Consider the forcing $\Bbb P$ whose conditions are partial functions $p\colon\omega\to2$ with $\operatorname{dom}(p)$ a co-infinite subset of $\omega$, ordered by reverse inclusion.
Does $\Bbb P$ collapse the continuum?
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asked Jan 26, 2015 at 12:54
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2 Answers
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This is the forcing to add a Prikry-Silver real, discussed on page 17 of Jech's book Multiple Forcing. A fusion argument shows that it satisfies the countable cover property (every new countable set of ordinals is covered by a ground model countable set), and so it does not collapse the continuum. A Prikry-Silver real is minimal.
answered Jan 26, 2015 at 13:14
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$\begingroup$ Aha! I knew I'd seen it before. And just as luck would have it, I have a copy of that book from the library with me. I wonder why I didn't check it before! $\endgroup$– Asaf Karagila ♦Commented Jan 26, 2015 at 13:15
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$\begingroup$ Looking at Jech's book it strikes me that we really have two tricks up our sleeve for adding a new subset to some $X$. Fix some ideal on $X$ and forcing with partial functions from $X$ to $2$; either the domains are from the ideal, or the domains are co-positive sets. The ideal of finite sets gives us Cohen in the first case and Prikry-Silver in the latter case. $\endgroup$– Asaf Karagila ♦Commented Jan 26, 2015 at 14:09
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$\begingroup$ I'm sure that it's possible to investigate properties of the new sets inherited from the ideal and the choice of zero or co-positive domains, in a context much more general than ideals on $\omega$ (but I do expect things to get much more complicated, of course... and depend on additional hypotheses like cardinal arithmetics and large cardinals). $\endgroup$– Asaf Karagila ♦Commented Jan 26, 2015 at 14:10
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$\begingroup$ I'd encourage you to go for it! And what of other natural ideals, such as the ideal of all $A\subset\omega$ for which $\sum_{n\in A}\frac1n$ converges, or the ideal of all asymptotic density zero sets? $\endgroup$ Commented Jan 26, 2015 at 14:14
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2$\begingroup$ @Asaf The generalized Cohen forcing and generalized Silver forcing were introduced by Serge Grigorieff Combinatorics on ideals and forcing, Ann. Math. Logic 3 (1971), 363-394; MR0297560. $\endgroup$ Commented Jan 26, 2015 at 19:10
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While Prikry-Silver forcing satisfies Axiom A and hence does not collapse $\omega_{1}$, it is independent of $MA + \neg CH$ whether Prikry-Silver forcing preserves the continuum. Under $MA + \neg CH$, this question is equivalent to whether the additivity of the Silver ideal is the continuum. So, for example, PFA implies that Prikry-Silver forcing does indeed preserve the continuum. For the corresponding question for Sacks forcing, see:
H. Judah, A. W. Miller and S. Shelah, Sacks forcing, Laver forcing, and Martin's axiom. Arch. Math. Logic 31 (1992), 145–161.
answered Feb 27, 2019 at 23:37