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The following concept is due to Shelah and I have some issues with a claim using this notion: Suppose that $\nu$ is a limit ordinal and that $P_\nu$ is an iteration of forcing notions. We say that a $P_\nu$ name $\dot{\alpha}$ of an ordinal is $prompt$ iff the following two things hold:

  1. $\Vdash_\nu \dot{\alpha} \le \nu$

  2. If p \Vdash_\nu "\dot{\alpha} = \xi" then even $p \upharpoonright \xi ^\smallfrown 1_\nu \upharpoonright [\xi, \nu) \Vdash_\nu \dot{\alpha} = \xi$ ( $1_\nu$ should be the largest element of the iteration, and $\xi$ is the hacek name of an ordinal though I refused to write the hacek)

Then the following two things should hold:

  1. If $\dot{\alpha}$ is prompt $\eta \le \nu$ and if $p \Vdash_\nu \eta \le \dot{\alpha}$ then $p \upharpoonright \xi ^\smallfrown 1_\nu \upharpoonright [\xi, \nu) \Vdash_\nu \eta \le \dot{\alpha}$
  2. If $\dot{\alpha_i}$ are prompt then so is the supremum $sup$ and the minimum $min$

I have problems proving those two assertions so any help would be highly appreciated. Thank you!

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    Seeing how it was answered (twice!) on MO, I'm voting to close this. Also, it's a shame you're not coming to Italy after all! I'll see you in September, I hope!2013-05-31

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