Computably enumerable partial orders

TL;DR

This paper explores the degree spectra of computably enumerable partial orders, establishing their logical strength and showing their spectra can match nonzero degrees, with implications for reverse mathematics.

Contribution

It introduces new formulations of principles for c.e. partial orders and demonstrates their strength and spectrum properties, linking them to nonzero degrees.

Findings

Chain/antichain principle is strictly stronger than ascending/descending sequence principle.

Every $ ext{0'}$-computable structure has the same degree spectrum as some c.e. partial order.

There exists a c.e. partial order with spectrum equal to the nonzero degrees.

Abstract

We study the degree spectra and reverse-mathematical applications of computably enumerable and co-computably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the former is strictly stronger than the latter. We then show that every $\emptyset'$-computable structure (or even just of c.e.\ degree) has the same degree spectrum as some computably enumerable (co-c.e.)\ partial order, and hence that there is a c.e.\ (co-c.e.)\ partial order with spectrum equal to the set of nonzero degrees.