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.-
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? -
\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? -
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)}? -
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? -
\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 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.