A Stronger System of Object Theory as a Prototype of Set Theory

We have introduced in our former work [1] a theory of mathematical objects which can be regarded as a prototype of set theory. We have been successful to imbed the Zermelo set-theory [3] without the axiom of choice in our system. However, it seems impossible to imbed the Fraenkel set-theory [4] in our system even without the axiom of choice. In this work, we introduce another system of object theory in which we can imbed the Fraenkel set-theory without the axiom of choice. We shall denote our former system by OZ (object theory in the manner of the Zermelo set-theory) and the new system we are going to introduce in this work by OF (object theory in the manner of the Fraenkel set-theory). We shall also denote the Zermelo set-theory without the axiom of choice by SZ and the Fraenkel set-theory without the axiom of choice by SF.