Narrow Results

We identify a particular mouse, $M^{\mathrm{\text{ld}}}$, the minimal ladder mouse, that sits in the mouse order just past $M_n^{♯}$ for all $n$, and we show that $ℝ\cap M^{\mathrm{\text{ld}}}=Q_{\omega +1}$, the set of reals that are ${\mathrm{\Delta }}_{\omega +1}^1$ in a countable ordinal. Thus $Q_{\omega +1}$ is a mouse set.

This is analogous to the fact that $ℝ\cap M_1^{♯}=Q_3$ where $M_1^{♯}$ is the sharp for the minimal inner model with a Woodin cardinal, and $Q_3$ is the set of reals that are ${\mathrm{\Delta }}_3^1$ in a countable ordinal.

More generally $ℝ\cap M_{2n+1}^{♯}=Q_{2n+3}$. The mouse $M^{\mathrm{\text{ld}}}$ and the set $Q_{\omega +1}$ compose the next natural pair to consider in this series of results. Thus we are proving the mouse set theorem just past projective.

Some of this is not new. $ℝ\cap M^{\mathrm{\text{ld}}}\subseteq Q_{\omega +1}$ was known in the 1990s. But $Q_{\omega +1}\subseteq M^{\mathrm{\text{ld}}}$ was open until Woodin found a proof in 2018. The main goal of this paper is to give Woodin’s proof.

Let $b\ge 2$ be an integer. We show that the set of real numbers that are Poisson generic in base $b$ is ${\mathrm{\text{Π}}}_3^0$-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base $b$ and not Poisson generic in base $b$ is complete for the class given by the differences between ${\mathrm{\text{Π}}}_3^0$ sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.

Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216(1) (2007) 89–117] and Steel [PFA implies AD^L(ℝ), J. Symbolic Logic70(4) (2005) 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L(ℝ) of the core model induction that is aimed at getting a model of AD_ℝ + "Θ is regular" from the failure of square at a singular strong limit cardinal or PFA.

We provide a new criterion for embedding 𝔼₀, and apply it to equivalence relations in model theory. This generalize the results of the authors and Pierre Simon on the Borel cardinality of Lascar strong types equality, and Newelski's results about pseudo F_σ groups.

We introduce the notion of $(\mathrm{\Gamma },E)$-determinacy for $\mathrm{\Gamma }$ a pointclass and $E$ an equivalence relation on a Polish space $X$. A case of particular interest is the case when $E=E_G$ is the (left) shift-action of $G$ on $S^G$ where $S=2=\{0,1\}$ or $S=\omega$. We show that for all shift actions by countable groups $G$, and any “reasonable” pointclass $\mathrm{\Gamma }$, that $(\mathrm{\Gamma },E_G)$-determinacy implies $\mathrm{\Gamma }$-determinacy. We also prove a corresponding result when $E$ is a subshift of finite type of the shift map on $2^{\mathbb{Z}}$.

We generalize the notion of analytic/Borel equivalence relations, orbit equivalence relations, and Borel reductions between them to their continuous and quantitative counterparts: analytic/Borel pseudometrics, orbit pseudometrics, and Borel reductions between them. We motivate these concepts on examples and we set some basic general theory. We illustrate the new notion of reduction by showing that the Gromov–Hausdorff distance maintains the same complexity if it is defined on the class of all Polish metric spaces, spaces bounded from below, from above, and from both below and above. Then we show that $E_1$ is not reducible to equivalences induced by orbit pseudometrics, generalizing the seminal result of Kechris and Louveau. We answer in negative a question of Ben Yaacov, Doucha, Nies, and Tsankov on whether balls in the Gromov–Hausdorff and Kadets distances are Borel. In appendix, we provide new methods using games showing that the distance-zero classes in certain pseudometrics are Borel, extending the results of Ben Yaacov, Doucha, Nies, and Tsankov. There is a complementary paper of the authors where reductions between the most common pseudometrics from functional analysis and metric geometry are provided.