Compactness, interpolation and Friedman's third problem

Let Robinson's consistency theorem hold in logic L: then L will satisfy all the usual interpolation and definability properties, together with coutable compactness, provided L is reasonably small. The latter assumption can be weakened ro removed by using special set-theoretical assumptions. Thus, if Robinson's consistency theorem holds in L, then (i) L is countably compact if its Löwenheim number is < μ0 = the smallest uncountable measurable cardinal; (ii) if ω is the only measurable cardinal, L is countably compact, or the theories of L characterize every structure up to isomorphism. As a corollary, a partial answer is given to H. Friedman's third problem, by proving that no logic L strictly between L∞ω and L∞∞ satisfies interpolation (or Robinson's consistency), unless K-elementary equivalence coincides with isomorphism.