Proof of the Solovay Theorem in Jech
1
$begingroup$
The Solovay Theorem says:
Let $kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95]
One step in Jech's proof of this theorem I do not understand (underlined in red in the picture below). Why is $D$ a closed unbounded set?
set-theory cardinals
asked Jan 5 at 11:48
JoriJori
513224
$endgroup$
|
show 1 more comment
1
$begingroup$
The Solovay Theorem says:
Let $kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95]
One step in Jech's proof of this theorem I do not understand (underlined in red in the picture below). Why is $D$ a closed unbounded set?
set-theory cardinals
asked Jan 5 at 11:48
JoriJori
513224
$endgroup$
4
$begingroup$
Do you understand why $D$ is closed? Or why it's unbounded?
$endgroup$
– Asaf Karagila♦
Jan 5 at 14:00
$begingroup$
@AsafKaragila Maybe unboundedness comes from the assumption that $W$ is a stationary set of regular cardinals? And closure by the definition of $D$? But... I find the concepts pretty hard to get an intuitive grasp!
$endgroup$
– Jori
Jan 6 at 16:44
$begingroup$
Why is $C$ a set of regular cardinals?
$endgroup$
– Asaf Karagila♦
Jan 6 at 17:19
$begingroup$
@AsafKaragila My idea was: $W$ is a stationary set, hence $sup W = kappa$ (unbounded). Also, because $gamma in W$ is regular, $eta(xi) < gamma$ for all $xi < gamma$. But I now realize that is not true. So I'm back to square one.
$endgroup$
– Jori
Jan 7 at 19:21
$begingroup$
I feel I just do not know enough about the $eta(xi)$.
$endgroup$
– Jori
Jan 7 at 20:02
|
show 1 more comment
1
1
1
$begingroup$
The Solovay Theorem says:
Let $kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95]
One step in Jech's proof of this theorem I do not understand (underlined in red in the picture below). Why is $D$ a closed unbounded set?
set-theory cardinals
asked Jan 5 at 11:48
JoriJori
513224
$endgroup$
The Solovay Theorem says:
Let $kappa$ be a regular uncountable cardinal. Then every stationary subset is the disjoint union of $kappa$-many stationary subsets. [Jech, Theorem 8.8, p. 95]
One step in Jech's proof of this theorem I do not understand (underlined in red in the picture below). Why is $D$ a closed unbounded set?
set-theory cardinals
set-theory cardinals
asked Jan 5 at 11:48
JoriJori
513224
asked Jan 5 at 11:48
JoriJori
513224
edited Jan 6 at 16:46
Jori
asked Jan 5 at 11:48
JoriJori
513224
asked Jan 5 at 11:48
JoriJori
513224
asked Jan 5 at 11:48
JoriJori
513224
513224
4
$begingroup$
Do you understand why $D$ is closed? Or why it's unbounded?
$endgroup$
– Asaf Karagila♦
Jan 5 at 14:00
$begingroup$
@AsafKaragila Maybe unboundedness comes from the assumption that $W$ is a stationary set of regular cardinals? And closure by the definition of $D$? But... I find the concepts pretty hard to get an intuitive grasp!
$endgroup$
– Jori
Jan 6 at 16:44
$begingroup$
Why is $C$ a set of regular cardinals?
$endgroup$
– Asaf Karagila♦
Jan 6 at 17:19
$begingroup$
@AsafKaragila My idea was: $W$ is a stationary set, hence $sup W = kappa$ (unbounded). Also, because $gamma in W$ is regular, $eta(xi) < gamma$ for all $xi < gamma$. But I now realize that is not true. So I'm back to square one.
$endgroup$
– Jori
Jan 7 at 19:21
$begingroup$
I feel I just do not know enough about the $eta(xi)$.
$endgroup$
– Jori
Jan 7 at 20:02
|
show 1 more comment
4
$begingroup$
Do you understand why $D$ is closed? Or why it's unbounded?
$endgroup$
– Asaf Karagila♦
Jan 5 at 14:00
$begingroup$
@AsafKaragila Maybe unboundedness comes from the assumption that $W$ is a stationary set of regular cardinals? And closure by the definition of $D$? But... I find the concepts pretty hard to get an intuitive grasp!
$endgroup$
– Jori
Jan 6 at 16:44
$begingroup$
Why is $C$ a set of regular cardinals?
$endgroup$
– Asaf Karagila♦
Jan 6 at 17:19
$begingroup$
@AsafKaragila My idea was: $W$ is a stationary set, hence $sup W = kappa$ (unbounded). Also, because $gamma in W$ is regular, $eta(xi) < gamma$ for all $xi < gamma$. But I now realize that is not true. So I'm back to square one.
$endgroup$
– Jori
Jan 7 at 19:21
$begingroup$
I feel I just do not know enough about the $eta(xi)$.
$endgroup$
– Jori
Jan 7 at 20:02
4
4
$begingroup$
Do you understand why $D$ is closed? Or why it's unbounded?
$endgroup$
– Asaf Karagila♦
Jan 5 at 14:00
$begingroup$
Do you understand why $D$ is closed? Or why it's unbounded?
$endgroup$
– Asaf Karagila♦
Jan 5 at 14:00
$begingroup$
@AsafKaragila Maybe unboundedness comes from the assumption that $W$ is a stationary set of regular cardinals? And closure by the definition of $D$? But... I find the concepts pretty hard to get an intuitive grasp!
$endgroup$
– Jori
Jan 6 at 16:44
$begingroup$
@AsafKaragila Maybe unboundedness comes from the assumption that $W$ is a stationary set of regular cardinals? And closure by the definition of $D$? But... I find the concepts pretty hard to get an intuitive grasp!
$endgroup$
– Jori
Jan 6 at 16:44
$begingroup$
Why is $C$ a set of regular cardinals?
$endgroup$
– Asaf Karagila♦
Jan 6 at 17:19
$begingroup$
Why is $C$ a set of regular cardinals?
$endgroup$
– Asaf Karagila♦
Jan 6 at 17:19
$begingroup$
@AsafKaragila My idea was: $W$ is a stationary set, hence $sup W = kappa$ (unbounded). Also, because $gamma in W$ is regular, $eta(xi) < gamma$ for all $xi < gamma$. But I now realize that is not true. So I'm back to square one.
$endgroup$
– Jori
Jan 7 at 19:21
$begingroup$
@AsafKaragila My idea was: $W$ is a stationary set, hence $sup W = kappa$ (unbounded). Also, because $gamma in W$ is regular, $eta(xi) < gamma$ for all $xi < gamma$. But I now realize that is not true. So I'm back to square one.
$endgroup$
– Jori
Jan 7 at 19:21
$begingroup$
I feel I just do not know enough about the $eta(xi)$.
$endgroup$
– Jori
Jan 7 at 20:02
$begingroup$
I feel I just do not know enough about the $eta(xi)$.
$endgroup$
– Jori
Jan 7 at 20:02
|
show 1 more comment
0
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4
$begingroup$
Do you understand why $D$ is closed? Or why it's unbounded?
$endgroup$
– Asaf Karagila♦
Jan 5 at 14:00
$begingroup$
@AsafKaragila Maybe unboundedness comes from the assumption that $W$ is a stationary set of regular cardinals? And closure by the definition of $D$? But... I find the concepts pretty hard to get an intuitive grasp!
$endgroup$
– Jori
Jan 6 at 16:44
$begingroup$
Why is $C$ a set of regular cardinals?
$endgroup$
– Asaf Karagila♦
Jan 6 at 17:19
$begingroup$
@AsafKaragila My idea was: $W$ is a stationary set, hence $sup W = kappa$ (unbounded). Also, because $gamma in W$ is regular, $eta(xi) < gamma$ for all $xi < gamma$. But I now realize that is not true. So I'm back to square one.
$endgroup$
– Jori
Jan 7 at 19:21
$begingroup$
I feel I just do not know enough about the $eta(xi)$.
$endgroup$
– Jori
Jan 7 at 20:02