arXiv subjects

Leo Harrington

Publications and source records attributed to Leo Harrington.

3 recordsLinked to original sources

arXiv2007

On the Orbits of Computably Enumerable Sets

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

math.LO↗

arXiv2007

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 $Σ^1_1$-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 $α\geq 9$, there is a properly $Δ^0_α$ orbit (from the proof). A few small corrections made in this version

math.LO↗

arXiv2005

Extensions Theorems, Orbits, and Automorphisms of the Computably Enumerable Sets

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.

math.LO↗