An Impossible Question
A while ago, I asked: "Could Frege's ontology be a Henkin model?". I now believe that this question doesn't make sense: A standard model of second-order logic is a (standard) Henkin model. I should have asked: "Could Frege's ontology be a non-standard Henkin model?". Even this question is, uh, questionable, because the late Frege would have certainly rejected both a standard and a Henkin semantics, as both of these employ singular terms to denote the semantic values of function expressions. So I should rather have asked: "Are Frege's logical and semantical theses satisfiable in a non-standard Henkin model?" But now, I guess, the answer is trivially Yes, because nothing you can say in higher-order logic rules out a non-standard Henkin interpretation. However, my question was not meant to be trivial. I wanted to know whether Frege is comitted to there being more concepts (values of second-order quantifiers) than objects (values of first-order quantifiers), a claim that is true in standard models, but not in some non-standard models of any (really?)* second-order theory. Unfortunately, this question can't even be asked without violating Frege's semantical theses. As he himself notes in a letter to Russell:
[D]er Beweis, dass es keine eindeutige Beziehung zwischen allen Gegenständen und allen Funktionen gebe, ist mir bedenklich. Ich glaube, dass schon der Gedanke nicht ganz klar ist, dass Gegenstände zu Funktionen in eindeutiger Beziehung stehen. Bei der Eindeutigkeit wird nämlich die Gleichheit (Identität) vorausgesetzt, und diese Beziehung der Gleichheit ist eine solche erster Stufe, in der nur Gegenstände, nicht aber Funktionen zu einander stehen können. (Wissenschaftlicher Briefwechsel, p.225f.)
* [Update: Obviously not for any theory. Consider { 
x(x=a),
Fa, 
Ga}. But by the Skolem-Löwenheim theorem, what I said should hold
for every theory that is satisfiable in an infinite domain (of
individuals).]