Does a $kappa$-Kurepa tree imply we have a slim $kappa$-Kurepa tree?The tree property for non-weakly compact $kappa$IF $kappa$ is weakly compact, then $kappa$ has the tree property.Tree, no uncountable antichainsViewing a $kappa$-tree as a set of functions.How can an $omega_1$-tree possibly be normal and yet not have any $omega_1$-branch?Kurepa trees and inaccessible cardinals in $L$For which $kappa$ can we have if $#alpha=#beta=kappa$, then the following also have cardinality $kappa$?Definition for “power” of a tree? (Kurepa trees)existence of a branch of length $kappa$ in a tree of height $kappa$ whose levels are finite$kappa^+$ tree with no $kappa^+$ branch
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Does a $kappa$-Kurepa tree imply we have a slim $kappa$-Kurepa tree?
The tree property for non-weakly compact $kappa$IF $kappa$ is weakly compact, then $kappa$ has the tree property.Tree, no uncountable antichainsViewing a $kappa$-tree as a set of functions.How can an $omega_1$-tree possibly be normal and yet not have any $omega_1$-branch?Kurepa trees and inaccessible cardinals in $L$For which $kappa$ can we have if $#alpha=#beta=kappa$, then the following also have cardinality $kappa$?Definition for “power” of a tree? (Kurepa trees)existence of a branch of length $kappa$ in a tree of height $kappa$ whose levels are finite$kappa^+$ tree with no $kappa^+$ branch
$begingroup$
Take a $kappa$-Kurepa tree to be a tree with more than $kappa$ branches, of height $kappa$, each level having cardinality less than $kappa$
A slim $kappa$-Kurepa tree is the same but the cardinality of each level $alpha<kappa$ has cardinality $leq|alpha|$.
Is it true in general the existence of a $kappa$-Kurepa tree implies a slim one exists?
set-theory
asked Mar 20 at 19:26
A.HA.H
184
$endgroup$
add a comment |
$begingroup$
Take a $kappa$-Kurepa tree to be a tree with more than $kappa$ branches, of height $kappa$, each level having cardinality less than $kappa$
A slim $kappa$-Kurepa tree is the same but the cardinality of each level $alpha<kappa$ has cardinality $leq|alpha|$.
Is it true in general the existence of a $kappa$-Kurepa tree implies a slim one exists?
set-theory
asked Mar 20 at 19:26
A.HA.H
184
$endgroup$
1
$begingroup$
What are your thoughts on the problem? What have you tried?
$endgroup$
– clathratus
Mar 20 at 20:02
2
$begingroup$
@clathratus: The question is one that is sufficiently advanced that your comment, although well-meaning, feels to me a bit out of touch.
$endgroup$
– Asaf Karagila♦
Mar 20 at 23:35
add a comment |
$begingroup$
Take a $kappa$-Kurepa tree to be a tree with more than $kappa$ branches, of height $kappa$, each level having cardinality less than $kappa$
A slim $kappa$-Kurepa tree is the same but the cardinality of each level $alpha<kappa$ has cardinality $leq|alpha|$.
Is it true in general the existence of a $kappa$-Kurepa tree implies a slim one exists?
set-theory
asked Mar 20 at 19:26
A.HA.H
184
$endgroup$
Take a $kappa$-Kurepa tree to be a tree with more than $kappa$ branches, of height $kappa$, each level having cardinality less than $kappa$
A slim $kappa$-Kurepa tree is the same but the cardinality of each level $alpha<kappa$ has cardinality $leq|alpha|$.
Is it true in general the existence of a $kappa$-Kurepa tree implies a slim one exists?
set-theory
set-theory
asked Mar 20 at 19:26
A.HA.H
184
asked Mar 20 at 19:26
A.HA.H
184
asked Mar 20 at 19:26
A.HA.H
184
asked Mar 20 at 19:26
A.HA.H
184
asked Mar 20 at 19:26
A.HA.H
184
184
1
$begingroup$
What are your thoughts on the problem? What have you tried?
$endgroup$
– clathratus
Mar 20 at 20:02
2
$begingroup$
@clathratus: The question is one that is sufficiently advanced that your comment, although well-meaning, feels to me a bit out of touch.
$endgroup$
– Asaf Karagila♦
Mar 20 at 23:35
add a comment |
1
$begingroup$
What are your thoughts on the problem? What have you tried?
$endgroup$
– clathratus
Mar 20 at 20:02
2
$begingroup$
@clathratus: The question is one that is sufficiently advanced that your comment, although well-meaning, feels to me a bit out of touch.
$endgroup$
– Asaf Karagila♦
Mar 20 at 23:35
1
1
$begingroup$
What are your thoughts on the problem? What have you tried?
$endgroup$
– clathratus
Mar 20 at 20:02
$begingroup$
What are your thoughts on the problem? What have you tried?
$endgroup$
– clathratus
Mar 20 at 20:02
2
2
$begingroup$
@clathratus: The question is one that is sufficiently advanced that your comment, although well-meaning, feels to me a bit out of touch.
$endgroup$
– Asaf Karagila♦
Mar 20 at 23:35
$begingroup$
@clathratus: The question is one that is sufficiently advanced that your comment, although well-meaning, feels to me a bit out of touch.
$endgroup$
– Asaf Karagila♦
Mar 20 at 23:35
add a comment |
1 Answer
1
active
oldest
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$begingroup$
Well, if $kappa$ is a successor, then any $kappa$-Kurepa tree is slim (or, at least, is slim above some level).
But the implication is not true in general. For example, if $kappa$ is a strong limit, then the full binary tree of height $kappa$ is a $kappa$-Kurepa tree. But if $kappa$ additionally has some compactness properties (for example, if it is measurable) then there cannot be any slim $kappa$-Kurepa trees.
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
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$begingroup$
Well, if $kappa$ is a successor, then any $kappa$-Kurepa tree is slim (or, at least, is slim above some level).
But the implication is not true in general. For example, if $kappa$ is a strong limit, then the full binary tree of height $kappa$ is a $kappa$-Kurepa tree. But if $kappa$ additionally has some compactness properties (for example, if it is measurable) then there cannot be any slim $kappa$-Kurepa trees.
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
$endgroup$
add a comment |
$begingroup$
Well, if $kappa$ is a successor, then any $kappa$-Kurepa tree is slim (or, at least, is slim above some level).
But the implication is not true in general. For example, if $kappa$ is a strong limit, then the full binary tree of height $kappa$ is a $kappa$-Kurepa tree. But if $kappa$ additionally has some compactness properties (for example, if it is measurable) then there cannot be any slim $kappa$-Kurepa trees.
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
$endgroup$
add a comment |
$begingroup$
Well, if $kappa$ is a successor, then any $kappa$-Kurepa tree is slim (or, at least, is slim above some level).
But the implication is not true in general. For example, if $kappa$ is a strong limit, then the full binary tree of height $kappa$ is a $kappa$-Kurepa tree. But if $kappa$ additionally has some compactness properties (for example, if it is measurable) then there cannot be any slim $kappa$-Kurepa trees.
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
$endgroup$
Well, if $kappa$ is a successor, then any $kappa$-Kurepa tree is slim (or, at least, is slim above some level).
But the implication is not true in general. For example, if $kappa$ is a strong limit, then the full binary tree of height $kappa$ is a $kappa$-Kurepa tree. But if $kappa$ additionally has some compactness properties (for example, if it is measurable) then there cannot be any slim $kappa$-Kurepa trees.
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
answered Mar 21 at 16:12
Miha HabičMiha Habič
5,66511744
5,66511744
add a comment |
add a comment |
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$begingroup$
What are your thoughts on the problem? What have you tried?
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– clathratus
Mar 20 at 20:02
2
$begingroup$
@clathratus: The question is one that is sufficiently advanced that your comment, although well-meaning, feels to me a bit out of touch.
$endgroup$
– Asaf Karagila♦
Mar 20 at 23:35