Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homogeneity of the Lévy collapse from the perspective of Fraïssé theory

Ziemowit Kostana

Abstract

Given a strongly inaccessible cardinal $λ$, we study the Fraïssé class of all Boolean algebras of size $<λ$, together with regular embeddings. We prove that this is indeed a Fraïsséclass, and its limit has the same completion as the Lévy collapse. We also give a direct proof that the collapsing algebra of density $κ$ is not the union of a $κ$-chain of regular sub-algebras of density $<κ$.

Homogeneity of the Lévy collapse from the perspective of Fraïssé theory

Abstract

Given a strongly inaccessible cardinal , we study the Fraïssé class of all Boolean algebras of size , together with regular embeddings. We prove that this is indeed a Fraïsséclass, and its limit has the same completion as the Lévy collapse. We also give a direct proof that the collapsing algebra of density is not the union of a -chain of regular sub-algebras of density .
Paper Structure (16 sections, 19 theorems, 54 equations)

This paper contains 16 sections, 19 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Fraïssé-Jónsson theory
  3. Collapsing algebras
  4. Lévy collapse
  5. Main definitions and notation
  6. Partial orders
  7. Regular Boolean sub-algebras
  8. Forcing arguments
  9. Fraïssé-Jónsson Theory
  10. Fraïssé-Jónsson classes
  11. Classes of Boolean algebras
  12. Collapsing algebras
  13. Lévy collapse
  14. Universality of the group $\operatorname{Aut}{\mathbb{B}_\lambda}$
  15. The collapsing algebra is not in $\mathop{\mathrm{BoolSeq}}\nolimits$
  16. ...and 1 more sections

Key Result

Proposition 1.1

Any regular embedding $e:A\longrightarrow B$ extends uniquely to a complete embedding $\bar{e}:\overline{A}\longrightarrow \overline{B}$.

Theorems & Definitions (42)

  • Proposition 1.1
  • Proposition 1.2
  • proof
  • Definition 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Fraïssé Theorem, fraisse,jonsson,kub
  • proof
  • ...and 32 more