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2. Problems in High Forcing

High forcing has to do with the combinatorics of compactness versus non-compactness.

$\text{TP}_\kappa$ refers to the tree property at $\kappa$: Every tree $T$ of height $\kappa$ with levels of size $<\kappa$ has a cofinal branch of length $\kappa$.

$\text{SCH}_\kappa$ is the singular cardinal hypothesis at $\kappa$: $2^{\text{cf}(\kappa)} < \kappa$ imples $\kappa^{\text{cf}(\kappa)} = \kappa^+$.

$\square_\kappa^\ast$ is the weak (Jensen) square, which asserts the existence of a sequence $ \langle {\mathcal C}_\alpha : \alpha < \kappa^+ \rangle$ such that $1 \le |{\mathcal C}_\alpha| \le \kappa$, and such that for all $\alpha$, every $C \in {\mathcal C}_\alpha$ is a club in $\alpha$ of order-type $\le \kappa$ such that for every $\beta \in \lim C$, $C \cap \beta \in {\mathcal C}_\beta$.

$\text{AP}_\kappa$ is the approachability property, which asserts the existence of a sequence $\langle C_\alpha : \alpha < \kappa^+ \rangle$ such that every $C_\alpha$ is a club in $\alpha$ such that $\text{o.t.}(C_\alpha) = \text{o.t.}(\alpha)$ and for club-many $\alpha$, $\forall \beta<\alpha, \exists \gamma < \alpha, C_\alpha \cap \beta = C_\gamma$.


    1. Add remark

      Problem 2.05.

      Is it consistent (relative to large cardinals) that $\text{TP}_{\aleph_{\omega+1}}$ holds along with reflection for all stationary subsets of $\aleph_{\omega+1}$?
          This is consistent at $\aleph_{\omega^2+1}$.


        • Add remark

          Problem 2.1.

          Is it consistent to have the failure of both $\text{SCH}$ and $\square_{\aleph_\omega}^\ast$?
              A positive answer is necessary to get $\text{AP}_{\aleph_{\omega+1}}+\text{AP}_{\aleph_{\omega+2}}$. Since $\text{TP}_{\kappa^+}$ implies $\neg \square_\kappa^\ast$, this is also necessary to get the failure of $\text{SCH}_{\aleph_\omega}$ together with $\text{TP}_{\aleph_{\omega+1}}$. Furthermore, $\neg \text{SCH}_{\aleph_\omega}$ is required for $\text{TP}_{\aleph_{\omega+2}}$ where $\aleph_\omega$ is a strong limit.


            • Add remark

              Problem 2.15.

              Is $\neg \square_\kappa^\ast + \neg SCH_\kappa + GCH_{<\kappa}$ consistent for $\kappa$ singular?
                  This is known without $\text{GCH}$ from Gitik-Sharon, where Laver preparation disrupts GCH below the cardinal.


                • Add remark

                  Problem 2.2.

                  Is it consistent that there is a poset turning $\aleph_{\omega+1}$ into $\aleph_2$?


                    • Add remark

                      Problem 2.25.

                      Find a model of $\text{TP}_\kappa$ for all regular $\kappa \in [\aleph_2,\aleph_{\omega^2+2}]$ where $\aleph_{\omega^2}$ is the first strong limit.


                        • Add remark

                          Problem 2.3.

                          Is it consistent that every poset either adds a real or collapses a cardinal?
                              This question is related to the tree property.

                            •     A (thin) $(\kappa,\lambda)$-tree is a set $F \subset \{f: x \to 2 | x \in P_\kappa(\lambda)\}$ such that:

                              1. $\forall f \in F, \forall x \subset \text{dom}(f), f \upharpoonleft x \in F$;
                              2. $\forall x \in P_\kappa(\lambda), \exists f \in F$ such that $\text{dom}(f)=x$;
                              3. $\forall x \in P_\kappa(\lambda), |\text{Lev}_x(F)|<\kappa$ where $\text{Lev}_x(F) = \{f \in F: \text{dom}(f)=x \}$.


                              A cofinal branch of a $(\kappa,\lambda)$-tree $F$ is a function $b:\lambda \to 2$ such that $\forall x \in P_\kappa(\lambda)$, $b \upharpoonright x \in \text{Lev}_x(F)$. Given an assignment $x \mapsto f_x \in \text{Lev}_x(F)$, a cofinal branch $b$ is ineffable if $\{x \in P_\kappa(\lambda): b \upharpoonright x = f_x\}$ is stationary.

                              If $\kappa$ is regular it has the strong tree property if $\forall \lambda \ge \kappa$, every $(\kappa,\lambda)$-tree has a cofinal branch. $\kappa$ has the super tree property if $\forall \lambda \ge \kappa$ and every $(\kappa,\lambda)$-tree, there is a cofinal branch $b$ which is ineffable.

                              Add remark

                              Problem 2.35.

                              [Laura Fontanella] Can we obtain the super tree property at $\aleph_{\omega+1}$?

                                •     There is a lemma of Magidor and Shelah that if $\lambda = \sup_{n<\omega}\kappa_n$ for supercompact cardinals $\kappa_n$, then $\lambda^+$ has the tree property. Fontanella proved that if the $\kappa_n$’s are strongly compact then $\lambda^+$ has the strong tree property.

                                  Add remark

                                  Problem 2.4.

                                  [Laura Fontanella] Is it possible to obtain the super tree property at a successor of a singular cardinal?


                                    • Add remark

                                      Problem 2.45.

                                      [Laura Fontanella] Suppose $\forall n<\omega$, $kappa_n$ has the super tree property property. If $\lambda = \sup_{n<\omega}$, does $\lambda^+$ have the tree property?


                                        • Add remark

                                          Problem 2.5.

                                          [Laura Fontanella] Suppose $\kappa$ has the strong tree property. Does SCH hold above $\kappa$? (If yes, then the strong tree property at $\aleph_2$ implies SCH.)


                                            • Add remark

                                              Problem 2.55.

                                              [Monroe Eskiw] Is it consistent to obtain $TP_{\aleph_2}$ together with the existence of a saturated ideal on $\aleph_2$?

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