A Road To Compactness Through Guessing Models

TL;DR

This paper surveys the role of guessing models in set theory, highlighting their compactness properties, known results, generalizations, and applications, and discusses open problems in the field.

Contribution

It reformulates, generalizes, and expands existing results on guessing models, emphasizing their significance in set-theoretic compactness phenomena and related applications.

Findings

Guessing models exhibit significant compactness properties.

Many known results about guessing models are reformulated and generalized.

Open problems in the application and theory of guessing models are presented.

Abstract

The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that structure. With this phenomenon and its twin sibling "reflection", modern set theory has settled many mathematical statements left undecided by the conventionally accepted formalism of mathematics, $\rm ZFC$. A broad research program investigates whether a notion of compactness can universe-widely emerge without running into contradictions. These notes are a survey about guessing models whose existence provides intriguing compactness phenomena. Most of the results in the manuscript are well-known. We shall reformulate, generalise and expand some of them. We also present some known applications of guessing models and state some open problems.