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We give a new and effective classification of all Borel Wadge classes of subsets of Baire space. This relies on the true stage machinery originally developed by Montalb\'an. We use this machinery to give a new proof of Louveau and Saint-Raymond's separation theorem for Borel Wadge classes. This gives a proof of Borel Wadge determinacy in the subsystem $\text{ATR}_0+\Pi^1_1$-I of second-order arithmetic.

We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff-Kuratowski and Wadge theorems on the structure of ${\mathbf \Delta}^0_\xi$, Louveau and Saint-Raymond's separation theorem, and Louveau's separation theorem.

A central theme in set theory is to find universes with extreme, well-understood behaviour. The case we are interested in is assuming GCH and has a strong forcing axiom of higher order than usual. Instead of "for every suitable forcing notion for~$\lambda$" we shall say "for every such family of forcing notions, depending on stationary $S\subseteq \lambda$, for some such stationary set we have\dots". Such notions of forcing are important for Abelian group theory, but this application is delayed for a sequel.

Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $\mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to $\mathsf{WKL}_0$; (ii) Cousin's lemma for Baire class 1 functions is equivalent to $\mathsf{ACA}_0$; (iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to $\mathsf{ATR}_0$ (modulo some induction)., Comment: 13 pages, no figures

A set is introreducible if it can be computed by every infinite subset of itself. Such a set can be thought of as coding information very robustly. We investigate introreducible sets and related notions. Our two main results are that the collection of introreducible sets is $\Pi^1_1$-complete, so that there is no simple characterization of the introreducible sets; and that every introenumerable set has an introreducible subset., Comment: 30 pages

We investigate what collections of c.e.\ Turing degrees can be realised as the collection of elements of a separating $\Pi^0_1$ class of c.e.\ degree. We show that for every c.e.\ degree $\mathbf{c}$, the collection $\{\mathbf{c}, \mathbf{0}'\}$ can be thus realized. We also rule out several attempts at constructing separating classes realizing a unique c.e.\ degree. For example, we show that there is no \emph{super-maximal} pair: disjoint c.e.\ sets $A$ and $B$ whose separating class is infinite, but every separator of c.e.\ degree is a finite variant of either $A$ or $\overline{B}$.

Suppose we are given a computably enumerable object arise from algorithmic randomness or computable analysis. We are interested in the strength of oracles which can compute an object that approximates this c.e. object. It turns out that, depending on the type of object, the resulting highness property is either close to, or equivalent to being PA-complete. We examine, for example, dominating a c.e. martingale by an oracle-computable martingale, computing compressions functions for two variants of Kolmogorov complexity, and computing subtrees of positive measure of a given $\Pi^0_1$ tree of positive measure without dead ends. We prove a separation result from PA-completeness for the latter property, called the \emph{continuous covering property}. We also separate the corresponding principles in reverse mathematics.

Many constructions in computability theory rely on "time tricks". In the higher setting, relativising to some oracles shows the necessity of these. We construct an oracle~$A$ and a set~$X$, higher Turing reducible to~$X$, but for which $\Psi(A)\ne X$ for any higher functional~$\Psi$ which is consistent on all oracles. We construct an oracle~$A$ relative to which there is no universal higher ML-test. On the other hand, we show that badness has its limits: there are no higher self-PA oracles, and for no~$A$ can we construct a higher $A$-c.e.\ set which is also higher $A$-ML-random. We study various classes of bad oracles and differentiate between them using other familiar classes. For example, bad oracles for consistent reductions can be higher ML-random, whereas bad oracles for universal tests cannot.

A problem is a multivalued function from a set of \emph{instances} to a set of \emph{solutions}. We consider only instances and solutions coded by sets of integers. A problem admits preservation of some computability-theoretic weakness property if every computable instance of the problem admits a solution relative to which the property holds. For example, cone avoidance is the ability, given a non-computable set $A$ and a computable instance of a problem $\mathsf{P}$, to find a solution relative to which $A$ is still non-computable. In this article, we compare relativized versions of computability-theoretic notions of preservation which have been studied in reverse mathematics, and prove that the ones which were not already separated by natural statements in the literature actually coincide. In particular, we prove that it is equivalent to admit avoidance of 1 cone, of $\omega$ cones, of 1 hyperimmunity or of 1 non-$\Sigma^0_1$ definition. We also prove that the hierarchies of preservation of hyperimmunity and non-$\Sigma^0_1$ definitions coincide. On the other hand, none of these notions coincide in a non-relativized setting., Comment: 19 pages

We show that a sequence has effective Hausdorff dimension 1 if and only if it is coarsely similar to a Martin-L\"{o}f random sequence. More generally, a sequence has effective dimension $s$ if and only if it is coarsely similar to a weakly $s$-random sequence. Further, for any $s

We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending on the cardinality. For example, we show, under the assumption $V=L$, that there is a first-order definable free abelian group with no first-order definable basis., Comment: 26 pages

Martin-L\"of (ML)-reducibility compares $K$-trivial sets by examining the Martin-L\"of random sequences that compute them. We show that every $K$-trivial set is computable from a c.e.\ set of the same ML-degree. We investigate the interplay between ML-reducibility and cost functions, which are used to both measure the number of changes in a computable approximation, and the type of null sets used to capture ML-random sequences. We show that for every cost function there is a c.e.\ set ML-above the sets obeying it (called an ML-complete set for the cost function). We characterise the $K$-trivial sets computable from a fragment of the left-c.e.\ random real~$\Omega$. This leads to a new characterisation of strong jump-traceability.

We study the sets that are computable from both halves of some (Martin-L\"of) random sequence, which we call \emph{$1/2$-bases}. We show that the collection of such sets forms an ideal in the Turing degrees that is generated by its c.e.\ elements. It is a proper subideal of the $K$-trivial sets. We characterise $1/2$-bases as the sets computable from both halves of Chaitin's $\Omega$, and as the sets that obey the cost function $\mathbf c(x,s) = \sqrt{\Omega_s - \Omega_x}$. Generalising these results yields a dense hierarchy of subideals in the $K$-trivial degrees: For $k< n$, let $B_{k/n}$ be the collection of sets that are below any $k$ out of $n$ columns of some random sequence. As before, this is an ideal generated by its c.e.\ elements and the random sequence in the definition can always be taken to be $\Omega$. Furthermore, the corresponding cost function characterisation reveals that $B_{k/n}$ is independent of the particular representation of the rational $k/n$, and that $B_p$ is properly contained in $B_q$ for rational numbers $p< q$. These results are proved using a generalisation of the Loomis--Whitney inequality, which bounds the measure of an open set in terms of the measures of its projections. The generality allows us to analyse arbitrary families of orthogonal projections. As it turns out, these do not give us new subideals of the $K$-trivial sets, we can calculate from the family which $B_p$ it characterises. We finish by showing that the the union of $B_p$ for $p<1$ is the collection of sets which are robustly computable from a random, a class previously studied by Hirschfeldt, Jockusch, Kuyper, and Schupp.

We construct an increasing $\omega$-sequence $(a_n)$ of Turing degrees which forms an initial segment of the Turing degrees, and such that each~$a_{n+1}$ is diagonally noncomputable relative to $a_n$. It follows that the~$\mathsf{DNR}$ principle of reverse mathematics does not imply the existence of Turing incomparable degrees.

We investigate the role of continuous reductions and continuous relativisation in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van-Lambalgen's theorem and the Miller-Yu / Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterisations of lowness for Martin-L\"of randomness. We also characterise computing higher $K$-trivial sets by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak-2-randomness and $\Pi^1_1$-randomness. To do so we investigate classes of functions computable from Kleene's~$O$ based on strong forms of the higher limit lemma., Comment: 45 pages

We show that in contrast with the Cohen version of Solovay's model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.

Every K-trivial set is computable from an incomplete Martin-L\"of random set, i.e., a Martin-L\"of random set that does not compute 0'.

We prove that there is a structure, indeed a linear ordering, whose degree spectrum is the set of all non-hyperarithmetic degrees. We also show that degree spectra can distinguish measure from category., Comment: 21 pages

We show that every strongly jump-traceable set obeys every benign cost function. Moreover, we show that every strongly jump-traceable set is computable from a computably enumerable strongly jump-traceable set. This allows us to generalise properties of c.e.\ strongly jump-traceable sets to all such sets. For example, the strongly jump-traceable sets induce an ideal in the Turing degrees; the strongly jump-traceable sets are precisely those that are computable from all superlow Martin-L\"{o}f random sets; the strongly jump-traceable sets are precisely those that are a base for $\text{Demuth}_{\text{BLR}}$-randomness; and strong jump-traceability is equivalent to strong superlowness., Comment: 25 pages

In contrast with the notion of complexity, a set $A$ is called anti-complex if the Kolmogorov complexity of the initial segments of $A$ chosen by a recursive function is always bounded by the identity function. We show that, as for complexity, the natural arena for examining anti-complexity is the weak-truth table degrees. In this context, we show the equivalence of anti-complexity and other lowness notions such as r.e.\ traceability or being weak truth-table reducible to a Schnorr trivial set. A set $A$ is anti-complex if and only if it is reducible to another set $B$ with \emph{tiny use}, whereby we mean that the use function for reducing $A$ to $B$ can be made to grow arbitrarily slowly, as gauged by unbounded nondecreasing recursive functions. This notion of reducibility is then studied in its own right, and we also investigate its range and the range of its uniform counterpart., Comment: 20

There are noncomputable c.e.\ sets, computable from every SJT-hard c.e.\ set. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone., Comment: 34 pages

We show that if a set $A$ is computable from every superlow 1-random set, then $A$ is strongly jump-traceable. This theorem shows that the computably enumerable (c.e.) strongly jump-traceable sets are exactly the c.e.\ sets computable from every superlow 1-random set. We also prove the analogous result for superhighness: a c.e.\ set is strongly jump-traceable if and only if it is computable from every superhigh 1-random set. Finally, we show that for each cost function $c$ with the limit condition there is a 1-random $\Delta^0_2$ set $Y$ such that every c.e.\ set $A \le_T Y$ obeys $c$. To do so, we connect cost function strength and the strength of randomness notions. This result gives a full correspondence between obedience of cost functions and being computable from $\Delta^0_2$ 1-random sets., Comment: 41 pages

We solve the covering problem for Demuth randomness, showing that a computably enumerable set is computable from a Demuth random set if and only if it is strongly jump-traceable. We show that on the other hand, the class of sets which form a base for Demuth randomness is a proper subclass of the class of strongly jump-traceable sets., Comment: 28 pages