A virtual five element basis for the uncountable linear orders
John Krueger, Justin Tatch Moore
TL;DR
This work proves that for any Aronszajn line $A$ and Countryman line $C$, there exists a proper forcing extension in which $A$ embeds a copy of $C$ or $C^*$, yielding powerful corollaries: under an inaccessible cardinal, uncountable linear orders have a five-element basis; BPFA similarly implies such a basis and is equiconsistent with BPFA plus Aronszajn tree saturation. Central to the results is a new preservation lemma for $oldsymbol{}$-tree subtrees under countable-support iterations, enabling controlled forcing constructions that force $oldsymbol{ atvarphi}$ (and $oldsymbol{ atpsi}$) and, in the predense case, $oldsymbol{ atvarphi}$ implies $oldsymbol{ atpsi}$. The paper develops the families $oldsymbol{ R}_n$ and $oldsymbol{ R}_n^ot$, stability frameworks for $T$ and $K$, and CAT-type forcings, culminating in equiconsistency results and a pathway from BPFA to Shelah’s conjecture and the five-element basis. These results significantly reduce the large-cardinal strength previously associated with Shelah’s conjecture and connect forcing axioms to precise structural bases for uncountable linear orders.
Abstract
We prove that for every Aronzsajn line A and every Countryman line C, there is a proper forcing extension in which A contains an isomorphic copy of either C or its converse C*. As a corollary, we obtain answers to several related questions asked by the second author in the literature: if there is an inaccessible cardinal, then there is a proper forcing extension in which the uncountable linear orders have a five element basis; BPFA implies the existence of a five element basis for the uncountable linear orders; BPFA is equiconsistent with the conjunction of BPFA and Aronszajn tree saturation. These results are derived from new preservation results concerning subtrees of Aronszajn trees, proper forcings, and countable support iterations, generalizing work of Miyamoto, Abraham, and Shelah.