Forcing a Basis into ℵ₁-Free Groups

2022/01/17 by Daniel Bossaller, Bossaller, Daniel, Daniel Herden +3
Mathematics · #03E40 #20K20 #20K25 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Primary: 13C10 #Secondary: 03E35 #math.GR #math.LO #msc:03E35 #msc:03E40 #msc:13C10 #msc:20K20 #msc:20K25

paper · pdf · doi:10.48550/arxiv.2201.06634

12 pages

arxiv created 2022/01/17 · arxiv updated 2022/01/19

Abstract

In this paper, we address the question of when a non-free ℵ₁-free group H can be be free in a transitive cardinality-preserving model extension. Using the Γ-invariant, denoted Γ(H), we present a necessary and sufficient condition resolving this question for ℵ₁-free groups of cardinality ℵ₁. Specifically, if Γ(H) = [ℵ₁], then H will be free in a transitive model extension if and only if ℵ₁ collapses, while for Γ(H) ≠ [ℵ₁] there exist cardinality-preserving forcings that will add a basis to H. In particular, for Γ(H) ≠ [ℵ₁], we provide a poset (\mathcal P_{\rm pb}, ≤) of partial bases for adding a basis to H without collapsing ℵ₁.

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