Back-and-forth method

In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove thatany two countably infinite densely ordered sets without endpoints are isomorphic. An isomorphism between linear orders is simply a strictly increasing bijection. This result implies, for example, that there exists a strictly increasing bijection between the set of all rational numbers and the set of all real algebraic numbers. any two countably infinite atomless Boolean algebras are isomorphic to each other. any two equivalent countable atomic models of a theory are isomorphic. the Erdős–Rényi model of random graphs, when applied to countably infinite graphs, almost surely produces a unique graph, the Rado graph. any two many-complete recursively enumerable sets are recursively isomorphic.

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