Loading [MathJax]/jax/element/mml/optable/BasicLatin.js
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

1. Problems in Low Forcing

"Low Forcing" refers to the method of side conditions, specifically where the side conditions are elementary submodels of some large transitive structure.
    1.     Let \mathbb{C} = (C,\le) be a linear order of size \kappa and consider its Cartesian square under the ordering (x_1,y_1) \le (x_2,y_2) iff x_1 \le x_2 and y_1 \le y_2. \mathbb C is a Countryman line if this Cartesian square is the union of less than \kappa-many chains (i.e. linearly ordered subsets).

      Problem 1.05.

      [Justin Moore] Is it consistent (with the continuum hypothesis) that there is a minimal \aleph_2-Countryman line—that is, an \aleph_2-Countryman line that order-embeds into all others?
        •     Two linear orders \mathbb{C}_1 and \mathbb{C}_2 are near if there is another linear order \mathbb{C}_0 that embeds into both of them. \mathbb{C}_1 and \mathbb{C}_2 are co-near if there is a linear order embedding into \mathbb{C}_1 and \mathbb{C}_2^\ast, where \mathbb{C}_2^\ast is the reverse of \mathbb{C}_2.

          Problem 1.1.

          [Justin Moore] Is it consistent with the continuum hypothesis that any two \aleph_2-Countryman lines are near or co-near?
            •     Strong homology is a homology theory satisfying the Eilenberg-Steenrod axioms and is invariant under strong shape.

              Problem 1.15.

              [Justin Moore] Is strong homology consistently additive for closed subspaces of Euclidean space, or more generally for locally compact metric spaces?
                  CH and PFA both imply a negative answer to this question. It is likely that a positive answer to this question implies that \mathfrak{d}<\aleph_\omega, where \mathfrak{d} is the dominating number of the continuum.
                •     \mathbb A = \langle A_f : f \in \omega^\omega \rangle is the following inverse system of abelian groups:

                  A_f = \bigoplus_{n \in \omega} \bigoplus_{i < f(n)} \mathbb Z = \bigoplus_{n \in \omega} {\mathbb Z}^{f(n)}

                  Let {\mathbb Z}^{\omega \times \omega}/fin be {\mathbb Z}^{\omega \times \omega} modulo finite equivalence, let G_f = \prod_{n \in \omega} {\mathbb Z}^{f(n)}/fin, and consider the chain complex

                  {\mathbb Z}^{\omega \times \omega}/fin \xrightarrow{\delta} \prod_{f_0 \in \omega^\omega} G_{f_0} \xrightarrow{\delta} \prod_{f_0 \le f_1} G_{f_0} \xrightarrow{\delta} \prod_{f_0 \le f_1 \le f_2} G_{f_0} \xrightarrow{\delta} \ldots

                  where

                  \delta_s(f_0,f_1,\ldots,f_n) = \sum^n_{i = 0}(-1)^i s(f_0,\ldots,\hat{f}_i,\ldots,f_n)

                  for s \in \prod_{f_0 \le \ldots \le f_{n-1}} G_{f_0}.

                  We define \lim^p \mathbb A \cong \ker(\delta^p)/\text{im}(\delta^{p-1}).

                  Problem 1.2.

                  [Justin Moore] Is it consistent to have \lim^p \mathbb{A} = 0 for all p?
                      If strong homology is consistent for locally compact metric spaces, then the answer to this question is positive.
                    • Problem 1.25.

                      Is it consistent that there is a poset which adds a club in \omega_3 which contains no infinite ground model subset over a model of GCH?
                        • Problem 1.3.

                          Is there a large cardinal hypothesis that proves the bounded forcing axiom for Namba forcing?
                            • Problem 1.35.

                              Can Namba forcing be iterated with side conditions? Does this work if we replace Namba forcing with a forcing satisfying Shelah’s S-condition?
                                • Problem 1.4.

                                  Assume CH. Is there a strongest forcing axiom of \sigma-closed posets meeting \aleph_2-many dense sets? What if we do not assume CH?
                                    • Conjecture 1.45.

                                      The following is inconsistent: “For every \Sigma_1 statement \varphi with \aleph_1 and \aleph_2 as parameters, if \varphi can be forced by an \aleph_1, \aleph_2-preserving forcing, then \varphi holds.”
                                        •     If X is a topological space, let s(X)=\sup\{|Y|:Y \text{ is a discrete subspace of }X\}. It is a theorem that |X| \le 2^{2^{s(X)}}.

                                          Problem 1.5.

                                          [Stevo Todorčević] When can one obtain |X|\le 2^{s(X)}?
                                              The answer is negative under \Diamond, even for compact Hausdorff spaces. PFA implies that the answer is negative for X with s(X)=\aleph_0. PFA also implies that |X| \le 2^{\aleph_0} for every Hausdorff space with no uncountable discrete subspaces.
                                            • Problem 1.55.

                                              [Stevo Todorčević] Do any of the analogs of PFA at \aleph_2 give us |X| \le 2^{\aleph_1} for every Hausdorff space with no discrete subspaces of size \aleph_2?
                                                •     It is a fact that a compact Hausdorff space X is countably tight if and only if X contains no \omega_1-free sequence.

                                                  Problem 1.6.

                                                  [Stevo Todorčević] Suppose X is a compact Hausdorff space of tightness \aleph_1 (i.e. there are no \omega_2-free sequences) and density \aleph_1. Is it true that |X| \le 2^{\aleph_1}?
                                                    • Problem 1.65.

                                                      [Stevo Todorčević] Under a given analog of PFA for \aleph_2, how many cofinal types of directed sets of cardinality \aleph_2 are there?
                                                          Under PFA, 1, \omega, \omega_1, \omega \times \omega_1, and (\omega_1)^{<\omega} are the cofinal types of directed sets of cardinality \aleph_1.
                                                        •     \kappa \rightarrow (\alpha,\beta)^n means that for every coloring F:[\kappa]^n \rightarrow \{0,1\}, then then there is either a homogeneous set H_0 of order-type \alpha such that F"H_0=\{0\}, or else a homogeneous set H_1 such that F"H_1=\{1\}.

                                                          Problem 1.7.

                                                          [Stevo Todorčević] Can \omega_2 \rightarrow (\omega_2,\alpha)^2 be deduced from one of the forcing axioms at \aleph_2.
                                                              It is known that PFA implies \omega_1 \rightarrow (\omega_1,\alpha)^2 for all \alpha<\omega_1.
                                                            •     It is known under PFA that the gaps-spectrum of P(\omega)/fin consists of (\omega_1,\omega_1^\ast),(\omega_2,\omega^\ast), and (\omega,\omega_2^\ast).

                                                              Problem 1.75.

                                                              [Stevo Todorčević] Assuming an appropriate generalization of PFA, determine the gaps spectrum of P(\omega)/fin.

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