Anti-basis theorems for higher Aronszajn lines, part 1

Aronszajn (1935) proved that there exists a special w1-Aronszajn tree and Jensen (1972) proved that, more generally, the existence of a special (mu^+)-Aronszajn tree is equivalent to the weak square principle holding at mu. We shall present a far-reaching extension of Jensen's theorem. As an application, we get that if there exists a special w2-Aronszajn tree, then any basis for the class of w2-Aronszajn lines must be of the maximal possible size 2^{w2}. This is in contrast with Justin Moore's theorem (2006) that the Proper Forcing Axiom implies that the class of w1-Aronszajn lines admits a basis of size 2.

The proof combines walks on ordinals, club guessing, strong colorings, vanishing levels of trees and a bit of finite combinatorics. The results are not limited to w2 (or to successor cardinals).

This is joint work with Tanmay Inamdar.