Effective Prime Uniqueness

Abstract

Assuming the obvious definitions (see paper) we show the a decidable model that is effectively prime is also effectively atomic. This implies that two effectively prime (decidable) models are computably isomorphic. This is in contrast to the theorem that there are two atomic decidable models which are not computably isomorphic. The implications of this work in reverse mathematics is that "effectively prime implies effectively atomic" holds in topped models. But due to an observation of David Belanger, "effectively prime implies effectively atomic" fails for some Scott sets. The reserve mathematical strength of "Prime Uniqueness" remains open.

This is now the final accepted version. In early March 2016, Richard Shore and Leo Harrington pointed out to us that one of our corollaries in first version was incorrect. This new version has new results addressing those errors within the body of the paper