Is there a reason why it’d be different in second-order logic?
Second-order set theory is beyond my expertise too, but I’m going by this paper, which on page 8 says:
We have managed to give a formal semantics for the second-order language of
set theory without expanding our ontology to include classes that are not sets. The
obvious alternative is to invoke the existence of proper classes. One can then tinker
with the definition of a standard model so as to allow for a model with the (proper)
class of all sets as its domain and the class of all ordered-pairs x, y (for x an element
of y) as its interpretation function.12 The existence of such a model is in fact all
it takes to render the truth of a sentence of the language of set theory an immediate
consequence of its validity.
So I was taking the “obvious alternative” of proper class of all sets to be the standard model for second order set theory. I don’t quite understand the paper’s own proposed model, but I don’t think it’s a set either.
I’m not sure I believe in proper classes and in particular, I’m not sure there’s a proper class of all sets that could be the model of a second-order theory such that you could not describe any set larger than the model, and as for pinning down that model using axioms I’m pretty sure you shouldn’t be able to do that. There are analogues of the Lowenheim-Skolem theorem for sufficiently large infinities in second-order logic, I seem to recall reading.
Second-order set theory is beyond my expertise too, but I’m going by this paper, which on page 8 says:
So I was taking the “obvious alternative” of proper class of all sets to be the standard model for second order set theory. I don’t quite understand the paper’s own proposed model, but I don’t think it’s a set either.
I’m not sure I believe in proper classes and in particular, I’m not sure there’s a proper class of all sets that could be the model of a second-order theory such that you could not describe any set larger than the model, and as for pinning down that model using axioms I’m pretty sure you shouldn’t be able to do that. There are analogues of the Lowenheim-Skolem theorem for sufficiently large infinities in second-order logic, I seem to recall reading.