Effective Prime Uniqueness

TL;DR

This paper explores the relationship between effectively prime and effectively atomic models, showing that in certain contexts, effectively prime models are also effectively atomic, with implications for computable isomorphism and reverse mathematics.

Contribution

It proves that effectively prime models are effectively atomic in decidable models and discusses the implications for reverse mathematics and Scott sets.

Findings

Effectively prime models are effectively atomic in decidable models.

Two effectively prime models are computably isomorphic.

The equivalence fails in some Scott sets.

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.