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 that any two countably infinite densely ordered sets (i.e., linearly ordered in such a way that between any two members there is another) 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.

• It can be used to prove that any two countably infinite atomless Boolean algebras are isomorphic to each other.

• It can be used to prove that any two equivalent countable atomic models of a theory are isomorphic.

• It can be used to prove that the Erdős–Rényi model of random graphs, when applied to countably infinite graphs, always produces a unique graph, the Rado graph.

Application to densely ordered sets

Suppose that

• (A, ≤_A) and (B, ≤_B) are linearly ordered sets;
• They are both unbounded, in other words neither A nor B has either a maximum or a minimum;
• They are densely ordered, i.e. between any two members there is another;
• They are countably infinite.

Fix enumerations (without repetition) of the underlying sets:

A = { a₁, a₂, a₃, … },

B = { b₁, b₂, b₃, … }.

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.

(1) Let i be the smallest index such that a_i is not yet paired with any member of B. Let j be some index such that b_j is not yet paired with any member of A and a_i can be paired with b_j consistently with the requirement that the pairing be strictly increasing. Pair a_i with b_j.

(2) Let j be the smallest index such that b_j is not yet paired with any member of A. Let i be some index such that a_i is not yet paired with any member of B and b_j can be paired with a_i consistently with the requirement that the pairing be strictly increasing. Pair b_j with a_i.

(3) Go back to step (1).

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 a_p and a_q in A corresponding to b_p and b_q in B respectively such that a_p < a_i < a_q and b_p < b_q, we choose b_j in between b_p and b_q 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.

If we iterated only step (1), rather than going back and forth, then in some cases the resulting function from A to B would fail to be surjective. In the easy case of unbounded dense totally ordered sets it is possible to avoid step 2 by choosing the element b_j more carefully (by choosing j as small as possible), but this does not work for more complicated examples such as atomless Boolean algebras where steps 1 and 2 are both needed.

History

According to Hodges (1993):

Back-and-forth methods are often ascribed to Cantor, Bertrand Russell and C. H. Langford […], but there is no evidence to support any of these attributions.

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 Huntington (1904) and Hausdorff (1914). Later it was applied in other situations, most notably by Roland Fraïssé in model theory.

See also: Ehrenfeucht–Fraïssé game.

References

• Huntington, E. V. (1904), The continuum and other types of serial order, with an introduction to Cantor's transfinite numbers, Harvard University Press 
• Hausdorff, F. (1914), Grundzüge der Mengenlehre 
• Hodges, Wilfrid (1993), Model theory, Cambridge University Press, ISBN 978-0-521-30442-9 
• Marker, David (2002), Model Theory: An Introduction, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98760-6