PlanetMath: Aronszajn tree

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Aronszajn tree

(Definition)

A $\kappa$ -tree for which $\vert T_\alpha\vert<\kappa$ for all $\alpha<\kappa$ and which has no cofinal branches is called a $\kappa$ -Aronszajn tree. If $\kappa=\omega_1$ then it is referred to simply as an Aronszajn tree.

If there are no $\kappa$ -Aronszajn trees for some $\kappa$ then we say $\kappa$ has the tree property. $\omega$ has the tree property, but no singular cardinal has the tree property.

"Aronszajn tree" is owned by Henry.

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Also defines:

Aronszajn tree, $\kappa$ -Aronszajn tree, tree property

Attachments:: proof that $\omega$ has the tree property (Proof) by Henry
example of Aronszajn tree (Example) by Henry

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Cross-references: singular cardinal, cofinal branches
There are 4 references to this entry.

This is version 7 of Aronszajn tree, born on 2002-07-27, modified 2006-06-24.
Object id is 3217, canonical name is Aronszajn.
Accessed 6163 times total.

Classification:

AMS MSC:	05C05 (Combinatorics :: Graph theory :: Trees)
	03E05 (Mathematical logic and foundations :: Set theory :: Other combinatorial set theory)

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