  The theory of addition in the domains of natural (N), integer (Z), rational
(Q), real (R) and complex (C) numbers is decidable, so is the theory of
multiplication in all those domains. By Godel's Incompleteness Theorem the
theory of addition and multiplication is undecidable in the domains of N, Z and
Q, though Tarski proved that this theory is decidable in the domains of R and
C. The theory of multiplication and order (x,<) behaves differently in the
above mentioned domains of numbers. By a theorem of Robinson, addition is
definable by multiplication and order in the domain of natural numbers, thus
the theory (N;x,<) is undecidable. By a classical theorem in mathematical
logic, addition is not definable in terms of multiplication and order in R. In
this paper, we extend Robinson's theorem to the domain of integers (Z) by
showing the definability of addition in (Z;x,<), this implies that (Z;x,<) is
undecidable. We also show the decidability of (Q;x,<) by the method of
quantifier elimination. Whence, addition is not definable in (Q;x,<).

Logic;Logic in Computer Science