 This is a rather difficult theory. A model of the alternative set theory in the usual set theory is a nonstandard model of \(V_{\omega}\) of size \(\omega_1\) in which every countable external function extends to a function in the model. It might be best to suppose that this model is constructed inside \(L\) (the constructible universe) so that the axiom of cardinalities will be satisfied. The axiom of extensional coding follows from Choice in the ambient set theory.