Back-and-forth method

Definition

We establish a language \mathcal{L} and we consider two \mathcal{L}-structures \mathcal{M} and \mathcal{N} of domains respectively M and N.

We call a partial isomorphism between \mathcal{M} and \mathcal{N} any isomorphism between two \mathcal{L}-substructures of \mathcal{M} and \mathcal{N}.

A non-empty family \mathcal{I} of partial isomorphisms between \mathcal{M} and \mathcal{N} is called a back-and-forth if both of the following properties hold:

• (FORTH) \forall \sigma \in \mathcal{I};;\forall c \in M;;\exists \sigma' \in \mathcal{I};\bigl(\sigma \subseteq \sigma' ;\land; c \in \mathrm{dom}(\sigma')\bigr)

• (BACK) \forall \sigma \in \mathcal{I};;\forall d \in N;;\exists \sigma' \in \mathcal{I};\bigl(\sigma \subseteq \sigma' ;\land; d \in \mathrm{im}(\sigma')\bigr)

In other words, each partial isomorphism of the family admits an extension which still belongs to the family itself. Moreover, one can find such an extension more precisely for each partial isomorphism, by imposing which new element must belong to the domain of the extension, or to its image (codomain).

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.{{citation

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 i q and b**p 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.

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

References