A set of integers $A$ is computably encodable if every infinite set of integers has an infinite subset computing $A$. By a result of Solovay, the computably encodable sets are exactly the hyperarithmetic ones. In this article, we extend this notion of computable encodability to subsets of the Baire space, and we characterize the ${\Pi }_1^0$-encodable compact sets as those which admit a nonempty ${\Sigma }_1^1$-subset. Thanks to this equivalence, we prove that weak weak König’s lemma is not strongly computably reducible to Ramsey’s theorem. This answers a question of Hirschfeldt and Jockusch.

We exhibit a family of computably enumerable sets which can be learned within polynomial resource bounds given access only to a teacher but which requires exponential resources to be learned given access only to a membership oracle. In general, we compare the families that can be learned with and without teachers and oracles for four measures of efficient learning.

We develop the theory of layered posets and use the notion of layering to prove a new iteration theorem (Theorem 6: if $\kappa$ is weakly compact, then any universal Kunen iteration of $\kappa$-cc posets (each possibly of size $\kappa$) is $\kappa$-cc, as long as direct limits are used sufficiently often. This iteration theorem simplifies and generalizes the various chain condition arguments for universal Kunen iterations in the literature on saturated ideals, especially in situations where finite support iterations are not possible. We also provide two applications:

1 For any $n\ge 1$, a wide variety of $<{\omega }_{n-1}$-closed, ${\omega }_{n+1}$-cc posets of size ${\omega }_{n+1}$ can consistently be absorbed (as regular suborders) by quotients of saturated ideals on ${\omega }_n$ (see Theorem 7 and Corollary 8).

2 For any $n\in \omega$, the tree property at ${\omega }_{n+3}$ is consistent with Chang’s conjecture $({\omega }_{n+3},{\omega }_{n+1})\twoheadrightarrow ({\omega }_{n+1},{\omega }_n)$ (Theorem 9).

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous “multivalued functions.” This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

This article improves two existing theorems of interest to neologicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation $E$ is defined without nonlogical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine’s theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan’s relative categoricity theorem.

We introduce a hierarchy of degree structures between the Medvedev and Muchnik lattices which allow varying amounts of nonuniformity. We use these structures to introduce the notion of the uniformity of a Muchnik reduction, which expresses how uniform a reduction is. We study this notion for several well-known reductions from algorithmic randomness. Furthermore, since our new structures are Brouwer algebras, we study their propositional theories. Finally, we study if our new structures are elementarily equivalent to each other.

Disjoint $n$-amalgamation is a condition on a complete first-order theory specifying that certain locally consistent families of types are also globally consistent. In this article, we show that if a countably categorical theory $T$ admits an expansion with disjoint $n$-amalgamation for all $n$, then $T$ is pseudofinite. All theories which admit an expansion with disjoint $n$-amalgamation for all $n$ are simple, but the method can be extended, using filtrations of Fraïssé classes, to show that certain nonsimple theories are pseudofinite. As case studies, we examine two generic theories of equivalence relations, $T_{\mathrm{feq}}^{\ast }$ and $T_{\mathrm{CPZ}}$, and show that both are pseudofinite. The theories $T_{\mathrm{feq}}^{\ast }$ and $T_{\mathrm{CPZ}}$ are not simple, but they have NSOP${}_1$. This is established here for $T_{\mathrm{CPZ}}$ for the first time.