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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. 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}.
        • 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.
            • 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.
                • Problem 2.2.

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

                              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.

                                  Problem 2.4.

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