Killing meager towers

[theorem]Proposition [theorem]Lemma [theorem]Claim [theorem]Fact [theorem]Corollary [theorem]Observation [theorem]Definition [theorem]Example [theorem]Question

Tower

Let \(\mathcal T = \{T_\alpha \colon \alpha \in {\omega}_1\}\) be an increasing tower generating (together with \({\mathord\mathrm{Fin}}\)) a meager ideal.

We can assume that \({\omega}\) is partitioned into intervals \({\omega}= \bigcup_{n \in {\omega}} I_n\), such that for each \(\alpha \in {\omega}_1\) and \(n \in {\omega}\) is \(I_n \not{\subseteq}T_\alpha\). (Make finite modification to \(T_\alpha\) if necessary.) Denote \(J_n = \bigcup_{i \leq n} I_i\). We can moreover assume that \(\mathcal T\) is \({\omega}_1\)-dense.

[theorem]Proposition [theorem]Lemma [theorem]Claim [theorem]Fact [theorem]Corollary [theorem]Observation [theorem]Definition [theorem]Example [theorem]Question

Forcing

To each \(F \in \mathcal T^{< {\omega}}\) we can assign (and fix) several characteristics.

\(|F| \in {\omega}\), the cardinality of \(F\)

denote \(e_n(F) = |J_n \setminus \bigcup F|\)

denote \(t_n(F) = \{ J_n \cap T \colon T\in F \}\)

\(l(F) \in {\omega}\), some integer such that \(|t_{l(F)}(F)| = |F|\)

\(s(F) \in {\omega}\), some integer such that \(s(F) \geq l(F)\) and \(e_{s(F)} > 2 |F| \)

The forcing is \(\mathbb P = \{F \in \mathcal T^{< {\omega}} \colon (\forall k \in {\omega}) |t_k(F)| < e_k(F)\}\), the ordering is inverse inclusion.

For each \(\alpha \in{\omega}_1\) is the set \(D_\alpha = \{F \in \mathbb P \colon (\exists \beta>\alpha) T_\beta \in F \}\) dense.

Use \({\omega}_1\)-density.

If \(\mathcal C \subset \mathbb P\) is centered, then \(\bigcup \bigcup \mathcal C\) is co-infinite!

Let \(G \subset \mathbb P\) be a generic filter. For \(T = \bigcup \bigcup \mathcal G\) is \({\omega}\setminus T\) infinite and \(T_\alpha \subset^* T\) for each \(\alpha \in {\omega}_1\). Thus \(\mathbb P\) destroys tallness of \(\mathcal T\).

\(\mathbb P\) is \(\sigma\)-linked.

A piece of the decomposition of \(\mathbb P\) is set of some \(F\)’s, such that all characteristics \(|F|\), \(l(F)\), \(s(F)\) and \(t_{s(F)}\) are constant.

\(\mathbb P\) is almost \({{{\vphantom{{\omega}}}^{{\omega}\!\!}{{\omega}}}}\) bounding.

We are given a condition \(p\) and a name \(\dot h\) for a function in \({{{\vphantom{{\omega}}}^{{\omega}\!\!}{{\omega}}}}\). We are supposed to find \(g \in {{{\vphantom{{\omega}}}^{{\omega}\!\!}{{\omega}}}}\) such that for each \(X \in [{\omega}]^{\omega}\) there is a condition \(q<p\) such that \[q \Vdash g\restriction X \text{ is not } <^* \text{-bounded by } \dot h\restriction X.\] For simplicity assume \(p = \emptyset\).

For each \(k \in {\omega}\) let \(\mathcal A_k \subset \mathbb P\) be the set of all conditions deciding \(\dot{h(k)}\).

For \(k \in {\omega}\) let \(Y_k\) be the finite set \(\{t_k(F) \colon F \in \mathbb P, s(F) \leq k \}.\)