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3. Problems in the Overlap of High and Low Forcing


    1. Add remark

      Problem 3.1.

      Is there a model for $\text{TP}_{\kappa^{++}}$, with $\kappa$ a singular strong limit of countable cofinality, where we use side conditions in place of the Mitchell poset?


        • Add remark

          Problem 3.2.

          Is there a model for $\text{TP}_{\aleph_2} + \text{TP}_{\aleph_3}$ using side conditions?


            • Add remark

              Problem 3.3.

              Is there a model where both $I[\aleph_2]$ and $I[\aleph_3]$ are trivial? In other words, is $\text{AP}_{\aleph_2} + \text{AP}_{\aleph_3}$ consistent?


                • Add remark

                  Problem 3.4.

                  Is the forcing axiom for proper posets of size $\aleph_1$ with $\aleph_2$-many dense sets consistent? If so, what if we replace the size-$\aleph_2$ requirement with the $\aleph_2$-chain condition?

                    •     There is a coloring $F:[\aleph_2]^2 \rightarrow {0,1}$ of pairs from $\aleph_2$ in $2$ colors so that the poset of finite approximations to a $0$- or $1$-homogeneous set has the $\aleph_2$-chain condition. (We mean either the poset $\mathbb P$ of finite functions $f:\omega \rightarrow F^{-1}(0)$ or the poset of $f:\omega \rightarrow F^{-1}(0)$, ordered by inclusion)

                      Add remark

                      Problem 3.5.

                      Can this forcing be made proper using side conditions? Does this preserve the $\aleph_2$-chain condition?
                          If we do obtain properness, then the fprcing axiom for $\aleph_2$-many dense sets proper posets of size $\aleph_1$—and for proper posets with the $\aleph_2$-chain condition—is inconsistent.


                        • Add remark

                          Problem 3.6.

                          [David Aspero] Suppose $\aleph_\omega$ is a strong limit. Is there a poset ${\mathbb P}$ of size $<\aleph_\omega$, which adds a club guessing sequence for $S^{\omega_2}_{\omega_1}$, preserving $\omega_1$, $\omega_2$, and $\omega_3$.
                              If yes than by PCF theory, in ZFC it would be provable that $2^{\aleph_\omega}<\aleph_{\omega_3}$ when $\aleph_\omega$ is a strong limit.

                              Cite this as: AimPL: High and Low forcing, available at http://aimpl.org/highlowforcing.