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

Reverse mathematics and properties of finite character

We study the reverse mathematics of the principle stating that, for every property of finite character, every set has a maximal subset satisfying the property. In the context of set theory, this variant of Tukey's lemma is equivalent to the axiom of choice. We study its behavior in the context of second-order arithmetic, where it applies to sets of natural numbers only, and give a full characterization of its strength in terms of the quantifier structure of the formula defining the property. We then study the interaction between properties of finite character and finitary closure operators, and the interaction between these properties and a class of nondeterministic closure operators.Comment: This paper corresponds to section 4 of arXiv:1009.3242, "Reverse mathematics and equivalents of the axiom of choice", which has been abbreviated and divided into two pieces for publicatio

Reverse mathematics and equivalents of the axiom of choice

We study the reverse mathematics of countable analogues of several maximality principles that are equivalent to the axiom of choice in set theory. Among these are the principle asserting that every family of sets has a $\subseteq$-maximal subfamily with the finite intersection property and the principle asserting that if $P$ is a property of finite character then every set has a $\subseteq$-maximal subset of which $P$ holds. We show that these principles and their variations have a wide range of strengths in the context of second-order arithmetic, from being equivalent to $\mathsf{Z}_2$ to being weaker than $\mathsf{ACA}_0$ and incomparable with $\mathsf{WKL}_0$. In particular, we identify a choice principle that, modulo $\Sigma^0_2$ induction, lies strictly below the atomic model theorem principle $\mathsf{AMT}$ and implies the omitting partial types principle $\mathsf{OPT}$