Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
forcing; descriptive set theory; large cardinals
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 $V[s]=V[r]$.
References:
[B] Bartoszynski T., Judah H.: Set Theory: On the Structure of the Real Line. (1995), A K Peters Wellesley, Massachusetts. MR 1350295 | Zbl 0834.04001
[J] Jech T.: Set Theory. (1978), Academic Press New York. MR 0506523 | Zbl 0419.03028
[M] Martin D.A., Steel J.: A proof of projective determinacy. J. Amer. Math. Soc. (1989), 85 6582-6586. MR 0959109 | Zbl 0668.03021
[N] Neeman I., Zapletal J.: Proper forcings and absoluteness in $L(\Bbb R)$. Comment. Math. Univ. Carolinae (1998), 39 281-301. MR 1651950 | Zbl 0939.03054
[S] Solecki S.: Covering analytic sets by families of closed sets. J. Symbolic Logic 59 (1994), 1022-1031. MR 1295987 | Zbl 0808.03031
[W] Woodin W.H.: Supercompact cardinals, sets of reals and weakly homogeneous trees. Proc. Natl. Acad. Sci. USA 85 6587-6591 (1988). MR 0959110 | Zbl 0656.03037
[Z1] Zapletal J.: Isolating cardinal invariants. J. Math. Logic accepted. Zbl 1025.03046
[Z2] Zapletal J.: Countable support iteration revisited. J. Math. Logic submitted.
Search
Partner of
