Mixing
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
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
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add a comment |
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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
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
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3
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Abraham, Uri; Shelah, Saharon_, Forcing closed unbounded sets, J. Symb. Log. 48, 643-657 (1983). ZBL0568.03024.
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– Asaf Karagila♦
Jan 19 at 23:12
add a comment |
$begingroup$
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
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
$endgroup$
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
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
asked Jan 19 at 20:52
JunderscoreHJunderscoreH
164
164
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
add a comment |
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
add a comment |
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$begingroup$
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