# Locating generic filters in the Lévy collapse

Let $$\operatorname{Col}(\omega,<\kappa)$$ denote the Lévy collapse of an inaccessible cardinal $$\kappa$$. A variant of the Factor Lemma is as follows:

Lemma. Suppose that $$\kappa$$ is an inaccessible cardinal and that $$\mathbb{P}$$ is a poset of size $$<\kappa$$. Let $$G$$ be $$\operatorname{Col}(\omega,<\kappa$$)-generic over $$V$$. If in $$V[G]$$ there is a filter $$h \subseteq \mathbb{P}$$ that is $$\mathbb{P}$$-generic over $$V$$, then there is $$G^* \in V[G]$$ that is $$\operatorname{Col}(\omega,<\kappa)$$-generic over $$V[h]$$ and such that $$V[h][G^*] = V[G]$$.

For those interested, I took this variant from the paper Happy and mad families in $$L(\mathbb{R})$$, Lemma 17 of Page 10.

This lemma shows that, in particular, $$h \in V[G]$$. Is it true that there must exist some ordinal $$\beta < \kappa$$ such that $$h \in V[G\upharpoonright\beta]$$?

Note that the lemma doesn't show that $$h$$ is in $$V[G]$$, it assumes this. But yes, if $$h\in V[G]$$ is a subset of a set $$X\in V$$ such that $$|X| < \kappa$$, then for some $$\beta < \kappa$$, $$h\in V[G\restriction \beta]$$. The argument to follow is by no means original, but I don't remember where I saw it. Fix a name $$\dot h$$ such that $$h = \dot h_G$$, and for each $$x\in X$$, let $$A_x\subseteq \text{Col}(\omega,{<}\kappa)$$ be a maximal antichain consisting of conditions that force either $$\check{x}\in \dot h$$ or its negation. Recall that $$\text{Col}(\omega,{<}\kappa)$$ has the $$\kappa$$-cc, so $$|A_x| < \kappa$$. Let $$A = \bigcup_{x\in X} A_x$$. Since $$|X| < \kappa$$, $$|A| < \kappa$$, and therefore there is some $$\beta < \kappa$$ be such that $$A\subseteq \text{Col}(\omega,{<}\beta)$$. We have $$h\in V[G\restriction \beta]$$ since $$h = \{x\in X : \exists q\in G\restriction \beta\, (q\Vdash \check{x} \in \dot h)\}$$.
• I think that, in the definition of $A_x$, $\check p\in\dot h$ should be $\check x\in\dot h$, and two lines later "Since $|\mathbb P|<\kappa$" should be "Since $|X|<\kappa$". Jan 26 at 0:45