Quite free complicated abelian group, PCF and Black Boxes

2014/04/10 by Saharon Shelah, Shelah, Saharon · 1 citation
Computer Science · Mathematics · #03E04 #03E75 #20K20 #20K30 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Logic (math.LO) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.1404.2775

openalex publication_date 2014/04/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We like to build Abelian groups (or R-modules) which on the one hand are quite free, say ℵ_{ω+ 1}-free, and on the other hand, are complicated in suitable sense. We choose as our test problem having no non-trivial homomorphism to Z (known classically for ℵ₁-free, recently for ℵ_n-free). We succeed to prove the existence of even ℵ_{ω1 ⋅ n}-free ones. This requires building n-dimensional black boxes, which are quite free. Thus combinatorics is of self interest and we believe will be useful also for other purposes. On the other hand, modulo suitable large cardinals, we prove that it is consistent that every ℵ_{ω1 ⋅ ω}-free Abelian group has non-trivial homomorphisms to Z.

Citations

• aleph_n-Free abelain group with no non-zero homomorphism Z
• Non-reflection of the bad set for check I_theta[lambda] and pcf

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• Failure of singular compactness for Hom

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