We will formulate the arithmetic of complex numbers in a system of finite types, which is a conservative extension of Peano Arithmetic. The rational complex numbers are objects of type 0.
Definition 1.1.The system T.
Symbols: N, r, 0, 1,i, +, ., -, ÷, =, <.
The logical system: The system of finite types defined in Chapter1.
Axioms:(1)The equality axiom Eq.
(2)Axioms of Peano Arithmetic relativized to N.
(3)MI, the mathematical induction relativized to N.
(4)Axioms on0, 1, +, ·, -, ÷,and<relativized to r.
(5)\forall x\exists !y\exists !z(r(y)\wedge r(z)\wedge x=y+iz)where x, y, z...