Algebra Seminar talk
2025-05-23
Tatsuya Goto
Preservation of $\mathcal{E}$-positivity along some proper forcing notions
Abstract:
A preservation theorem has been obtained by J. Zapletal for preserving $\mathrm{non}(\mathcal{E})$ by countable support iterations of proper forcing notions. In this talk, we prove that the forcing $\mathbf{PT}_{f,g}$ (which increases $\mathrm{non}(\mathcal{M})$), the Miller forcing (which increases $\mathfrak{d}$), and the forcing $\mathbf{S}_{g,g*}$ (which increases $\mathrm{non}(\mathcal{N})$) preserve $\mathcal{E}$-positivity, and we apply Zapletal's theorem to construct a model satisyfying $\max \{ \mathrm{non}(\mathcal{E}), \mathrm{cov}(\mathcal{M})\} < \min \{ \mathrm{non}(\mathcal{M}), \mathfrak{d}, \mathrm{non}(\mathcal{N})\}$.