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To this question in Kunen's introduction to independence proofs

on the page 379, in the proof of lemma V.7.3 you never mention $q$ in the proof, only $p$; why? I cannot make sense of this simple proof and lemma.

I've received this answer:

The claim is that there is a q extending p and a countable subset B of kappa such that q forces a certain property of B.

The proof says that if there is no such q and B, then p forces that in the generic extension, there is no countable ground-model subset B of kappa satisfying this property.

And this last statement (which doesn't mention q) leads to a contradiction.

BUT I still do not understand why we can get rid off q and say

then p forces that in the generic extension, there is no countable ground-model subset B of kappa satisfying this property.

I may write down the details for those who do not have the book opened on the table, but I think that the context is almost clear from the question and the partial answer of it.

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The inductive definition of forcing (by complexity of formulas) gives in particular that $p$ forces $\lnot\psi$ if and only if no extension of $p$ forces $\psi$. That is exactly what is being claimed.

As for why this general fact about forcing of negations holds, it is either immediate from the fact that a statement holds in a generic extension if and only if some member of the generic forces it, or else one needs to delve into the ground model characterization of the forcing relation. The latter is done in detail in Kunen's book.

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  • $\begingroup$ Could you please give me the number of the relevant page of that "detail in Kunen's book" ? $\endgroup$ – user122424 Jan 25 at 19:50
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    $\begingroup$ There is a whole chapter devoted to details of forcing. It is more beneficial if you actually look through it than if I do. $\endgroup$ – Andrés E. Caicedo Jan 25 at 19:51
  • $\begingroup$ I really cannot skim through almost 400 pages to find such a detail. Can you mention here at least the title of the relevant subchapter? Thank you a lot. $\endgroup$ – user122424 Jan 25 at 19:55
  • $\begingroup$ Is it the "Truth lemma" on the page 251?? $\endgroup$ – user122424 Jan 25 at 20:12
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    $\begingroup$ @user122424 p.244--263 are relevant to your question. $\endgroup$ – Stefan Mesken Jan 25 at 20:19

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