## Forcing with ideals generated by closed sets.(English)Zbl 1069.03037

The author shows some methods connecting descriptive set theory with definable proper forcing. He proves in (ZFC + large cardinals) the following result:
Let $$I$$ be a $$\sigma$$-ideal $$\sigma$$-generated by a projective collection of closed sets. The poset $$P_I = \text{Borel}(\mathbb R) \setminus I$$ is proper and adds a single real $$r_{\text{gen}}$$ of an almost minimal degree: If $$V \subseteq V[s] \subseteq V[r_{\text{gen}}]$$ is an intermediate model for some real $$s$$, then $$V[s]$$ is a Cohen extension of $$V$$ or else $$V[s] = V[r_{\text{gen}}]$$.

### MSC:

 3e+15 Descriptive set theory 3e+40 Other aspects of forcing and Boolean-valued models 3e+55 Large cardinals
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