Aronszajn tree

A set-theoretic tree of height ${\omega }_1$ with countable levels and no cofinal branch. More generally, a $\kappa$-Aronszajn tree has height a regular uncountable $\kappa$, levels of size less than $\kappa$, and no cofinal branch.

Given coherent coinfinite injections into omega, use all restrictions $e_{\alpha }\upharpoonright \beta$, where $\beta \le \alpha <{\omega }_1$, ordered by proper extension. Every level is countable because its nodes differ finitely from a fixed function on a countable domain. An uncountable chain in a partial order would have unbounded domain heights, and its union would inject ${\omega }_1$ into $\omega$, which is impossible.

An Aronszajn tree with no uncountable tree antichain. A normal splitting Suslin tree yields a Suslin line by a lexicographic ordering followed by Dedekind completion.

A well-pruned set-theoretic tree that is an Aronszajn tree and a Suslin tree gives a forcing with the countable chain condition for forcing: stronger nodes extend weaker ones. The ${\omega }_1$ dense subsets of a forcing order of nodes at or above each level cannot all be met by a filter in an ordered set, since that would produce a cofinal branch. Thus ${\mathrm{MA}}_{{\aleph }_1}$ fails. When the continuum exceeds ${\aleph }_1$, full Martin axiom includes this instance.

An Aronszajn tree that is a union of countably many tree antichains. Equivalently, it admits a map into a countable set that is injective on each chain.

Construct countable levels of bounded increasing rational sequences, maintaining the bounded rational extension property. At successor stages append every allowed rational. At a countable limit, handle each earlier-node/rational-cap pair by a cofinal sequence of extensions with values approaching a rational strictly below the cap. Its union has that rational supremum. Countably many requirements give a countable new level. Strictly increasing labels exclude uncountable chains, and label fibers are tree antichains, so the result is a special Aronszajn tree.

For every earlier node and rational cap above its label, an extension at each later level remains below the cap. In the rationally labelled Aronszajn tree construction this reserves space for limit-stage unions. Choosing an intermediate rational and forcing successive values toward it ensures a rational supremum and strictly increasing labels even at limit levels.

A set-theoretic tree of height ${\omega }_2$, levels of size at most ${\aleph }_1$, and no cofinal branch. The Continuum hypothesis supplies one through the minimal-walk tree.