15 problems

Steel's conjecture. Assume $AD$. Every $T$-invariant function is equivalent to a uniformly $T$-invariant function on a cone.

Martin's conjecture. Assume $AD$. Then:

Let $2^{\mathbb N}$ be Cantor space, and let $x\equiv_T y$ mean that $x$ and $y$ have the same Turing degree. Regard $(\equiv_T)$ as a countable Borel equivalence relation under Bo…

For $p\in\mathbb{N}^{\mathbb{N}}$, let $\mathsf{TDINC}(p)$ consist of sequences $\langle p_0,p_1,p_2,\ldots\rangle$ such that, for every $i$,…

Let $\mathsf{NON}$ be the problem discussed in the paper, and let $\mathsf{NON}^\omega$ and $\mathsf{NON}^\infty$ denote its countable and inverse-limit iterations, respectively. T…

Let ${\mathcal{L}}$ be a finite lattice. A finite lattice is join-semidistributive if it satisfies the join-semidistributive law, and it is interval dismantlable if it belongs to t…

A partial order $(P,\leq_P)$ is locally countable if every element has at most countably many predecessors. The continuum is the cardinality of the real numbers, and the Turing deg…

C.e. Sigma-intermediate/Delta-low conjecture. There is a computably enumerable set that is intermediate for $\Sigma$-Feiner and low for $\Delta$-Feiner.

Differing Feiner behavior conjecture. Some intermediate Turing degrees behave differently for $\Delta$-Feiner and for $\Sigma$-Feiner.

For a Turing degree $x$, let $\mathds{R}_{x}$ denote the set of real numbers of degree at most $x$, and for Turing degrees $x$ and $y$, let $x\oplus y$ be their join and…

Let $e\in\mathbb{N}$ and let $A$ be a real such that … For a real $B$, define the $e$-th pseudo-hyperjump by $\operatorname{HJ}_e(B)=B\oplus V_e^B$, where $V_e^B$ is the $e$-th mem…

Let $X_1,X_2\in 2^{\mathbb{N}}$. They are simultaneously continuously random if there exist a real $Z$ and a measure $\mu$ such that $Z$ computes $\mu$ and both $X_1$ and $X_2$ are…

A set $A$ has coarse computability bound $b3(A)=1$ when its coarse computability bound is one, but is not coarsely computable. The surrounding results show that if either the Turin…

Lowness conjecture. Under these assumptions, $[\mathbf a,\mathbf g]$ is $\Sigma^0_1(\mathbf x)$-presentable.

Let $A$ be a set of natural numbers, let $A'$ denote its Turing jump, and let $0”$ denote the second Turing jump of the computable degree. A set is almost everywhere dominating whe…