Homogeneity of the L\'evy collapse from the perspective of Fra\"iss\'e theory

TL;DR

This paper explores the structure of Boolean algebras related to Levy collapse using Fra"iss"e theory, establishing their homogeneity and properties of their limits, with implications for set-theoretic forcing.

Contribution

It demonstrates that the class of Boolean algebras of size less than a strongly inaccessible cardinal forms a Fra"iss"e class and characterizes its limit's completion, linking it to Levy collapse.

Findings

The class of Boolean algebras of size < λ forms a Fra"iss"e class.

The limit of this class has the same completion as Levy collapse.

The collapsing algebra of density κ cannot be expressed as a union of a κ-chain of smaller regular sub-algebras.

Abstract

Given a strongly inaccessible cardinal $\lambda$, we study the Fra\"iss\'e class of all Boolean algebras of size $<\lambda$, together with regular embeddings. We prove that this is indeed a Fra\"iss\'eclass, and its limit has the same completion as the L\'evy collapse. We also give a direct proof that the collapsing algebra of density $\kappa$ is not the union of a $\kappa$-chain of regular sub-algebras of density $<\kappa$.