[Paper Review] Models of some cardinal invariants with large continuum

Diego A. Mejía|arXiv (Cornell University)|May 21, 2013

Advanced Topology and Set Theory13 references9 citations

TL;DR

This paper constructs models of set theory with a large continuum (greater than ℵ₂) where various cardinal invariants of the continuum—such as add(𝒩), cov(𝒩), p, non(𝒫), cov(𝒫), d, non(𝒩), and 𝔠—take arbitrary regular values. Using finite support iterations and matrix iteration techniques, the author extends previous methods to achieve arbitrary consistent values for these invariants, including non-Cichon’s diagram invariants like 𝔰 and 𝔯.

ABSTRACT

We extend the applications of the techniques used in Arch Math Logic 52:261-278, 2013, to present various examples of consistency results where some cardinal invariants of the continuum take arbitrary regular values with the size of the continuum being bigger than $\aleph_2$.

Motivation & Objective

• To extend forcing techniques to construct models where the continuum is larger than ℵ₂ and specific cardinal invariants take arbitrary regular values.
• To address the consistency of arbitrary regular values for cardinal invariants beyond those in Cichon’s diagram, such as 𝔰 and 𝔯.
• To investigate the preservation of lower and upper bounds of cardinal invariants under forcing extensions using generalized preservation results.
• To explore the limitations and conditions under which invariants like 𝔲 and 𝔯 can be controlled in matrix iteration constructions.
• To identify open questions regarding the possibility of achieving specific values for 𝔲 and 𝔯 in such models.

Proposed method

• Utilizes finite support iteration (fsi) techniques from [8] and matrix iteration methods introduced by Blass and Shelah.
• Employs forcing notions including Cohen (𝓒), random (𝓑), Hechler (𝓓), eventually different reals (𝓔), and Mathias forcing (𝓜𝓕) with filter bases.
• Applies preservation results from Section 2 to maintain bounds on invariants through forcing extensions.
• Constructs matrix iterations of dimensions 𝜅 × 𝜅𝜈 with specific forcing conditions at each stage, parameterized by sequences of names for suborders and filter bases.
• Uses the iteration 𝔼_{𝜅,𝜆𝜅𝜈} to control invariants such as add(𝒩) = 𝜇₁, cov(𝒩) = 𝜇₂, p = non(𝒫) = 𝜈, d = 𝜅, non(𝒩) = 𝔠 = 𝜆.
• Applies Lemma 2.25 on Laver forcing with ultrafilters to extend the framework to new forcing contexts.

Experimental results

Research questions

• RQ1 Can the matrix iteration method be extended to achieve 𝔰 = 𝜅 or 𝔯 = 𝜅 in models with large continuum?
• RQ2 Is it possible to achieve 𝔲 = 𝜈 instead of 𝔡 = 𝜅 in the consistency results of Theorem 4.5?
• RQ3 Does Lemma 2.23 hold for the relation ⊥ (pitchfork), which would allow control over 𝔰 and 𝔯 in new constructions?
• RQ4 Can Laver forcing be used to achieve 𝔲 = 𝜅 or 𝔲 = 𝜆 in models with large continuum?
• RQ5 What are the limitations of the matrix iteration method when forcing notions like 𝓑 and 𝓓 are used cofinally in columns?

Key findings

• In the model 𝑉_{𝜅,𝜆𝜅𝜈}, add(𝒩) = 𝜇₁, cov(𝒩) = 𝜇₂, p = non(𝒫) = 𝜈, and cov(𝒫) ≥ 𝜈.
• The dominating number satisfies 𝔡 = 𝜅 and the continuum is ℵ_𝜆, with non(𝒩) = 𝔠 = 𝜆.
• The ultrafilter number satisfies 𝔲 ≤ 𝜈, witnessed by a sequence of Mathias reals that generate an ultrafilter.
• The model satisfies the Generalized Continuum Hypothesis at 𝜆, denoted GCH_𝜆.
• The construction ensures that the values of the invariants are preserved and can be assigned arbitrary regular values.
• The method fails to achieve 𝔯 = 𝜅 due to the non-preservation of ( +_{𝓑,∝} ), limiting the control over 𝔯 in such models.