and 5.4).

math.LO↗

arXiv2020

Closure properties of measurable ultrapowers

We study closure properties of measurable ultrapowers with respect to Hamkin's notion of "freshness" and show that the extent of these properties highly depends on the combinatorial properties of the underlying model of set theory. In one direction, a result of Sakai shows that, by collapsing a strongly compact cardinal to become the double successor of a measurable cardinal, it is possible to obtain a model of set theory in which such ultrapowers possess the strongest possible closure properties. In the other direction, we use various square principles to show that measurable ultrapowers of canonical inner models only possess the minimal amount of closure properties. In addition, the techniques developed in the proofs of these results also allow us to derive statements about the consistency strength of the existence of measurable ultrapowers with non-minimal closure properties.

math.LO↗

arXiv2020

Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (that is, all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

math.GN↗

arXiv2019

Structural Properties of the Stable Core

The stable core, an inner model of the form $\langle L[S],\in, S\rangle$ for a simply definable predicate $S$, was introduced by the first author in [Fri12], where he showed that $V$ is a class forcing extension of its stable core. We study the structural properties of the stable core and its interactions with large cardinals. We show that the $\operatorname{GCH}$ can fail at all regular cardinals in the stable core, that the stable core can have a discrete proper class of measurable cardinals, but that measurable cardinals need not be downward absolute to the stable core. Moreover, we show that, if large cardinals exist in $V$, then the stable core has inner models with a proper class of measurable limits of measurables, with a proper class of measurable limits of measurable limits of measurables, and so forth. We show this by providing a characterization of natural inner models $L[C_1, \dots, C_n]$ for specially nested class clubs $C_1, \dots, C_n$, like those arising in the stable core, generalizing recent results of Welch [Wel19].

math.LO↗

arXiv2019

The Axiom of Determinacy Implies Dependent Choices in Mice

We show that the Axiom of Dependent Choices, $\operatorname{DC}$, holds in countably iterable, passive premice $\mathcal{M}$ construced over their reals which satisfy the Axiom of Determinacy, $\operatorname{AD}$, in a $\operatorname{ZF}+\operatorname{DC}_{\mathbb{R}^{\mathcal{M}}}$ background universe. This generalizes an argument of Kechris for $L(\mathbb{R})$ using Steel's analysis of scales in mice. In particular, we show that for any $n \leq ω$ and any countable set of reals $A$ so that $M_n(A) \cap \mathbb{R} = A$ and $M_n(A) \vDash \operatorname{AD}$, we have that $M_n(A) \vDash \operatorname{DC}$.

math.LO↗

arXiv2019

The consistency strength of long projective determinacy

We determine the consistency strength of determinacy for projective games of length $ω^2$. Our main theorem is that $\boldsymbolΠ^1_{n+1}$-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 $M_n(A)$, the canonical inner model for $n$ Woodin cardinals constructed over $A$, satisfies $A = \mathbb{R}$ 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 $\Game^\mathbb{R} \boldsymbolΠ^1_1$ or with $σ$-projective payoff.

math.LO↗

arXiv2019

Mice with finitely many Woodin cardinals from optimal determinacy hypotheses

We prove the following result which is due to the third author. Let $n \geq 1$. If $\boldsymbolΠ^1_n$ determinacy and $Π^1_{n+1}$ determinacy both hold true and there is no $\boldsymbolΣ^1_{n+2}$-definable $ω_1$-sequence of pairwise distinct reals, then $M_n^\#$ exists and is $ω_1$-iterable. The proof yields that $\boldsymbolΠ^1_{n+1}$ determinacy implies that $M_n^\#(x)$ exists and is $ω_1$-iterable for all reals $x$. A consequence is the Determinacy Transfer Theorem for arbitrary $n \geq 1$, namely the statement that $\boldsymbolΠ^1_{n+1}$ determinacy implies $\Game^{(n)}(<ω^2 - \boldsymbolΠ^1_1)$ determinacy.

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