Theories of real addition with and without a predicate for integers
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We show that it is decidable whether or not a relation on the reals definable in the structure 〈R, +,<, Z〉 can be defined in the structure 〈R, +,<, 1 〉. This result is achieved by obtaining a topological characterization of 〈R, +,<, 1 〉-definable relations in the family of 〈R, +,<, 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 〈R, +,<,1, X 〉. The above characterization allows us to prove that there is no intermediate structure between 〈R, +,<, Z〉 and 〈R, +,<, 1 〉. We also show that a 〈R, +,<, Z〉-definable relation is 〈R, +,<, 1 〉-definable if and only if its intersection with every 〈R, +,<, 1 〉-definable line is 〈R, +,<, 1 〉-definable. This gives a noneffective but simple characterization of 〈R, +,<, 1 〉-definable relations.
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