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 that

Application to densely ordered sets

As an example, the back-and-forth method can be used to prove Cantor's isomorphism theorem, although this was not Georg Cantor's original proof. This theorem states that two unbounded countable dense linear orders are isomorphic. Suppose that Fix enumerations (without repetition) of the underlying sets: Now we construct a one-to-one correspondence between A and B that is strictly increasing. Initially no member of A is paired with any member of B. It still has to be checked that the choice required in step (1) and (2) can actually be made in accordance to the requirements. Using step (1) as an example: If there are already ap and aq in A corresponding to bp and bq in B respectively such that ap < ai < aq and bp < bq, we choose bj in between bp and bq using density. Otherwise, we choose a suitable large or small element of B using the fact that B has neither a maximum nor a minimum. Choices made in step (2) are dually possible. Finally, the construction ends after countably many steps because A and B are countably infinite. Note that we had to use all the prerequisites.

History

According to Hodges (1993): While the theorem on countable densely ordered sets is due to Cantor (1895), the back-and-forth method with which it is now proved was developed by Edward Vermilye Huntington (1904) and Felix Hausdorff (1914). Later it was applied in other situations, most notably by Roland Fraïssé in model theory.