The Monadic Theory of $\omega^1_2$
Yuri Gurevich, Menachem Magidor, and Saharon Shelah
Source: J. Symbolic Logic Volume 48, Issue 2 (1983), 387-398.
Abstract
Assume ZFC + "There is a weakly compact cardinal" is consistent. Then: (i) For every $S \subseteq \omega, \mathrm{ZFC} +$ "$S$ and the monadic theory of $\omega_2$ are recursive each in the other" is consistent; and (ii) ZFC + "The full second-order theory of $\omega_2$ is interpretable in the monadic theory of $\omega_2$" is consistent.
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Mathematical Reviews number (MathSciNet):
MR704093
