Henkin's completeness proof: Forty years later (Q1183715)

The paper --- contrary to what the title might suggest to some --- is not a historic assessment of the role of Henkin's completeness proof (though there are some historical remarks in section 6). The authors investigate the question: When can a set \(T\) of sentences of a (countable) first- order language \(L\) (with denumerably many individual constants and no operation symbols) be extended to a Henkin set in the original language \(L\)? One of their tools is ``instantial logic'' (essentially omega-logic, a term which the authors prefer to reserve to the case when the constants are (or can be?) interpreted as numerals).