A Sacks real out of nowhere. (English) Zbl 1187.03040

The authors investigate proper countable support forcing iterations where all the iterands add no new reals (NNR). An example of Jensen’s shows that the limit can add a new real. It is known (Shelah) that the added-in-the-limit real must be bounded by a real in the ground model; it cannot be a Cohen, random, Laver or Mathias real. But it can be a Sacks real. For the authors prove the following: There is an iteration \((P_n,Q_n)_{n<\omega}\) such that each \(Q_n\) is forced to be proper and NNR and such that the countable support limit \(P_{\omega}\) adds a Sacks real. Moreover, \(P_{\omega}\) is equivalent to \(S\ast P'\), where \(S\) is Sacks forcing and \(P'\) is NNR.


03E40 Other aspects of forcing and Boolean-valued models
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[1] Set theory of the reals (Ramat Gan, 1991) 6 pp 305– (1993)
[2] The Souslin problem 405 (1974) · Zbl 0289.02043
[3] The covering property axiom, CPA: A combinatorial core of the iterated perfect set model 164 (2004) · Zbl 1066.03047
[4] Set theory: On the structure of the real line (1995)
[5] DOI: 10.1002/malq.200510018 · Zbl 1087.03031
[6] DOI: 10.2307/1970860 · Zbl 0244.02023
[7] Proper and improper forcing (1998) · Zbl 0889.03041
[8] Proper forcing 940 (1982)
[9] Forcing idealized 174 (2008) · Zbl 1140.03030
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