Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology

This paper is a brief overview of the concepts involved in measuring the degree of contextuality and detecting contextuality in systems of binary measurements of a finite number of objects. We discuss and clarify the main concepts and terminology of the theory called "contextuality-by-default," and then discuss a possible generalization of the theory from binary to arbitrary measurements.Comment: Lecture Notes in Computer Science 9535 (with the corrected list of authors) (2016

Cohesive avoidance and arithmetical sets

An open question in reverse mathematics is whether the cohesive principle, \COH, is implied by the stable form of Ramsey's theorem for pairs, \SRT^2_2, in $\omega$-models of \RCA. One typical way of establishing this implication would be to show that for every sequence $\vec{R}$ of subsets of $\omega$, there is a set $A$ that is $\Delta^0_2$ in $\vec{R}$ such that every infinite subset of $A$ or $\bar{A}$ computes an $\vec{R}$-cohesive set. In this article, this is shown to be false, even under far less stringent assumptions: for all natural numbers $n \geq 2$ and $m < 2^n$, there is a sequence \vec{R} = \sequence{R_0,...,R_{n-1}} of subsets of $\omega$ such that for any partition $A_0,...,A_{m-1}$ of $\omega$ arithmetical in $\vec{R}$, there is an infinite subset of some $A_j$ that computes no set cohesive for $\vec{R}$. This complements a number of previous results in computability theory on the computational feebleness of infinite sets of numbers with prescribed combinatorial properties. The proof is a forcing argument using an adaptation of the method of Seetapun showing that every finite coloring of pairs of integers has an infinite homogeneous set not computing a given non-computable set