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Consider the forcing notion(s) introduced by Friedman (or Mitchell or Neeman) for adding a club subset of $\omega_2$ by finite conditions. In the generic extension CH fails, but I can't see the reals added by the forcing. Would you please give an explicit construction of $\aleph_2$-many reals in the generic extension by these forcings. Can we determine if the added reals are Cohen, Random, or ....

Remark. The question applies to many similar forcing constructions, in particular to the forcings introduced in Neeman's paper.

References.

1) Friedman, Forcing with finite conditions. Set theory, 285–295, Trends Math., Birkhäuser, Basel, 2006.

2) Mitchell, Adding closed unbounded subsets of $\omega_2$ with finite forcing. Notre Dame J. Formal Logic 46 (2005), no. 3, 357–371.

3) Neeman, Forcing with sequences of models of two types.

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According to your remark this is a really interesting question. –  Saint Georg Nov 5 '13 at 9:42

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up vote 9 down vote accepted

Each of the posets you mention adds $\omega_2$ many Cohen reals. Let $G$ be generic for any of the 3 posets you mentioned. The point is that any collection of $\omega_1$-many reals in $V[G]$ can be captured by an intermediate extension of the form $V[G \cap M]$ where $M$ is a sufficiently elementary substructure of $V$ containing $\omega_1$ as a subset and $G$ contains a strong master condition (in the sense of Mitchell) for the model $M$. Then one can show that the quotient forcing $\mathbb{P}/(G \cap M)$ is strongly proper with respect to all countable models of the form $N[G \cap M]$, where $N$ is a countable model from $V$. This is a general phenomenon, but in the case of these forcings it can be shown rather directly; Neeman's preprint gives the details. The fact that the quotient is strongly proper with respect to stationarily many countable models abstractly implies that the quotient adds lots of Cohen reals; Mitchell discusses this in his paper "On the Hamkins Approximation Property".

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Thanks for your nice answer. –  Mohammad Golshani Nov 6 '13 at 3:21

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