Some results on $\mathbb{R}$-computable structures

TL;DR

This survey explores the effective model theory of uncountable structures using the BSS model, covering topics like computable ordinals, infinitary formulas, forcing, and effective topology.

Contribution

It provides a comprehensive overview of the effective model theory for uncountable structures within the BSS real number computation framework.

Findings

Analysis of computable ordinals in the BSS model

Insights into satisfaction of computable infinitary formulas

Connections between effective topology and uncountable structures

Abstract

This survey paper examines the effective model theory obtained with the BSS model of real number computation. It treats the following topics: computable ordinals, satisfaction of computable infinitary formulas, forcing as a construction technique, effective categoricity, effective topology, and relations with other models for the effective theory of uncountable structures.