Abstract In their logical analysis of theorems about disjoint rays in graphs, Barnes, Shore, and the author (hereafter BGS) introduced a weak choice scheme in second-order arithmetic, called the $\Sigma ^1_1$ axiom of finite choice (hereafter finite choice). This is a special case of the $\Sigma ^1_1$ axiom of choice ( $\Sigma ^1_1\text {-}\mathsf {AC}_0$ ) introduced by Kreisel. BGS showed that $\Sigma ^1_1\text {-}\mathsf {AC}_0$ suffices for proving many of the aforementioned theorems in graph theory. While it is not known if these implications reverse, BGS also showed that those theorems imply finite choice (in some cases, with additional induction assumptions). This motivated us to study the proof-theoretic strength of finite choice. Using a variant of Steel forcing with tagged trees, we show that finite choice is not provable from the $\Delta ^1_1$ -comprehension scheme (even over $\omega $ -models). We also show that finite choice is a consequence of the arithmetic Bolzano–Weierstrass theorem (introduced by Friedman and studied by Conidis), assuming $\Sigma ^1_1$ -induction. Our results were used by BGS to show that several theorems in graph theory cannot be proved using $\Delta ^1_1$ -comprehension. Our results also strengthen results of Conidis.

Abstract Recall that B is PA relative to A if B computes a member of every nonempty $\Pi ^0_1(A)$ class. This two-place relation is invariant under Turing equivalence and so can be thought of as a binary relation on Turing degrees. Miller and Soskova [23] introduced the notion of a $\Pi ^0_1$ class relative to an enumeration oracle A, which they called a $\Pi ^0_1{\left \langle {A}\right \rangle }$ class. We study the induced extension of the relation B is PA relative to A to enumeration oracles and hence enumeration degrees. We isolate several classes of enumeration degrees based on their behavior with respect to this relation: the PA bounded degrees, the degrees that have a universal class, the low for PA degrees, and the ${\left \langle {\text {self}\kern1pt}\right \rangle }$ -PA degrees. We study the relationship between these classes and other known classes of enumeration degrees. We also investigate a group of classes of enumeration degrees that were introduced by Kalimullin and Puzarenko [14] based on properties that are commonly studied in descriptive set theory. As part of this investigation, we give characterizations of three of their classes in terms of a special sub-collection of relativized $\Pi ^0_1$ classes—the separating classes. These three can then be seen to be direct analogues of three of our classes. We completely determine the relative position of all classes in question.

Abstract In this work we investigate the Weihrauch degree of the problem Decreasing Sequence ( $\mathsf {DS}$ ) of finding an infinite descending sequence through a given ill-founded linear order, which is shared by the problem Bad Sequence ( $\mathsf {BS}$ ) of finding a bad sequence through a given non-well quasi-order. We show that $\mathsf {DS}$ , despite being hard to solve (it has computable inputs with no hyperarithmetic solution), is rather weak in terms of uniform computational strength. To make the latter precise, we introduce the notion of the deterministic part of a Weihrauch degree. We then generalize $\mathsf {DS}$ and $\mathsf {BS}$ by considering $\boldsymbol {\Gamma }$ -presented orders, where $\boldsymbol {\Gamma }$ is a Borel pointclass or $\boldsymbol {\Delta }^1_1$ , $\boldsymbol {\Sigma }^1_1$ , $\boldsymbol {\Pi }^1_1$ . We study the obtained $\mathsf {DS}$ -hierarchy and $\mathsf {BS}$ -hierarchy of problems in comparison with the (effective) Baire hierarchy and show that they do not collapse at any finite level.