Journal of Mathematical Logic

For $A\subseteq \omega$, the coarse similarity class of A, denoted by $[A]$, is the set of all $B\subseteq \omega$ such that the symmetric difference of A and B has asymptotic density 0. There is a natural metric $\delta$ on the space ${𝒮}$ of coarse similarity classes defined by letting $\delta ([A],[B])$ be the upper density of the symmetric difference of A and B. We study the metric space of coarse similarity classes under this metric, and show in particular that between any two distinct points in this space there are continuum many geodesic paths. We also study subspaces of the form $\{[A]:A\in {𝒰}\}$ where ${𝒰}$ is closed under Turing equivalence, and show that there is a tight connection between topological properties of such a space and computability-theoretic properties of ${𝒰}$.

We then define a distance between Turing degrees based on Hausdorff distance in the metric space $({𝒮},\delta )$. We adapt a proof of Monin to show that the Hausdorff distances between Turing degrees that occur are exactly 0, $1/2$, and 1, and study which of these values occur most frequently in the senses of Lebesgue measure and Baire category. We define a degree a to be attractive if the class of all degrees at distance $1/2$ from a has measure 1, and dispersive otherwise. In particular, we study the distribution of attractive and dispersive degrees. We also study some properties of the metric space of Turing degrees under this Hausdorff distance, in particular the question of which countable metric spaces are isometrically embeddable in it, giving a graph-theoretic sufficient condition for embeddability.

Motivated by a couple of issues arising in the above work, we also study the computability-theoretic and reverse-mathematical aspects of a Ramsey-theoretic theorem due to Mycielski, which in particular implies that there is a perfect set whose elements are mutually 1-random, as well as a perfect set whose elements are mutually 1-generic.

Finally, we study the completeness of $({𝒮},\delta )$ from the perspectives of computability theory and reverse mathematics.

Let K be an NIP field and let v be a Henselian valuation on K. We ask whether $(K,v)$ is NIP as a valued field. By a result of Shelah, we know that if v is externally definable, then $(K,v)$ is NIP. Using the definability of the canonical p-Henselian valuation, we show that whenever the residue field of v is not separably closed, then v is externally definable. In the case of separably closed residue field, we show that $(K,v)$ is NIP as a pure valued field.

We prove that any proof of a $\forall {\mathrm{\Sigma }}_2^0$ sentence in the theory ${\mathrm{\text{WKL}}}_0+{\mathrm{\text{RT}}}_2^2$ can be translated into a proof in ${\mathrm{\text{RCA}}}_0$ at the cost of a polynomial increase in size. In fact, the proof in ${\mathrm{\text{RCA}}}_0$ can be obtained by a polynomial-time algorithm. On the other hand, ${\mathrm{\text{RT}}}_2^2$ has nonelementary speedup over the weaker base theory ${\mathrm{\text{RCA}}}_0^{*}$ for proofs of ${\mathrm{\Sigma }}_1$ sentences. We also show that for $n\ge 0$, proofs of ${\mathrm{\Pi }}_{n+2}$ sentences in $\mathrm{\text{B}}{\mathrm{\Sigma }}_{n+1}+exp$ can be translated into proofs in $\mathrm{\text{I}}{\mathrm{\Sigma }}_n+exp$ at a polynomial cost in size. Moreover, the ${\mathrm{\Pi }}_{n+2}$-conservativity of $\mathrm{\text{B}}{\mathrm{\Sigma }}_{n+1}+exp$ over $\mathrm{\text{I}}{\mathrm{\Sigma }}_n+exp$ can be proved in $\mathrm{\text{PV}}$, a fragment of bounded arithmetic corresponding to polynomial-time computation. For $n\ge 1$, this answers a question of Clote, Hájek and Paris.

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie “non-computational” proofs from the mainstream literature. To this end, we establish logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On the one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches.

We introduce a new method for building models of $CH$, together with ${\mathrm{\Pi }}_2$ statements over $H({\omega }_2)$, by forcing. Unlike other forcing constructions in the literature, our construction adds new reals, although only ${\aleph }_1$-many of them. Using this approach, we build a model in which a very strong form of the negation of Club Guessing at ${\omega }_1$ known as $Measuring$ holds together with $CH$, thereby answering a well-known question of Moore. This construction can be described as a finite-support weak forcing iteration with side conditions consisting of suitable graphs of sets of models with markers. The $CH$-preservation is accomplished through the imposition of copying constraints on the information carried by the condition, as dictated by the edges in the graph.