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 \(V[s] = V[r_{\text{gen}}]\).


03E15 Descriptive set theory
03E40 Other aspects of forcing and Boolean-valued models
03E55 Large cardinals
Full Text: EuDML EMIS