Theories of real addition with and without a predicate for integers

Abstract

We show that it is decidable whether or not a relation on the reals definable in the structure $⟨\mathbb{R},+,<,\mathbb{Z}⟩$ can be defined in the structure $⟨\mathbb{R},+,<,1⟩$. This result is achieved by obtaining a topological characterization of $⟨\mathbb{R},+,<,1⟩$-definable relations in the family of $⟨\mathbb{R},+,<,\mathbb{Z}⟩$-definable relations and then by following Muchnik's approach of showing that the characterization of the relation $X$ can be expressed in the logic of $⟨\mathbb{R},+,<,1,X⟩$. The above characterization allows us to prove that there is no intermediate structure between $⟨\mathbb{R},+,<,\mathbb{Z}⟩$ and $⟨\mathbb{R},+,<,1⟩$. We also show that a $⟨\mathbb{R},+,<,\mathbb{Z}⟩$-definable relation is $⟨\mathbb{R},+,<,1⟩$-definable if and only if its intersection with every $⟨\mathbb{R},+,<,1⟩$-definable line is $⟨\mathbb{R},+,<,1⟩$-definable. This gives a noneffective but simple characterization of $⟨\mathbb{R},+,<,1⟩$-definable relations.

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