Suppose $\kappa$ is a cardinal preserved in the generic extension $V[G]$. Let $Y \subseteq \kappa$ be an unbounded set in $V[G]$. Does there always exist an $X \in V$ such that $X \subseteq Y$ and $X$ is unbounded?
This question comes from Section 18 of James Cummings' Singular Cardinal Arithmetic. In this context, he forced with the Levy collapses $\operatorname{Col}(\omega,\delta^{+\omega}) \times \operatorname{Col}(\delta^{+\omega+2},<\kappa)$, and fixed "$X \subseteq \gamma$ unbounded of order type $\delta_V^{+\omega+1}$...". In the next line, he then says that "However, it is easy to see that there is $Y \in V$ with $Y \subseteq X$ unbounded".
This observation is certainly not easy for me, so my questions are:
- Why do such a $Y$ exist?
- Is the existence of such a $Y$ limited to forcing with Levy collapse?
Any help is appreciated.
EDIT: Fixed the broken link.
EDIT 2: I re-asked the question on MathOverflow, and have accepted an answer.