Mass problems and intuitionistic higher-order logic

In this paper we study a model of intuitionistic higher-order logic which we call \emph{the Muchnik topos}. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, \emph{the Muchnik reals}, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a \emph{choice principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists w\,\forall x\,A(x,wx)$ and a \emph{bounding principle} $(\forall x\,\exists y\,A(x,y))\Rightarrow\exists z\,\forall x\,\exists y\,(y\le_{\mathrm{T}}(x,z)\land A(x,y))$ where $x,y,z$ range over Muchnik reals, $w$ ranges over functions from Muchnik reals to Muchnik reals, and $A(x,y)$ is a formula not containing $w$ or $z$. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higher-order logic, Turing degrees, mass problems, Muchnik degrees, and Kolmogorov's calculus of problems. We also provide an English translation of Muchnik's 1963 paper on Muchnik degrees.Comment: 44 page

Generalized cohesiveness

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ramsey's Theorem. An infinite set $A$ of natural numbers is $n$--cohesive (respectively, $n$--r--cohesive) if $A$ is almost homogeneous for every computably enumerable (respectively, computable) $2$--coloring of the $n$--element sets of natural numbers. (Thus the $1$--cohesive and $1$--r--cohesive sets coincide with the cohesive and r--cohesive sets, respectively.) We consider the degrees of unsolvability and arithmetical definability levels of $n$--cohesive and $n$--r--cohesive sets. For example, we show that for all $n \ge 2$, there exists a $\Delta^0_{n+1}$ $n$--cohesive set. We improve this result for $n = 2$ by showing that there is a $\Pi^0_2$ $2$--cohesive set. We show that the $n$--cohesive and $n$--r--cohesive degrees together form a linear, non--collapsing hierarchy of degrees for $n \geq 2$. In addition, for $n \geq 2$ we characterize the jumps of $n$--cohesive degrees as exactly the degrees {\bf \geq \jump{0}{(n+1)}} and show that each $n$--r--cohesive degree has jump {\bf > \jump{0}{(n)}}

The Complexity of Orbits of Computably Enumerable Sets

The goal of this paper is to announce there is a single orbit of the c.e. sets with inclusion, \E, such that the question of membership in this orbit is $\Sigma^1_1$-complete. This result and proof have a number of nice corollaries: the Scott rank of \E is \wock +1; not all orbits are elementarily definable; there is no arithmetic description of all orbits of \E; for all finite $\alpha \geq 9$, there is a properly $\Delta^0_\alpha$ orbit (from the proof). A few small corrections made in this versionComment: To appear in the Bulletion of Symbolic Logi