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# Article

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Keywords:
forcing; descriptive set theory; large cardinals
Summary:
Consider the poset $P_I=\text{\rm Borel}(\Bbb R)\setminus I$ where $I$ is an arbitrary $\sigma$-ideal $\sigma$-generated by a projective collection of closed sets. Then the $P_I$ extension is given by a single real $r$ of an almost minimal degree: every real $s\in V[r]$ is Cohen-generic over $V$ or $V[s]=V[r]$.
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[Z1] Zapletal J.: Isolating cardinal invariants. J. Math. Logic accepted. Zbl 1025.03046
[Z2] Zapletal J.: Countable support iteration revisited. J. Math. Logic submitted.

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