[Paper Review] Kurepa-trees and Namba-forcing

Bernhard Koenig, Yasuo Yoshinobu|arXiv (Cornell University)|May 4, 2006

Advanced Topology and Set Theory3 references4 citations

TL;DR

This paper demonstrates that compact cardinals and Martin’s Maximum (MM) are sensitive to $λ$-closed forcing for arbitrarily large $λ$ by constructing $λ$-regressive $λ$-Kurepa-trees via $λ$-closed forcing, and shows that MM prevents such trees from existing. The key contribution is proving that Namba forcing is essential to destroy regressive Kurepa-trees under MM, revealing a fundamental gap between $λ$-closed and $λ$-directed-closed forcings in set-theoretic forcing axioms.

ABSTRACT

We show that compact cardinals and { m MM} are sensitive to $λ$-closed forcings for arbitrarily large $λ$. This is done by adding 'regressive' $λ$-Kurepa-trees in either case. We argue that the destruction of regressive Kurepa-trees with { m MM} requires the use of Namba forcing.

Motivation & Objective

• To investigate the sensitivity of large cardinal properties and forcing axioms like MM to $λ$-closed forcing for arbitrarily large $λ$.
• To construct $λ$-regressive $λ$-Kurepa-trees using $λ$-closed forcing, showing that such forcings can introduce new combinatorial structures.
• To demonstrate that MM prevents the existence of $ω_1$-regressive $λ$-Kurepa-trees for uncountable regular $λ$, establishing its sensitivity to $λ$-closed forcing.
• To clarify the role of Namba forcing in violating MM fragments, showing it is necessary for destroying regressive Kurepa-trees under MM.
• To contrast the behavior of $λ$-closed and $λ$-directed-closed forcings, revealing a significant gap in their impact on forcing axioms like MM and PFA.

Proposed method

• Introduces the concept of $γ$-regressive trees, where regressive functions on limit levels preserve meet-structure, to define $λ$-regressive $λ$-Kurepa-trees.
• Constructs a $λ$-closed forcing $Κ_{\text{reg}}^{\lambda}$ that adds a $λ$-regressive $λ$-Kurepa-tree for any uncountable regular $λ$.
• Uses Namba forcing to show that the destruction of regressive Kurepa-trees under MM requires its specific properties, as it preserves stationary subsets of $ω_1$ without the covering property.
• Applies an iteration $× = \text{Nm}*\text{CS}(\dot{E}_{\mathcal{B}})$ to shoot a club through a stationary set of ground model branches, preserving stationarity and enabling the use of MM.
• Employs a $ω_2$-directed-closed forcing $Χ$ to extract an $ω_2$-Kurepa-subtree from a weak $λ$-Kurepa-tree in a generic extension, preserving the $ω_1$-regressive property.
• Combines $ω_2$-directed-closed and $λ$-closed forcings to derive a contradiction under MM, proving that $ω_1$-regressive $λ$-Kurepa-trees cannot exist.

Experimental results

Research questions

• RQ1 Can $λ$-closed forcing for arbitrarily large $λ$ affect the existence of Kurepa-trees under compact cardinals?
• RQ2 Is Martin’s Maximum (MM) sensitive to $λ$-closed forcing for arbitrarily large $λ$?
• RQ3 Does Namba forcing play a unique role in destroying regressive Kurepa-trees under MM, or can other $λ$-closed forcings suffice?
• RQ4 What is the distinction between $λ$-closed and $λ$-directed-closed forcings in their impact on forcing axioms like MM and PFA?
• RQ5 Can the existence of $ω_1$-regressive $λ$-Kurepa-trees be consistently forced under MM for uncountable regular $λ$?

Key findings

• For every uncountable regular $λ$, there exists a $λ$-closed forcing $\mathcal{K}_{\text{reg}}^{\lambda}$ that adds a $λ$-regressive $λ$-Kurepa-tree.
• If $\kappa$ is a compact cardinal and $\lambda \geq \kappa$ is regular, then no $\kappa$-regressive $\lambda$-Kurepa-tree exists, showing compact cardinals are sensitive to $λ$-closed forcing.
• Under MM, there are no $\omega_1$-regressive $\lambda$-Kurepa-trees for any uncountable regular $\lambda$, proving MM’s sensitivity to $λ$-closed forcing.
• The failure of MM can be achieved using only $λ$-closed forcing for arbitrarily large $λ$, unlike previous results requiring stronger or more complex forcing.
• Namba forcing is essential for destroying regressive Kurepa-trees under MM, as it preserves stationary subsets of $\omega_1$ without the covering property.
• The composition of $ω_2$-directed-closed and $λ$-closed forcings preserves MM but leads to a contradiction if a $ω_1$-regressive $λ$-Kurepa-tree exists, confirming its non-existence under MM.