The Monadic Theory of w2

In a series of papers, Büchi proved the decidability of the monadic (second-order) theory of ω₀, of all countable ordinals, of ω₁, and finally of all ordinals < ω₂. Here, assuming the consistency of a weakly compact cardinal, we prove that, in different set-theoretic worlds, the monadic theory of ω₂ may be arbitrarily difficult (or easy).