Editing Tree (set theory) (section)

==Set-theoretic properties==
There are some fairly simply stated yet hard problems in infinite tree theory. Examples of this are the [[Kurepa conjecture]] and the [[Suslin conjecture]]. Both of these problems are known to be independent of [[Zermelo–Fraenkel set theory]]. By [[Kőnig's lemma]], every <math>\omega</math>-tree has an infinite branch. On the other hand, it is a theorem of [[ZFC]] that there are uncountable trees with no uncountable branches and no uncountable levels; such trees are known as [[Aronszajn tree]]s. Given a [[cardinal number]] <math>\kappa</math>, a ''<math>\kappa</math>-[[Suslin tree]]'' is a tree of height <math>\kappa</math> which has no chains or antichains of size <math>\kappa</math>. In particular, if <math>\kappa</math> is a [[singular cardinal]] then there exists a <math>\kappa</math>-Aronszajn tree and a <math>\kappa</math>-Suslin tree. In fact, for any infinite cardinal <math>\kappa</math>, every <math>\kappa</math>-Suslin tree is a <math>\kappa</math>-Aronszajn tree (the converse does not hold). One of the equivalent ways to define a [[weakly compact cardinal]] is that it is an [[inaccessible cardinal]] <math>\kappa</math> that has the [[tree property]], meaning that there is no <math>\kappa</math>-Aronszajn tree.<ref name=monk/>

The Suslin conjecture was originally stated as a question about certain [[totally ordered set|total orderings]] but it is equivalent to the statement: Every tree of whose height is the [[first uncountable ordinal]] <math>\omega_1</math> has an [[antichain]] of cardinality <math>\omega_1</math> or a branch of length <math>\omega_1</math>.

If <math>(T,<)</math> is a tree, then the [[reflexive closure]] <math>\le</math> of <math><</math> is a [[prefix order]] on <math>T</math>.
The converse does not hold: for example, the usual order <math>\le</math> on the set <math>\mathbb{Z}</math> of [[integer]]s is a [[total order|total]] and hence a prefix order, but <math>(\mathbb{Z},<)</math> is not a set-theoretic tree since e.g. the set <math>\{n\in\mathbb{Z}:n <0\}</math> has no least element.