2014/02/07 by Matteo Viale, Viale, Matteo, Giorgio Audrito +3 · 1 citation
Mathematics · Medicine · #03E40 #Advanced Topology and Set Theory #Diet and metabolism studies #FOS: Mathematics #Logic (math.LO) #Neurological and metabolic disorders
paper · pdf · doi:10.48550/arxiv.1402.1714
openalex publication_date 2014/02/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
These notes present a compact and self-contained approach to iterated forcing with a particular emphasis on semiproper forcing. We tried to make our presentation accessible to any scholar who has some familiarity with forcing and boolean valued models and full details of all proofs are given. We focus our presentation using the boolean algebra language and defining an iteration system as a directed and commutative system of complete and injective homomorphisms between complete and atomless boolean algebras. It is well known that the boolean algebra approach to forcing and iterations is fully equivalent to the standard one. While there are several monographs where forcing is introduced by means of boolean valued models, to our knowledge no detailed account of iterated forcing following a boolean algebraic approach has yet appeared. We believe that this different approach is fruitful since the richness of the algebraic language simplifies many calculations and definitions, among which that of RCS-limits. Some of the advantages of this approach have been already outlined by Donder and Fuchs in http://arxiv.org/abs/math/9207204. The first part of these notes present the general framework needed to develop the notion of limit of an iterated system of forcings in the boolean algebraic language. The second part contains a proof of the main result of Shelah on semiproper iterations, i.e. that RCS-limit of semiproper iterations are semiproper.