[Paper Review] Existence of almost free abelian groups and reflection of stationary set

Saharon Shelah|ArXiv.org|Jun 15, 1996

Advanced Topology and Set Theory20 citations

TL;DR

This paper establishes the existence of almost free abelian groups below the first fixed point of the aleph function, using forcing and reflection principles. It resolves key questions on non-reflection of stationary sets and shows that the NPT (non-proper transversal) property is not transitive, proving consistency results under weakly compact cardinals via iterated forcing with non-reflecting stationary sets.

ABSTRACT

section 2: We answer a question of Mekler Eklof on the closure operations of the incompactness spectrum. We answer a question of Foreman and Magidor on reflection of stationary subsets of S_{< aleph_2}(lambda) = {a subseteq lambda : |a| < aleph_2}]. section 3 - NPT is not transitive. We prove NPT(lambda, mu) + NPT(mu, kappa) not => NPT(lambda, kappa)

Motivation & Objective

• To establish the existence of λ-free but not free abelian groups for λ less than the first fixed point of the aleph function.
• To resolve a question by Foreman and Magidor on the reflection of stationary subsets of S<ℵ₂(λ).
• To answer Mekler and Eklof’s questions on closure operations in the incompactness spectrum.
• To demonstrate that the NPT (non-proper transversal) property is not transitive, even when assuming NPT(λ,μ) and NPT(μ,κ).
• To construct models where NPT(λ,μ) and NPT(μ,ℵ₀) hold but NPT(λ,ℵ₀) fails, using forcing with non-reflecting stationary sets.

Proposed method

• Uses forcing with non-reflecting stationary subsets S ⊆ {δ < λ : cf(δ) = κ} to preserve NPT properties while killing reflection.
• Applies iterated forcing with Easton support and ℵ₀-Cohen forcing to construct models where NPT(λ,ℵ₀) fails despite NPT(λ,μ) and NPT(μ,ℵ₀).
• Employs the ideal J^bd_κ and sequences ⟨f_α : α < μ⟩ in ∏_{i<κ} λ_i / J to define good, bad, and chaotic points for the sequence.
• Uses the concept of <_J-eub (exact upper bound modulo J) to characterize when a sequence has a transversal.
• Applies the principle that if a stationary set is disjoint from a non-reflecting stationary set and no bounded subset is added, then PT(λ,ℵ₀) holds.
• Relies on results from Shelah’s [Sh:355], [Sh:161], and [MgSh:204] to reduce the problem to known incompactness phenomena.

Experimental results

Research questions

• RQ1 Does there exist a λ-free abelian group that is not free for λ less than the first fixed point of the aleph function?
• RQ2 Can stationary subsets of S<ℵ₂(λ) fail to reflect, and under what conditions?
• RQ3 Is the NPT property transitive? That is, if NPT(λ,μ) and NPT(μ,κ), does it follow that NPT(λ,κ)?
• RQ4 Can one construct a model where NPT(λ,μ) and NPT(μ,ℵ₀) hold but NPT(λ,ℵ₀) fails, assuming consistency of two weakly compact cardinals?
• RQ5 What is the role of non-reflecting stationary sets in preserving or breaking transversal existence in abelian group constructions?

Key findings

• For λ less than the first fixed point α = ℵ_α with α > ℵ₀, there exists a λ-free abelian group that is not free, confirming a ZFC result from [Sh:161].
• The paper proves that NPT(λ,μ) and NPT(μ,ℵ₀) do not imply NPT(λ,ℵ₀), showing that the NPT property is not transitive.
• It constructs a model where NPT(λ,μ) and NPT(μ,ℵ₀) hold but NPT(λ,ℵ₀) fails, using forcing that adds a non-reflecting stationary subset of {δ < λ : cf(δ) = κ}.
• The consistency of this failure is established assuming the existence of two weakly compact cardinals above ℵ₀.
• The method preserves PT(κ,ℵ₀) and ensures no bounded subset of λ is added, so the incompactness persists in the generic extension.
• The result relies on the fact that if a stationary set is disjoint from a non-reflecting stationary set and no bounded subset is added, then PT(λ,ℵ₀) holds, which is preserved under the forcing extension.