Is there a property that isolates the formulas that remain valid as you move from the natural numbers to the integers?

Question

Is there a property that isolates the formulas that remain valid (or whose translations into the expanded language remain valid) as you move from the theory of the natural numbers to the theory of the integers and on to the reals?

Answer 1

The theories of natural numbers and integers (say, with addition and multiplication) are interpretable from each other: you can define the natural numbers as a subset of the integers with a first order formula; and you can define integers as equivalence classes of pairs of natural numbers. This implies that deciding the truth of a formula in natural numbers can be reduced to some other formula in the integers, and vice versa.

However, the first order theory of $(\mathbb Z,+,\times)$ is horribly difficult: there is no algorithm to decide whether a given sentence is true or not. The theory of real numbers is decidable, since $\mathbb R$ is a real-closed field. So the two theories you're asking about are very different.