Full Idea
We can give a semi-categorical axiomatisation of set-theory (all that remains undetermined is the size of the set of urelements and the length of the sequence of ordinals). The system is second-order in formalisation.
Gist of Idea
There is a semi-categorical axiomatisation of set-theory
Source
John Mayberry (What Required for Foundation for Maths? [1994], p.413-2)
Book Reference
'Philosophy of Mathematics: anthology', ed/tr. Jacquette,Dale [Blackwell 2002], p.413
A Reaction
I gather this means the models may not be isomorphic to one another (because they differ in size), but can be shown to isomorphic to some third ingredient. I think. Mayberry says this shows there is no such thing as non-Cantorian set theory.
Related Idea
Idea 17795 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]