User blog comment:LittlePeng9/FOOT is not as strong as I thought/@comment-5529393-20161225200516/@comment-1605058-20161226130623

The following arguments will probably not convince you, but with this we are precisely reaching the philosophical kind of argument I have mentioned in the post. Also let me mention that there most likely are people out there who would be jus tas hesitant to agree with your view point as you are to change it.

First a general remark - in Henkin semantics, if I understand them correctly, we don't necessarily quantify over the definable predicates. Rather, the set of predicates to be quantified over is a part of a (second-order) model, just like the objects of the underlying universe. Taking just the definable sets gives us sort of a minimal second-order model.

With that in mind, from a platonistic point of view, differentiating between Henkin and standard semantics is sort of meaningless, and here is why. A platonist believes in the "true" universe of sets \(V\) which is "the" model of first order set theory, and, by extension, believes it's a part of the "true" model of second order set theory. Now this model also ought to contain all the predicates (all classes, whatever) to be quantified. But at this point I don't see a reason a platonist would not believe that this "true" model contains all possible predicates over \(V\). So quantifying "Henkin-semantically" over predicates of this model would be the same as quantifying "standard-semantically" over all predicates.

Note that the above is a perspective of a somewhat "perfect platonist". I can imagine that someone might believe in platonistic \(V\) and yet be hesitant to believe in "true" second order model of set theory. Your last comment kinda hints towards such a perspective - you take a model of ZFC (which can be thought of as "the" model, but I see you don't call it so). I then admit that the process of taking just the definable classes is more "down to earth". But I also have a feeling that if someone doesn't accept the existence of the "true power set" of \(V\) might also don't accept the existence of the "definable power set" \(\text{Def}(V)\) as a single entity to work with.