Forcing with ideals generated by closed sets

Jindřich Zapletal

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 1, page 181-188
  • ISSN: 0010-2628

Abstract

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Consider the poset P I = Borel ( ℝ ) ∖ I where I is an arbitrary σ -ideal σ -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 ∈ V [ r ] is Cohen-generic over V or V [ s ] = V [ r ] .

How to cite

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Zapletal, Jindřich. "Forcing with ideals generated by closed sets." Commentationes Mathematicae Universitatis Carolinae 43.1 (2002): 181-188. <http://eudml.org/doc/248982>.

@article{Zapletal2002,
abstract = {Consider the poset $P_I=\text\{\rm Borel\}(\mathbb \{R\})\setminus I$ 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 $V[s]=V[r]$.},
author = {Zapletal, Jindřich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {forcing; descriptive set theory; large cardinals; forcing; descriptive set theory; large cardinals; -ideal},
language = {eng},
number = {1},
pages = {181-188},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Forcing with ideals generated by closed sets},
url = {http://eudml.org/doc/248982},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Zapletal, Jindřich
TI - Forcing with ideals generated by closed sets
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 1
SP - 181
EP - 188
AB - Consider the poset $P_I=\text{\rm Borel}(\mathbb {R})\setminus I$ 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 $V[s]=V[r]$.
LA - eng
KW - forcing; descriptive set theory; large cardinals; forcing; descriptive set theory; large cardinals; -ideal
UR - http://eudml.org/doc/248982
ER -

References

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  2. Jech T., Set Theory, (1978), Academic Press New York. (1978) Zbl0419.03028MR0506523
  3. Martin D.A., Steel J., A proof of projective determinacy, J. Amer. Math. Soc. (1989), 85 6582-6586. (1989) Zbl0668.03021MR0959109
  4. Neeman I., Zapletal J., Proper forcings and absoluteness in L ( ℝ ) , Comment. Math. Univ. Carolinae (1998), 39 281-301. (1998) Zbl0939.03054MR1651950
  5. Solecki S., Covering analytic sets by families of closed sets, J. Symbolic Logic 59 (1994), 1022-1031. (1994) Zbl0808.03031MR1295987
  6. Woodin W.H., Supercompact cardinals, sets of reals and weakly homogeneous trees, Proc. Natl. Acad. Sci. USA 85 6587-6591 (1988). (1988) Zbl0656.03037MR0959110
  7. Zapletal J., Isolating cardinal invariants, J. Math. Logic accepted. Zbl1025.03046
  8. Zapletal J., Countable support iteration revisited, J. Math. Logic submitted. 

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