Categoricity without Power

TL;DR

This paper establishes an analogue of Morley's categoricity theorem replacing cardinality with arithmetic degree, showing that $D$-categoricity for one nonzero degree implies it for all such degrees and relates to uncountable categoricity.

Contribution

It introduces the concept of $D$-categoricity based on arithmetic degrees and proves that $D$-categoricity for one nonzero degree implies it for all nonzero degrees, linking it to uncountable categoricity.

Findings

$D$-categoricity for one nonzero degree implies for all nonzero degrees

$D$-categoricity for some nonzero degree is equivalent to uncountable categoricity under ZFC

Introduces a new framework replacing cardinality with arithmetic degrees in categoricity

Abstract

We prove an analogue of Morley's categoricity theorem where cardinality is replaced by the recursion-theoretic notion of arithmetic degree. We say that a complete arithmetically definable theory $T$ is $D$-categorical if any two arithmetically extendible models of $T$ of arithmetic degree $D$, considered over a common elementary submodel with arithmetical elementary diagram, are isomorphic over that submodel by an isomorphism which preserves the complexity of sets of degree $D$. Here an arithmetically extendible model means an elementary substructure of a model whose elementary diagram is arithmetical. Our main result is: If $T$ is $D_1$-categorical for some nonzero arithmetic degree $D_1$, then $T$ is $D_2$-categorical for every nonzero arithmetic degree $D_2$. We also show that, assuming ZFC, $D$-categoricity for some nonzero arithmetic degree is equivalent to uncountable categoricity.