Generalized effective completeness for continuous logic

TL;DR

This paper proves a generalized effective completeness theorem for continuous logic, showing that every continuous theory has a structure with a matching Turing degree, extending prior partial results.

Contribution

It introduces a broad effective completeness theorem for continuous logic, unifying and extending earlier partial results by other researchers.

Findings

Every continuous theory has a structure with the same Turing degree.

Decidable theories are satisfied by computably presentable structures.

Extends previous partial effective completeness theorems.

Abstract

In this paper, we present a generalized effective completeness theorem for continuous logic. The primary result is that any continuous theory is satisfied in a structure which admits a presentation of the same Turing degree. It then follows that any decidable theory is satisfied by a computably presentable structure. This modifies and extends previous partial effective completeness theorems for continuous logic given by Calvert and Didehvar, Ghasemloo, and Pourmahdian.