# 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 $$\{t_k(F) \colon F \in \mathbb P, s(F) \leq k \}.$$