The problem of Horn and Tarski

Abstract

In 1948 A. Horn and A. Tarski asked whether the notions of a σ-finite cc and a σ-bounded cc ordering are equivalent. We give a negative answer to this question.When analyzing Boolean algebras carrying a measure, Horn and Tarski [HT48] defined the following two notions:Definition 1. An ordering P is called (i) σ-bounded cc if P = n∈ω P n , where each P n has the n + 2-cc.(ii) σ-finite cc if P = n∈ω P n , where each P n has the ω-cc.Here an ordering or its subset has the κ-cc (κ-chain condition) for a cardinal κ if it contains no antichain (set of pairwise orthogonal elements) of size κ.

Chain conditions in free Banach lattices

“…The σ-bounded chain condition was introduced by Horn and Tarski in connection with Boolean algebras carrying a strictly positive measures. It is worth recalling that the related Horn-Tarski problem, whether the condition σ-bcc is equivalent to its certain formally weaker version was solved in the negative only a few years ago by Thümmel [6] and Todorcevic [9].…”

Matrix Iterations with Vertical Support Restrictions

“…σfinite-cc) and, under b = ℵ 1 , a σ-Λ n -covered poset that is not Λ n+1 -Knaster. Todorčević [Tod14] and Thümmel [Thü14] constructed σ-finite-cc posets that are not σ-Γ bd-cc -covered (i.e. σ-bounded-cc).…”

Todorcevic orderings as examples of ccc forcings without adding random reals

“…In [13], Todorcevic introduced a Borel definable ccc forcing which consistently does not have property K. He defined it on a separable metric space. By generalizing it and applying it to other topological spaces, Thümmel discovered a forcing notion which has the σ-finite chain condition but does not have the σ-bounded chain condition, and so he solved the problem of Horn and Tarski [11]. (For Horn-Tarski's problem, see [5], [14].)…”