Hechler's tower and mad-family forcings at uncountable cardinals: equivalence, cross-addition, and the almost disjointness number

For a regular uncountable cardinal $\lambda$, Hechler's classical forcing $\mathbb{TOW}_\lambda$ adds a tower of length $\lambda$ and its sibling $\mathbb{MAD}_\lambda$ adds a maximal almost disjoint family of size $\lambda$, both by finite approximations on subsets of $\omega$. Fischer, Koelbing, and Wohofsky (2023) proved that both are finite-support iterations of Mathias forcings with respect to filters that are Canjar for every countably directed unbounded family of the ground model, so both preserve ground-model unbounded scales and force $\mathfrak{t}=\mathfrak{b}=\omega_1$. The task is the following triple of questions their paper poses in Section 6: (i) are $\mathbb{TOW}_\lambda$ and $\mathbb{MAD}_\lambda$ forcing equivalent for $\lambda>\omega_1$, given that for $\lambda\le\omega_1$ both coincide with adding $\lambda$ Cohen reals and the authors strongly conjecture inequivalence for larger $\lambda$; (ii) does $\mathbb{TOW}_\lambda$ add a mad family of size $\lambda$, and dually does $\mathbb{MAD}_\lambda$ add a tower of length $\lambda$; and (iii) do these forcings force $\mathfrak{a}=\omega_1$?

Background

Classical combinatorial set theory studies $[\omega]^\omega$, the infinite subsets of the natural numbers, modulo finite changes. For $a,b\in[\omega]^\omega$, $b\subseteq^* a$ means that $b\setminus a$ is finite (almost inclusion). A tower of length $\delta$ is a sequence $\langle a_\xi\mid\xi<\delta\rangle$ in $[\omega]^\omega$ such that $a_\eta\subseteq^* a_\xi$ whenever $\xi<\eta<\delta$ and there is no $b\in[\omega]^\omega$ with $b\subseteq^* a_\xi$ for all $\xi<\delta$ (no infinite pseudo-intersection); the tower number $\mathfrak{t}$ is the least length of a tower. Two infinite subsets of $\omega$ are almost disjoint if their intersection is finite; a family of pairwise almost disjoint infinite sets is maximal almost disjoint (a mad family) if every infinite $b\subseteq\omega$ has infinite intersection with some member of the family, and the almost disjointness number $\mathfrak{a}$ is the least size of a mad family. On $\omega^\omega$ one writes $f\leq^* g$ if $f(n)\leq g(n)$ for all but finitely many $n$; a family is unbounded if it is unbounded in this order, and the bounding number $\mathfrak{b}$ is the least size of an unbounded family.

In 1972, Stephen Hechler introduced two forcing notions, defined by finite approximations, that generically add a tower and a mad family, respectively. Fischer, Koelbing, and Wohofsky (2023) analyzed their versions $\mathbb{TOW}_\lambda$ and $\mathbb{MAD}_\lambda$ for a regular uncountable cardinal $\lambda$ (a cardinal whose cofinality equals itself). A condition of $\mathbb{TOW}_\lambda$ is a finite function $p$ with $\mathrm{dom}(p)\subseteq\lambda$ such that each value $p(\alpha)=(s_\alpha,f_\alpha)$ consists of a finite binary string $s_\alpha\in 2^{<\omega}$, a finite approximation to the $\alpha$-th set $a_\alpha$ of the future tower, together with a finite function $f_\alpha$ with domain $\subseteq\alpha$ recording thresholds that enforce the tower coherence: for $\beta\in\mathrm{dom}(f_\alpha)$ and $n\geq f_\alpha(\beta)$ in the common domain of $s_\beta$ and $s_\alpha$, $s_\beta(n)=0$ implies $s_\alpha(n)=0$, which in the limit yields $a_\alpha\subseteq^* a_\beta$ for $\beta<\alpha$. $\mathbb{MAD}_\lambda$ has exactly the same shape, except that the coherence clause becomes $s_\beta(n)=0$ or $s_\alpha(n)=0$ for $n\geq h_\alpha(\beta)$, which in the limit makes $a_\alpha\cap a_\beta$ finite. In the generic extension, $a_\alpha$ is the union of the stems $s_\alpha$ appearing in the generic filter; $\langle a_\alpha\mid\alpha<\lambda\rangle$ is then a tower of length $\lambda$, respectively $\{a_\alpha\mid\alpha<\lambda\}$ a mad family of size $\lambda$. Both posets are $\sigma$-centered (a countable union of centered subposets, hence ccc, i.e., every antichain is countable) whenever $\lambda\le\mathfrak{c}$, where $\mathfrak{c}=2^{\aleph_0}$.

The main theorem of the 2023 paper represents both forcings as finite-support iterations (iterations in which each condition mentions the forcing at only finitely many stages) of Mathias forcings. Mathias forcing with respect to a filter $\mathcal{F}$ on $\omega$ containing the Fréchet filter (the cofinite sets), denoted $\mathbb{M}(\mathcal{F})$, consists of pairs $(s,A)$ with $s\in 2^{<\omega}$ and $A\in\mathcal{F}$, ordered by end-extension of $s$ and shrinking of $A$. In the tower case the filter at stage $\alpha$ is generated by the initial segment of the generic tower; in the mad-family case, by the complements of the members of the initial segment of the generic mad family. A filter is $\mathcal{B}$-Canjar (for $\mathcal{B}\subseteq\omega^\omega$) when $\mathbb{M}(\mathcal{F})$ preserves the unboundedness of $\mathcal{B}$, and a family $\mathcal{B}$ is countably directed when every countable subfamily has a $\leq^*$-upper bound inside $\mathcal{B}$. Using the combinatorial characterization of $\mathcal{B}$-Canjarness due to Guzmán and Kalajdzievski, together with a preservation theorem of Judah and Shelah for finite-support iterations, the authors prove that all filters occurring in the two iterations are $\mathcal{B}$-Canjar for every countably directed unbounded family $\mathcal{B}$ of the ground model. Consequently both forcings preserve the unboundedness of every ground-model unbounded scale (a $\leq^*$-increasing unbounded sequence) and force $\mathfrak{b}=\omega_1$; in fact every extension by either forcing satisfies $\mathfrak{t}=\mathfrak{b}=\omega_1$ and contains a tower of length $\lambda$, respectively a mad family of size $\lambda$.

For $\lambda\le\omega_1$ the two forcings are forcing equivalent, meaning they produce the same generic extensions (for instance via an isomorphism of their Boolean completions, i.e., their regular open algebras): both decompose into finite-support iterations of Mathias forcings with respect to countably generated filters, each equivalent to Cohen forcing (the forcing adding a single Cohen real), so both amount to adding $\lambda$ Cohen reals. For $\lambda>\omega_1$ this argument collapses — in the $\mathbb{MAD}_\lambda$-iteration the filter at stage $\beta$ is never countably generated unless $\beta<\omega_1$ — and the authors state that they strongly conjecture the two forcings are not forcing equivalent for larger $\lambda$, without a proof. The three questions of Section 6 are the equivalence question, the cross-addition question (whether each forcing adds the other's characteristic object), and the question of what these forcings force about $\mathfrak{a}$.

Problem Statement

Let $\lambda$ be a regular uncountable cardinal, and let $\mathbb{TOW}_\lambda$ and $\mathbb{MAD}_\lambda$ be Hechler's forcing notions adding, respectively, a tower of length $\lambda$ and a mad family of size $\lambda$ of subsets of $\omega$, as defined by Fischer, Koelbing, and Wohofsky (Definitions 3.1 and 4.1 of the source paper; the explicit definitions are given in the background above). The problem is the triple of questions posed in Section 6 of that paper, which share one scientific objective: to distinguish $\mathbb{TOW}_\lambda$ from $\mathbb{MAD}_\lambda$ (or prove them equivalent) and thereby determine the effect of these classical Hechler-style forcings on the almost disjointness number $\mathfrak{a}$ (the values $\mathfrak{t}=\mathfrak{b}=\omega_1$ are already settled by the cited results); Questions 1 and 2 are asked for $\lambda>\omega_1$, where the Cohen-collapse argument available for $\lambda\le\omega_1$ no longer applies, while Question 3 is asked for every regular uncountable $\lambda$.

1. Equivalence: Are $\mathbb{TOW}_\lambda$ and $\mathbb{MAD}_\lambda$ forcing equivalent for $\lambda>\omega_1$? (For $\lambda\le\omega_1$ they are equivalent, both amounting to adding $\lambda$ Cohen reals; the source authors strongly conjecture a negative answer for $\lambda>\omega_1$.)

2. Cross-addition: For $\lambda>\omega_1$, does $\mathbb{TOW}_\lambda$ add a mad family of size $\lambda$? Does $\mathbb{MAD}_\lambda$ add a tower of length $\lambda$?

3. Almost disjointness number: For regular uncountable $\lambda$, does $\mathbb{TOW}_\lambda$ force $\mathfrak{a}=\omega_1$? Does $\mathbb{MAD}_\lambda$ force $\mathfrak{a}=\omega_1$? (Both forcings force $\mathfrak{t}=\mathfrak{b}=\omega_1$; the analogous statement for $\mathfrak{a}$ is what is asked.)

References

1. Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky: Towers, mad families, and unboundedness. Archive for Mathematical Logic 62(5-6):811-830, 2023. DOI: 10.1007/s00153-023-00861-x. https://doi.org/10.1007/s00153-023-00861-x
2. Vera Fischer, Marlene Koelbing, Wolfgang Wohofsky: Correction to: Towers, mad families, and unboundedness. Archive for Mathematical Logic 62(7-8):1159-1160, 2023. DOI: 10.1007/s00153-023-00870-w. https://doi.org/10.1007/s00153-023-00870-w
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