Tree property

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An infinite cardinal $\kappa$ has the tree property if every tree of height $\kappa$ whose levels has cardinality smaller than $\kappa$ has a branch of height $\kappa$ (a cofinal branch). Equivalently, there is no $\kappa$-Aronszajn tree, when a tree is $\kappa$-Aronszajn when it has height $\kappa$, levels with cardinality less than $\kappa$, yet has no cofinal branch.

Definition

A tree is a partially order set (poset) $(T,<)$ such that for all $x\in T$, the order $<$ is a well-order on the set $\{y|y<x\}$ (called a chain). The order type (length) of $<$ on that set is called the height of $x$. The height of $T$ is the supremum of the heights of all the sets $x\in T$. A $\alpha$th level of $T$ is a set that contains all $x\in T$ of height $\alpha$. A branch is a set $B$ well-ordered by $<$ such that any element of $T$ not in $B$ is incomparable with at least one element of $B$.

A tree is $\kappa$-Aronszajn if it has height $\kappa$, all its levels have cardinality smaller than $\kappa$, and every branch of $T$ has order type smaller than $\kappa$. An infinite cardinal $\kappa$ has the tree property if there is no $\kappa$-Aronszajn tree.

Properties

Konig's lemma states that $\aleph_0$ has the tree property. It is however provable that $\aleph_1$ does not have the tree property. Cummings and Foreman proved that, under suitable large cardinal assumptions (namely, the existence of many supercompacts), it is consistent with $ZFC$ all $\aleph_n$ cardinals have the tree property for $1<n<\omega$.

No cardinal can both be a successor cardinal in $L$ and have the tree property in $L$ (the constructible universe), thus the axiom of constructibility is incompatible with the existence of any successor cardinal with the tree property.

Weakly compact cardinals all have the tree property. Every cardinal that is inaccessible and has the tree property is weakly compact. Moreover, every uncountable cardinal with the tree property is weakly compact in the constructible universe, even if it is not inaccessible (in the universe of sets).

Foreman, Magidor and Schindler showed that if there exists infinitely many cardinals $\delta$ above the continuum such that both $\delta$ and $\delta^{+}$ has the tree property, then the axiom of projective determinacy holds. This hypothesis was shown to be consistent relative to the existence of infinitely many supercompact cardinals by Cummings and Foreman.

Magidor and Shelah showed, from the existence of a huge cardinals with infinitely many supercompact cardinals above it, the consistency of $\aleph_{\omega+1}$ having the tree property, and furthermore that the successor of a singular limit of strongly compact cardinals has the tree property.

Definable tree property

Strengthenings of the tree property

Special Aronszajn trees

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