Some new computable structures of high rank

TL;DR

This paper introduces new computable structures with high Scott rank, demonstrating diverse properties of their theories and answering a longstanding question about their categoricity.

Contribution

It provides the first example of a high Scott rank structure with a non-unique computable infinitary theory, and constructs structures with no indiscernible triples at high rank.

Findings

Constructed a computable structure of Scott rank _1^{CK} with non-isomorphic models.

Provided examples of high-rank structures lacking indiscernible triples.

Answered Millar and Sacks' question on _1^{CK}-categoricity.

Abstract

We give several new examples of computable structures of high Scott rank. For earlier known computable structures of Scott rank $\omega_1^{CK}$, the computable infinitary theory is $\aleph_0$-categorical. Millar and Sacks asked whether this was always the case. We answer this question by constructing an example whose computable infinitary theory has non-isomorphic countable models. The standard known computable structures of Scott rank $\omega_1^{CK}+1$ have infinite indiscernible sequences. We give two constructions with no indiscernible ordered triple.