The standard model that most set theorists have in mind is something like the Von Neumann Universe, V. Note that this is a proper class, so it's not a structure as usually considered in model theory.
We (hopefully) can't prove V is a model of ZF in ZF, because that would amount to proving consistency and fall foul of Gödel 2, but the axioms of ZF come from an attempt to axiomatise our understanding of set theory in the sense of it being the study of the objects that make up the Von Neumann Universe.
> Obviously, an inconsistent theory cannot have a standard model
Indeed. Paraconsistent logics are an attempt to deal with inconsistency from a proof-theoretic stance, but I'm far from an expert and I don't know what models of paraconistent theories look like.
The standard model that most set theorists have in mind is something like the Von Neumann Universe, V. Note that this is a proper class, so it's not a structure as usually considered in model theory.
We (hopefully) can't prove V is a model of ZF in ZF, because that would amount to proving consistency and fall foul of Gödel 2, but the axioms of ZF come from an attempt to axiomatise our understanding of set theory in the sense of it being the study of the objects that make up the Von Neumann Universe.
> Obviously, an inconsistent theory cannot have a standard model
Indeed. Paraconsistent logics are an attempt to deal with inconsistency from a proof-theoretic stance, but I'm far from an expert and I don't know what models of paraconistent theories look like.