Narrow Results

We answer a question of Slaman and Steel by showing that a version of Martin’s conjecture holds for all regressive functions on the hyperarithmetic degrees. A key step in our proof, which may have applications to other cases of Martin’s conjecture, consists of showing that we can always reduce to the case of a continuous function.

We show that in the ${\mathbb{P}}_{max}$ extension of a certain Chang-type model of determinacy, if $\kappa \in \{{\omega }_1,{\omega }_2,{\omega }_3\}$, then the restriction of the club filter on $\kappa \cap$ Cof$(\omega )$ to HOD is an ultrafilter in HOD. This answers Question 4.11 of [O. Ben-Neria and Y. Hayut, On $\omega$-strongly measurable cardinals, Forum Math. Sigma11 (2023) e19].

A four-party single-qubit operation sharing scheme is put forward by utilizing the Bell and Yeo–Chua product state in an entanglement structure as the composite quantum channel. Four features of the scheme are discussed and confirmed, including its determinacy, symmetry, and security as well as the scheme experimental feasibility. Moreover, some concrete comparisons between our present scheme and a previous scheme [H. Xing et al., Quantum Inf. Process. 13 (2014) 1553] are made from the aspects of quantum and classical resource consumption, necessary operation complexity, and intrinsic efficiency. It is found that our present scheme is more superior than that one. In addition, the essential reason why the employed state in the entanglement structure is applicable for sharing an arbitrary single-qubit operation among four parties is revealed via deep analyses. With respect to the essential reason, the capacity of the product state in quantum operation sharing (QOS) is consequently shown by simple presenting the corresponding schemes with the state in other entanglement structures.

We present a general lemma which allows proving determinacy from Woodin cardinals. The lemma can be used in many different settings. As a particular application we prove the determinacy of sets in , n ≥ 1. The assumption we use to prove determinacy is optimal in the base theory of determinacy.

Assuming AD + DC(ℝ), we characterize the self-dual boldface pointclasses which are strictly larger (in terms of cardinality) than the pointclasses contained in them: these are exactly the clopen sets, the collections of all sets of Wadge rank , and those of Wadge rank when ξ is limit.

We prove determinacy for open length ω₁ games. Going further we introduce, and prove determinacy for, a stronger class of games of length ω₁, with payoff conditions involving the entire run, the club filter on ω₁, and a sequence of ω₁ disjoint stationary subsets of ω₁. The determinacy proofs use an iterable model with a class of indiscernible Woodin cardinals, and we show that the games precisely capture the theory of the minimal model for this assumption.

Assume $ZF+{AD}^{+}+V=L(\mathcal{𝒫}(ℝ))$. Let $E$ be a ${\mathrm{\Sigma }}_1^1$ equivalence relation coded in $\mathrm{\text{HOD}}$. $E$ has an ordinal definable equivalence class without any ordinal definable elements if and only if $\mathrm{\text{HOD}}\vDash E$ is unpinned.

$ZF+{AD}^{+}+V=L(\mathcal{𝒫}(ℝ))$ proves $E$-class section uniformization when $E$ is a ${\mathrm{\Sigma }}_1^1$ equivalence relation on $ℝ$ which is pinned in every transitive model of $ZFC$ containing the real which codes $E$: Suppose $R$ is a relation on $ℝ$ such that each section $R_x=\{y:(x,y)\in R\}$ is an $E$-class, then there is a function $f:ℝ\to ℝ$ such that for all $x\in ℝ$, $R(x,f(x))$.

$ZF+AD$ proves that $ℝ\times \kappa$ is Jónsson whenever $\kappa$ is an ordinal: For every function $f:{[ℝ\times \kappa ]}_{=}^{<\omega }\to ℝ\times \kappa$, there is an $A\subseteq ℝ\times \kappa$ with $A$ in bijection with $ℝ\times \kappa$ and $f[{[A]}_{=}^{<\omega }]\ne ℝ\times \kappa$.

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

We study the notion of ${𝒥}$-MAD families where ${𝒥}$ is a Borel ideal on $\omega$. We show that if ${𝒥}$ is any finite or countably iterated Fubini product of the ideal of finite sets $\mathrm{\text{Fin}}$, then there are no analytic infinite ${𝒥}$-MAD families, and assuming Projective Determinacy and Dependent Choice there are no infinite projective ${𝒥}$-MAD families; and under the full Axiom of Determinacy $+$$V=L(ℝ)$ or under ${AD}^{+}$ there are no infinite ${𝒥}$-mad families. Similar results are obtained in Solovay’s model. These results apply in particular to the ideal $\mathrm{\text{Fin}}$, which corresponds to the classical notion of MAD families, as well as to the ideal $\mathrm{\text{Fin}}\otimes \mathrm{\text{Fin}}$. The proofs combine ideas from invariant descriptive set theory and forcing.

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 $2^{\mathbb{Z}}$.

Assume $ZF+AD$. If $𝜖$ is an ordinal and X is a set of ordinals, then ${[X]}_{\ast }^{𝜖}$ is the collection of order-preserving functions $f:𝜖\to X$ which have uniform cofinality $\omega$ and discontinuous everywhere. The weak partition properties on ${\omega }_1$ and ${\omega }_2$ yield partition measures on ${[{\omega }_1]}_{\ast }^{𝜖}$ when $𝜖<{\omega }_1$ and ${[{\omega }_2]}_{\ast }^{𝜖}$ when $𝜖<{\omega }_2$. The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every $𝜖<{\omega }_1$ and function $\mathrm{\Phi }:{[{\omega }_1]}^{𝜖}\to {\omega }_1$, there is a club $C\subseteq {\omega }_1$ and a $\zeta <𝜖$ so that for all $f,g\in {[C]}_{\ast }^{𝜖}$, if $f\upharpoonright \zeta =g\upharpoonright \zeta$ and $sup(f)=sup(g)$, then $\mathrm{\Phi }(f)=\mathrm{\Phi }(g)$. For every $𝜖<{\omega }_2$ and function $\mathrm{\Phi }:{[{\omega }_2]}^{𝜖}\to {\omega }_2$, there is an $\omega$-club $C\subseteq {\omega }_2$ and a $\zeta <𝜖$ so that for all $f,g\in {[C]}_{\ast }^{𝜖}$, if $f\upharpoonright \zeta =g\upharpoonright \zeta$ and $sup(f)=sup(g)$, then $\mathrm{\Phi }(f)=\mathrm{\Phi }(g)$. The previous two continuity results will be used to distinguish the cardinalities of some important subsets of $\mathcal{𝒫}({\omega }_2):|{[{\omega }_1]}^{\omega }|<|{[{\omega }_1]}^{<{\omega }_1}|$. $|{[{\omega }_2]}^{\omega }|<|{[{\omega }_2]}^{<{\omega }_1}|<|{[{\omega }_2]}^{{\omega }_1}|<|{[{\omega }_2]}^{<{\omega }_2}|$. $\neg (|{[{\omega }_1]}^{<{\omega }_1}|\le |{[{\omega }_2]}^{\omega }|)$. $\neg (|{[{\omega }_1]}^{{\omega }_1}|\le |{[{\omega }_2]}^{<{\omega }_1}|)$. It will also be shown that ${[{\omega }_1]}^{\omega }$ has the Jónsson property: For every $\mathrm{\Phi }:{}^{<\omega }({[{\omega }_1]}^{\omega })\to {[{\omega }_1]}^{\omega }$, there is an $X\subseteq {[{\omega }_1]}^{\omega }$ with $|X|=|{[{\omega }_1]}^{\omega }|$ so that $\mathrm{\Phi }[{}^{<\omega }X]\ne {[{\omega }_1]}^{\omega }$.

In Ref. 8, we introduced a new axiom scheme for multiple inductive definitions to capture the exact strength of -determinacy in second order arithmetic. In this paper, we show this complicated axiom turns out to be equivalent to an axiom of single inductive definitions. This observation is useful to characterize and related systems.