A remark on Martin's conjecture (Q2732288)

It is shown that, for the class of infinite atomic Boolean algebras, the classification of their \((\omega+ \omega)\)-elementary theories can be reduced to the classification of the elementary theories of their quotient algebras modulo the Frechet ideals. Then, by Ershov-Tarski's analysis, the following result is obtained:NEWLINENEWLINENEWLINEThere is a complete consistent theory \(T\) in \(L_{\omega, \omega}\) such that \(T\) has \(2^{\aleph_0}\) countable models, and there are at most countably many models of \(T\) with distinct complete theories in \(L_{\omega+ \omega,\omega}\).NEWLINENEWLINENEWLINEIn particular, the strong Martin conjecture is false.