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

Zapletal, Jindřich

Commentat. Math. Univ. Carol. 43, No. 1, 181-188 (2002).

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

Reviewer: Miroslav Zelený (Praha)

MSC:

03E15	Descriptive set theory
03E40	Other aspects of forcing and Boolean-valued models
03E55	Large cardinals

Keywords:

forcing; descriptive set theory; large cardinals; \(\sigma \)-ideal