No Suslin trees but a non-special Aronszajn tree exists by a side condition method : compact version (Iterated Forcing Theory and Cardinal Invariants)

Abstract

Let us fix a weakly Suslin tree T^{*} that is upward‐absolutely Aronszajn. Let us assume 2^{ $omega$} =$omega$_{1} and 2^{tlrcorner}1 = $omega$_{2} . We construct an Aspero-Mota type iterated forcing langle P_{$alpha$} | $alpha$ < $omega$_{2}rangle and take the direct limit P_{$omega$_{2}^{*} of the P_{$alpha$}mathrm{s}. In the generic extensions V^{P}backslash cdot 2, we have (1) 2^{ $omega$} = $omega$_{2}, (2) every Aronszajn tree gets an uncountable antichain and so no Suslin trees exist, while (3) T^{*} remains weakly Suslin and Aronszajn. In particular, T^{*} has no specializing maps. The idea of a weakly Suslin tree that is upward‐absolutely Aronszajn belongs to a work of S. Shelah. Combinatorics with Aronszajn trees, say, via R_{1, mathrm{N}_{1}} is due to T. Yorioka. An iterated forcing method that uses symmetric systems and markers is due to Aspero-Mota. It appears that the construction in this paper is sensitive to the length $omega$_{2}.