Walks along a weak square sequence and the non-semiproperness of Namba forcings

TL;DR

This paper explores the relationship between Namba forcing's semiproperness and square principles, revealing high consistency strength requirements and introducing minimal walk methods related to square sequences.

Contribution

It establishes a connection between the semiproperness of Namba forcing and the failure of certain square principles, using novel minimal walk techniques.

Findings

Semiproperness of Namba forcing implies failure of square principles.

High consistency strength exceeds that of infinitely many Woodin cardinals.

Introduces two-cardinal walks with naive C-sequences.

Abstract

In this paper, we demonstrate that if, for every $\kappa$-complete fine filter $F$ over $\mathcal{P}_{\kappa}\lambda$, the associated Namba forcing $\mathrm{Nm}(\kappa,\lambda,F)$ is semiproper, then $\square(\mu,{<}\aleph_1)$ fails for all regular $\mu \in [\lambda, 2^{\lambda}]$ under the certain cardinal arithmetic. In particular, this result establishes that the consistency strength of the semiproperness of $\mathrm{Nm}(\aleph_2,F)$ for every $\aleph_2$-complete filter $F$ over $\aleph_2$ exceeds the strength of infinitely many Woodin cardinals. Minimal walk methods associated with a square sequece play a central role in this paper. These observations introduce two-cardinal walks with naive $C$-sequences and show that the existence of non-reflecting stationary subsets implies $\mathcal{P}_{\kappa}\lambda \not\to [I_{\kappa\lambda}^{+}]^{3}_{\lambda}$.