Mixing
Confusion on a lemma concerning a club of substructures of $V_kappa$
$begingroup$
So i've reached this lemma in Kanamori's The higher infinite, which states:
Lemma. Suppose that $kappa$ is inaccessible and that $R subseteq V_kappa$. Then ${ alpha lt kappa mid langle V_alpha, in, Rcap V_alpharangle prec langle V_kappa, in, Rrangle }$ is a club in $kappa$.
Now my confusion arises from the fact that the book says that the reason this set is closed is clear. But for me it isn't because given an elementary embedding for each $ alpha_xi $, $xi lt theta lt kappa $, i don't know how to produce one for $lambda = cup_{xi lt theta}
alpha_xi$. Then i thought maybe he meant the elementary embeddings are the inclusion functions but that seems to be off aswell. I would really appreciate any help.
set-theory
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
$endgroup$
|
show 2 more comments
$begingroup$
So i've reached this lemma in Kanamori's The higher infinite, which states:
Lemma. Suppose that $kappa$ is inaccessible and that $R subseteq V_kappa$. Then ${ alpha lt kappa mid langle V_alpha, in, Rcap V_alpharangle prec langle V_kappa, in, Rrangle }$ is a club in $kappa$.
Now my confusion arises from the fact that the book says that the reason this set is closed is clear. But for me it isn't because given an elementary embedding for each $ alpha_xi $, $xi lt theta lt kappa $, i don't know how to produce one for $lambda = cup_{xi lt theta}
alpha_xi$. Then i thought maybe he meant the elementary embeddings are the inclusion functions but that seems to be off aswell. I would really appreciate any help.
set-theory
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
$endgroup$
1
$begingroup$
Do you know that the increasing union of elementary models is elementary?
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:16
$begingroup$
No. I haven't studied model theory that much. Can you please provide a reference or proof for this?
$endgroup$
– Shervin Sorouri
Feb 1 at 12:19
$begingroup$
I don't have one off hand. But it's fairly simple to prove directly from the definition. Especially in the case of these structures which are end-extensions of each other.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:30
$begingroup$
The statement mentioned by Asaf follows directly from the definitions. A reference will likely say no more than that. Better if you prove it on your own.
$endgroup$
– Andrés E. Caicedo
Feb 1 at 12:32
3
$begingroup$
You're thinking way too hard about this.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:38
|
show 2 more comments
$begingroup$
So i've reached this lemma in Kanamori's The higher infinite, which states:
Lemma. Suppose that $kappa$ is inaccessible and that $R subseteq V_kappa$. Then ${ alpha lt kappa mid langle V_alpha, in, Rcap V_alpharangle prec langle V_kappa, in, Rrangle }$ is a club in $kappa$.
Now my confusion arises from the fact that the book says that the reason this set is closed is clear. But for me it isn't because given an elementary embedding for each $ alpha_xi $, $xi lt theta lt kappa $, i don't know how to produce one for $lambda = cup_{xi lt theta}
alpha_xi$. Then i thought maybe he meant the elementary embeddings are the inclusion functions but that seems to be off aswell. I would really appreciate any help.
set-theory
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
$endgroup$
So i've reached this lemma in Kanamori's The higher infinite, which states:
Lemma. Suppose that $kappa$ is inaccessible and that $R subseteq V_kappa$. Then ${ alpha lt kappa mid langle V_alpha, in, Rcap V_alpharangle prec langle V_kappa, in, Rrangle }$ is a club in $kappa$.
Now my confusion arises from the fact that the book says that the reason this set is closed is clear. But for me it isn't because given an elementary embedding for each $ alpha_xi $, $xi lt theta lt kappa $, i don't know how to produce one for $lambda = cup_{xi lt theta}
alpha_xi$. Then i thought maybe he meant the elementary embeddings are the inclusion functions but that seems to be off aswell. I would really appreciate any help.
set-theory
set-theory
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
edited Feb 1 at 12:45
Shervin Sorouri
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
asked Feb 1 at 11:42
Shervin SorouriShervin Sorouri
519212
519212
1
$begingroup$
Do you know that the increasing union of elementary models is elementary?
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:16
$begingroup$
No. I haven't studied model theory that much. Can you please provide a reference or proof for this?
$endgroup$
– Shervin Sorouri
Feb 1 at 12:19
$begingroup$
I don't have one off hand. But it's fairly simple to prove directly from the definition. Especially in the case of these structures which are end-extensions of each other.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:30
$begingroup$
The statement mentioned by Asaf follows directly from the definitions. A reference will likely say no more than that. Better if you prove it on your own.
$endgroup$
– Andrés E. Caicedo
Feb 1 at 12:32
3
$begingroup$
You're thinking way too hard about this.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:38
|
show 2 more comments
1
$begingroup$
Do you know that the increasing union of elementary models is elementary?
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:16
$begingroup$
No. I haven't studied model theory that much. Can you please provide a reference or proof for this?
$endgroup$
– Shervin Sorouri
Feb 1 at 12:19
$begingroup$
I don't have one off hand. But it's fairly simple to prove directly from the definition. Especially in the case of these structures which are end-extensions of each other.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:30
$begingroup$
The statement mentioned by Asaf follows directly from the definitions. A reference will likely say no more than that. Better if you prove it on your own.
$endgroup$
– Andrés E. Caicedo
Feb 1 at 12:32
3
$begingroup$
You're thinking way too hard about this.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:38
1
1
$begingroup$
Do you know that the increasing union of elementary models is elementary?
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:16
$begingroup$
Do you know that the increasing union of elementary models is elementary?
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:16
$begingroup$
No. I haven't studied model theory that much. Can you please provide a reference or proof for this?
$endgroup$
– Shervin Sorouri
Feb 1 at 12:19
$begingroup$
No. I haven't studied model theory that much. Can you please provide a reference or proof for this?
$endgroup$
– Shervin Sorouri
Feb 1 at 12:19
$begingroup$
I don't have one off hand. But it's fairly simple to prove directly from the definition. Especially in the case of these structures which are end-extensions of each other.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:30
$begingroup$
I don't have one off hand. But it's fairly simple to prove directly from the definition. Especially in the case of these structures which are end-extensions of each other.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:30
$begingroup$
The statement mentioned by Asaf follows directly from the definitions. A reference will likely say no more than that. Better if you prove it on your own.
$endgroup$
– Andrés E. Caicedo
Feb 1 at 12:32
$begingroup$
The statement mentioned by Asaf follows directly from the definitions. A reference will likely say no more than that. Better if you prove it on your own.
$endgroup$
– Andrés E. Caicedo
Feb 1 at 12:32
3
3
$begingroup$
You're thinking way too hard about this.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:38
$begingroup$
You're thinking way too hard about this.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:38
|
show 2 more comments
0
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1
$begingroup$
Do you know that the increasing union of elementary models is elementary?
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:16
$begingroup$
No. I haven't studied model theory that much. Can you please provide a reference or proof for this?
$endgroup$
– Shervin Sorouri
Feb 1 at 12:19
$begingroup$
I don't have one off hand. But it's fairly simple to prove directly from the definition. Especially in the case of these structures which are end-extensions of each other.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:30
$begingroup$
The statement mentioned by Asaf follows directly from the definitions. A reference will likely say no more than that. Better if you prove it on your own.
$endgroup$
– Andrés E. Caicedo
Feb 1 at 12:32
3
$begingroup$
You're thinking way too hard about this.
$endgroup$
– Asaf Karagila♦
Feb 1 at 12:38