Question on logic with Mostowski cardinality quantifiers

[Guillermo Badia]

Dear all,

Is it still unknown whether it is consistent with ZFC that there is any alpha > 0 such that the abstract logic L(Q_alpha) is not recursively enumerable for validity? This claim appears in chapter V of Barwise & Feferman’s edited collection „Model theoretic logics“ (end of page 199). Many thanks in advance for your help!

Cheers,
Guillermo 

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