We demonstrate that the Weihrauch lattice can be used to classify the uniform computational content of computability-theoretic properties as well as the computational content of theorems in one common setting. The properties that we study include diagonal non-computability, hyperimmunity, complete consistent extensions of Peano arithmetic, 1-genericity, Martin-Löf randomness, and cohesiveness. The theorems that we include in our case study are the low basis theorem of Jockusch and Soare, the Kleene-Post theorem, and Friedberg's jump inversion theorem. It turns out that all the aforementioned properties and many theorems in computability theory, including all theorems that claim the existence of some Turing degree, have very little uniform computational content: they are located outside of the upper cone of binary choice (also known as LLPO); we call problems with this property indiscriminative. Since practically all theorems from classical analysis whose computational content has been classified are discriminative, our observation could yield an explanation for why theorems and results in computability theory typically have very few direct consequences in other disciplines such as analysis. A notable exception in our case study is the low basis theorem which is discriminative. This is perhaps why it is considered to be one of the most applicable theorems in computability theory. In some cases a bridge between the indiscriminative world and the discriminative world of classical mathematics can be established via a suitable residual operation and we demonstrate this in the case of the cohesiveness problem and the problem of consistent complete extensions of Peano arithmetic. Both turn out to be the quotient of two discriminative problems.

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We study the uniform computational content of different versions of the Baire Category Theorem in the Weihrauch lattice. The Baire Category Theorem can be seen as a pigeonhole principle that states that a complete (i.e., "large") metric space cannot be decomposed into countably many nowhere dense (i.e., "small") pieces. The Baire Category Theorem is an illuminating example of a theorem that can be used to demonstrate that one classical theorem can have several different computational interpretations. For one, we distinguish two different logical versions of the theorem, where one can be seen as the contrapositive form of the other one. The first version aims to find an uncovered point in the space, given a sequence of nowhere dense closed sets. The second version aims to find the index of a closed set that is somewhere dense, given a sequence of closed sets that cover the space. Even though the two statements behind these versions are equivalent to each other in classical logic, they are not equivalent in intuitionistic logic and likewise they exhibit different computational behavior in the Weihrauch lattice. Besides this logical distinction, we also consider different ways how the sequence of closed sets is "given". Essentially, we can distinguish between positive and negative information on closed sets. We discuss all the four resulting versions of the Baire Category Theorem. Somewhat surprisingly it turns out that the difference in providing the input information can also be expressed with the jump operation. Finally, we also relate the Baire Category Theorem to notions of genericity and computably comeager sets.

A quasi-order $Q$ induces two natural quasi-orders on $P(Q)$, but if $Q$ is a well-quasi-order, then these quasi-orders need not necessarily be well-quasi-orders. Nevertheless, Goubault-Larrecq showed that moving from a well-quasi-order $Q$ to the quasi-orders on $P(Q)$ preserves well-quasi-orderedness in a topological sense. Specifically, Goubault-Larrecq proved that the upper topologies of the induced quasi-orders on $P(Q)$ are Noetherian, which means that they contain no infinite strictly descending sequences of closed sets. We analyze various theorems of the form "if $Q$ is a well-quasi-order then a certain topology on (a subset of) $P(Q)$ is Noetherian'' in the style of reverse mathematics, proving that these theorems are equivalent to ACA_0 over RCA_0. To state these theorems in RCA_0 we introduce a new framework for dealing with second-countable topological spaces

We separate many of the basic fragments of classical logic which are used in reverse constructive mathematics. A group of related Kripke and topological models is used to show that various fragments of the Weak Law of the Excluded Middle, the Limited Principle of Omniscience, and Markov’s Principle, including Weak Markov’s Principle, do not imply each other.