What goes wrong shooting a club on $kappa >aleph_1$?












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I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $Ssubseteqomega_1$ can be generalized for regular $kappa>aleph_1$, but with additional restrictions on the stationary subset we use in order to preserve $kappa$.



Why do we need these restrictions? What goes wrong in the usual argument if we consider, say, $aleph_2$?



Is it that all club subsets of ordinals $alpha<omega_1$ have countable cofinality, while those $alpha<omega_2$ might have confinality $omega_1$ or $omega$?






set-theory forcing







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  • 3




    $begingroup$
    Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
    $endgroup$
    – Asaf Karagila♦
    Jan 19 at 23:12


















1












$begingroup$


I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $Ssubseteqomega_1$ can be generalized for regular $kappa>aleph_1$, but with additional restrictions on the stationary subset we use in order to preserve $kappa$.



Why do we need these restrictions? What goes wrong in the usual argument if we consider, say, $aleph_2$?



Is it that all club subsets of ordinals $alpha<omega_1$ have countable cofinality, while those $alpha<omega_2$ might have confinality $omega_1$ or $omega$?






set-theory forcing







share|cite|improve this question









$endgroup$








  • 3




    $begingroup$
    Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
    $endgroup$
    – Asaf Karagila♦
    Jan 19 at 23:12
















1












1








1





$begingroup$


I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $Ssubseteqomega_1$ can be generalized for regular $kappa>aleph_1$, but with additional restrictions on the stationary subset we use in order to preserve $kappa$.



Why do we need these restrictions? What goes wrong in the usual argument if we consider, say, $aleph_2$?



Is it that all club subsets of ordinals $alpha<omega_1$ have countable cofinality, while those $alpha<omega_2$ might have confinality $omega_1$ or $omega$?






set-theory forcing







share|cite|improve this question









$endgroup$




I'm reading through Jech's book, and in the section Stationary Sets in Generic Extensions (pages 444 and 445), he remarks that the poset used in shooting a club through a stationary $Ssubseteqomega_1$ can be generalized for regular $kappa>aleph_1$, but with additional restrictions on the stationary subset we use in order to preserve $kappa$.



Why do we need these restrictions? What goes wrong in the usual argument if we consider, say, $aleph_2$?



Is it that all club subsets of ordinals $alpha<omega_1$ have countable cofinality, while those $alpha<omega_2$ might have confinality $omega_1$ or $omega$?






set-theory forcing




set-theory forcing






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jan 19 at 20:52









JunderscoreHJunderscoreH

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  • 3




    $begingroup$
    Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
    $endgroup$
    – Asaf Karagila♦
    Jan 19 at 23:12
















  • 3




    $begingroup$
    Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
    $endgroup$
    – Asaf Karagila♦
    Jan 19 at 23:12










3




3




$begingroup$
Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
$endgroup$
– Asaf Karagila♦
Jan 19 at 23:12






$begingroup$
Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
$endgroup$
– Asaf Karagila♦
Jan 19 at 23:12












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