arXiv subjects
Publications and source records attributed to Peter Cholak.
As suggested by the title, this paper is a survey of recent results and questions on the collection of computably enumerable sets under inclusion. This is not a broad survey but one focused on the author's and a few others' current research.
This paper falls within the general program of investigating the proof theoretic strength (in terms of reverse mathematics) of combinatorial principals which follow from versions of Ramsey's theorem. We examine two statements in graph theory and one statement in lattice theory proved by Galvin, Rival and Sands \cite{GRS:82} using Ramsey's theorem for 4-tuples. Our main results are that the statements concerning graph theory are equivalent to Ramsey's theorem for 4-tuples over $\RCA$ while the statement concerning lattices is provable in $\RCA$. Revised 12/2010. To appear in Archive for Mathematical Logic
The goal of this paper is to show there is a single orbit of the c.e. sets with inclusion, $\mathcal{E}$, such that the question of membership in this orbit is $Σ^1_1$-complete. This result and proof have a number of nice corollaries: The Scott rank of $\mathcal{E}$ is $ω^{CK}_1+1; Not all orbits are elementarily definable; There is no arithmetic description of all orbits of $\mathcal{E}$; For all finite $α\geq 9$, there is a properly $Δ^0_α$ orbit (from the proof). April 6, 2007, minor changes Nov 20, 2007, minor changes
We explore the interaction between Lebesgue measure and dominating functions. We show, via both a priority construction and a forcing construction, that there is a function of incomplete degree that dominates almost all degrees. This answers a question of Dobrinen and Simpson, who showed that such functions are related to the proof-theoretic strength of the regularity of Lebesgue measure for $G_δ$ sets. Our constructions essentially settle the reverse mathematical classification of this principle. Revised Nov 13, 2005. Minor corrections made.
We prove an algebraic extension theorem for the computably enumerable sets, $\mathcal{E}$. Using this extension theorem and other work we then show if $A$ and $\hat{A}$ are automorphic via $Ψ$ then they are automorphic via $Λ$ where $Λ\restriction Ł^*(A) = Ψ$ and $Λ\restriction \E^*(A)$ is $Δ^0_3$. We give an algebraic description of when an arbitrary set $\Ahat$ is in the orbit of a \ce set $A$. We construct the first example of a definable orbit which is not a $Δ^0_3$ orbit. We conclude with some results which restrict the ways one can increase the complexity of orbits. For example, we show that if $A$ is simple and $\hat{A}$ is in the same orbit as $A$ then they are in the same $Δ^0_6$-orbit and furthermore we provide a classification of when two simple sets are in the same orbit.