The monadic theory of \((\omega{}_ 2,<)\) may be complicated (Q1190619)

It would be an ill-posed problem to ask for a characterization of those sets \(B\subseteq\omega\) which are recursive is the monadic theory of \((\omega_ 2,\leq)\). In fact, a classical result of \textit{Y. Gurevich, M. Magidor} and \textit{S. Shelah} [J. Symb. Log. 48, 387-398 (1983; Zbl 0549.03010)] states that under the assumption of the existence of a weakly compact cardinal in the universe \(V\) of set theory, for every subset \(B\) of \(\omega\) there is a generic extension \(W\) of \(V\) in which \(B\) is recursive in the monadic theory of \((\omega_ 2,\leq)\). Of course \(W\) depends on \(B\). In the present paper the same conclusion is obtained without the assumption of the existence of large cardinals. More precisely, the following main theorem is proved. There is a set of sentences \(\{\theta_ n; n\in\omega\}\) in the monadic language of order such that: if \(V\) satisfies the GCH, then for each \(B\subseteq\omega\) there exists a forcing notion \(P\) which is \(\omega_ 1\)-closed, satisfies the \(\omega_ 3\)-chain condition, preserves cardinals, cofinalities and the GCH, \(| P|=\aleph_ 3\), such that \(P\) forces \(B=\{n; (\omega_ 2,\leq)\models\theta_ n\}\).