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Where can I find a proof of Shelah mentioned in the book “Almost Free Modules†as Theorem VI.5.8?
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The following is stated as Theorem VI.5.8 in the newer edition of the book "Almost Free Modules" by Eklof and Mekler, but no proof is given and no precise reference is given. Presumably it comes from Shelah's book, "Cardinal Arithmetic," but that book has 500 pages. I would like to know precisely where in that book I can find the result.
Let $mu$ be a singular cardinal of cofinality $kappa$ and let $I$ be the ideal of subsets of $kappa$ of cardinality less than $kappa$. Let $<lambda_i: i<kappa>$ be a strictly increasing sequence of regular cardinals whose supremum if $mu$. Let $tau=mu^+$.
A "scale" for $mu$ is a sequence $bar f=<f_alpha:alpha<tau>$ such that each $f_alphainprod_i<kappalambda_i$ and some other properties hold.
We say $deltaintau$ is a "good point" of $bar f$ if $delta$ is a limit and there is a cofinal subset $Xi$ of $delta$ and there is $sigma<kappa$ such that for all $alpha<beta$ in $Xi$, $f_alpha(i)<f_beta(i)$ for all $i>sigma$.
Shelah's result is that if $delta<tau$ is a limit of cofinality greater than $lambda$ which is not good, then there is a club $C$ in $delta$ such that no $alphain C$ is good. [I do not know what $lambda$ is.]
set-theory cardinals
asked Jul 26 at 7:10
Tri
1596
add a comment |Â
up vote
2
down vote
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The following is stated as Theorem VI.5.8 in the newer edition of the book "Almost Free Modules" by Eklof and Mekler, but no proof is given and no precise reference is given. Presumably it comes from Shelah's book, "Cardinal Arithmetic," but that book has 500 pages. I would like to know precisely where in that book I can find the result.
Let $mu$ be a singular cardinal of cofinality $kappa$ and let $I$ be the ideal of subsets of $kappa$ of cardinality less than $kappa$. Let $<lambda_i: i<kappa>$ be a strictly increasing sequence of regular cardinals whose supremum if $mu$. Let $tau=mu^+$.
A "scale" for $mu$ is a sequence $bar f=<f_alpha:alpha<tau>$ such that each $f_alphainprod_i<kappalambda_i$ and some other properties hold.
We say $deltaintau$ is a "good point" of $bar f$ if $delta$ is a limit and there is a cofinal subset $Xi$ of $delta$ and there is $sigma<kappa$ such that for all $alpha<beta$ in $Xi$, $f_alpha(i)<f_beta(i)$ for all $i>sigma$.
Shelah's result is that if $delta<tau$ is a limit of cofinality greater than $lambda$ which is not good, then there is a club $C$ in $delta$ such that no $alphain C$ is good. [I do not know what $lambda$ is.]
set-theory cardinals
asked Jul 26 at 7:10
Tri
1596
It is Main claim 1.3 at the bottom of page 46.
– Gabriel Fernandes
Jul 27 at 20:07
Thank you. Do you know where I can find a simpler proof?
– Tri
Jul 27 at 20:22
Please, if someone is looking for a reference with a simpler proof this might help: math.stackexchange.com/questions/2873209/…
– Gabriel Fernandes
Aug 5 at 21:21
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
The following is stated as Theorem VI.5.8 in the newer edition of the book "Almost Free Modules" by Eklof and Mekler, but no proof is given and no precise reference is given. Presumably it comes from Shelah's book, "Cardinal Arithmetic," but that book has 500 pages. I would like to know precisely where in that book I can find the result.
Let $mu$ be a singular cardinal of cofinality $kappa$ and let $I$ be the ideal of subsets of $kappa$ of cardinality less than $kappa$. Let $<lambda_i: i<kappa>$ be a strictly increasing sequence of regular cardinals whose supremum if $mu$. Let $tau=mu^+$.
A "scale" for $mu$ is a sequence $bar f=<f_alpha:alpha<tau>$ such that each $f_alphainprod_i<kappalambda_i$ and some other properties hold.
We say $deltaintau$ is a "good point" of $bar f$ if $delta$ is a limit and there is a cofinal subset $Xi$ of $delta$ and there is $sigma<kappa$ such that for all $alpha<beta$ in $Xi$, $f_alpha(i)<f_beta(i)$ for all $i>sigma$.
Shelah's result is that if $delta<tau$ is a limit of cofinality greater than $lambda$ which is not good, then there is a club $C$ in $delta$ such that no $alphain C$ is good. [I do not know what $lambda$ is.]
set-theory cardinals
asked Jul 26 at 7:10
Tri
1596
The following is stated as Theorem VI.5.8 in the newer edition of the book "Almost Free Modules" by Eklof and Mekler, but no proof is given and no precise reference is given. Presumably it comes from Shelah's book, "Cardinal Arithmetic," but that book has 500 pages. I would like to know precisely where in that book I can find the result.
Let $mu$ be a singular cardinal of cofinality $kappa$ and let $I$ be the ideal of subsets of $kappa$ of cardinality less than $kappa$. Let $<lambda_i: i<kappa>$ be a strictly increasing sequence of regular cardinals whose supremum if $mu$. Let $tau=mu^+$.
A "scale" for $mu$ is a sequence $bar f=<f_alpha:alpha<tau>$ such that each $f_alphainprod_i<kappalambda_i$ and some other properties hold.
We say $deltaintau$ is a "good point" of $bar f$ if $delta$ is a limit and there is a cofinal subset $Xi$ of $delta$ and there is $sigma<kappa$ such that for all $alpha<beta$ in $Xi$, $f_alpha(i)<f_beta(i)$ for all $i>sigma$.
Shelah's result is that if $delta<tau$ is a limit of cofinality greater than $lambda$ which is not good, then there is a club $C$ in $delta$ such that no $alphain C$ is good. [I do not know what $lambda$ is.]
set-theory cardinals
asked Jul 26 at 7:10
Tri
1596
asked Jul 26 at 7:10
Tri
1596
asked Jul 26 at 7:10
Tri
1596
asked Jul 26 at 7:10
Tri
1596
1596
It is Main claim 1.3 at the bottom of page 46.
– Gabriel Fernandes
Jul 27 at 20:07
Thank you. Do you know where I can find a simpler proof?
– Tri
Jul 27 at 20:22
Please, if someone is looking for a reference with a simpler proof this might help: math.stackexchange.com/questions/2873209/…
– Gabriel Fernandes
Aug 5 at 21:21
add a comment |Â
It is Main claim 1.3 at the bottom of page 46.
– Gabriel Fernandes
Jul 27 at 20:07
Thank you. Do you know where I can find a simpler proof?
– Tri
Jul 27 at 20:22
Please, if someone is looking for a reference with a simpler proof this might help: math.stackexchange.com/questions/2873209/…
– Gabriel Fernandes
Aug 5 at 21:21
It is Main claim 1.3 at the bottom of page 46.
– Gabriel Fernandes
Jul 27 at 20:07
It is Main claim 1.3 at the bottom of page 46.
– Gabriel Fernandes
Jul 27 at 20:07
Thank you. Do you know where I can find a simpler proof?
– Tri
Jul 27 at 20:22
Thank you. Do you know where I can find a simpler proof?
– Tri
Jul 27 at 20:22
Please, if someone is looking for a reference with a simpler proof this might help: math.stackexchange.com/questions/2873209/…
– Gabriel Fernandes
Aug 5 at 21:21
Please, if someone is looking for a reference with a simpler proof this might help: math.stackexchange.com/questions/2873209/…
– Gabriel Fernandes
Aug 5 at 21:21
add a comment |Â
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It is Main claim 1.3 at the bottom of page 46.
– Gabriel Fernandes
Jul 27 at 20:07
Thank you. Do you know where I can find a simpler proof?
– Tri
Jul 27 at 20:22
Please, if someone is looking for a reference with a simpler proof this might help: math.stackexchange.com/questions/2873209/…
– Gabriel Fernandes
Aug 5 at 21:21