# Back-and-forth method > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Back-and-forth_method > Markdown URL: https://mediated.wiki/source/Back-and-forth_method.md > Source: https://en.wikipedia.org/wiki/Back-and-forth_method > Source revision: 1316656825 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) In [mathematical logic](/source/Mathematical_logic), especially [set theory](/source/Set_theory) and [model theory](/source/Model_theory), the **back-and-forth method** is a method for showing [isomorphism](/source/Isomorphism) between [countably infinite](/source/Countably_infinite) structures satisfying specified conditions. In particular it can be used to prove that: - any two [countably infinite](/source/Countably_infinite) [densely ordered](/source/Dense_order) 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](/source/Linear_order) is simply a strictly increasing [bijection](/source/Bijection). This result implies, for example, that there exists a strictly increasing bijection between the set of all [rational numbers](/source/Rational_number) and the set of all [real](/source/Real_number) [algebraic numbers](/source/Algebraic_number). - any two countably infinite atomless [Boolean algebras](/source/Boolean_algebra_(structure)) are isomorphic to each other. - any two equivalent countable [atomic models](/source/Atomic_model_(mathematical_logic)) of a theory are isomorphic. - the [Erdős–Rényi model](/source/Erd%C5%91s%E2%80%93R%C3%A9nyi_model) of [random graphs](/source/Random_graph), when applied to countably infinite graphs, [almost surely](/source/Almost_surely) produces a unique graph, the [Rado graph](/source/Rado_graph). - any two [many-complete](/source/Many-one_reduction) [recursively enumerable](/source/Recursively_enumerable) sets are [recursively](/source/Computable_function) isomorphic. ## Definition We establish a language \mathcal{L} and we consider two \mathcal{L}-[structures](/source/Structure_(mathematical_logic)) \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](/source/Substructure_(mathematics)) 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](/source/Extension_of_a_function) 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](/source/Cantor's_isomorphism_theorem), although this was not [Georg Cantor](/source/Georg_Cantor)'s original proof. This [theorem](/source/Theorem) states that two unbounded countable [dense linear orders](/source/Dense_order) are isomorphic.[1] 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* = { *a*1, *a*2, *a*3, ... }, - *B* = { *b*1, *b*2, *b*3, ... }. 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. ## History According to Hodges (1993): - *Back-and-forth methods are often ascribed to [Cantor](/source/Georg_Cantor), [Bertrand Russell](/source/Bertrand_Russell) and [C. H. Langford](/source/Cooper_Harold_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](/source/Edward_Vermilye_Huntington) (1904) and [Felix Hausdorff](/source/Felix_Hausdorff) (1914). Later it was applied in other situations, most notably by [Roland Fraïssé](/source/Roland_Fra%C3%AFss%C3%A9) in [model theory](/source/Model_theory). ## See also - [Ehrenfeucht–Fraïssé game](/source/Ehrenfeucht%E2%80%93Fra%C3%AFss%C3%A9_game) ## References 1. Silver, Charles L. (1994), ["Who invented Cantor's back-and-forth argument?"](https://projecteuclid.org/euclid.rml/1204835164), *Modern Logic*. **4** (1): 74–78, MR 1253680 - Hausdorff, F. (1914), "Grundzüge der Mengenlehre" - Hodges, Wilfrid (1993), [*Model theory*](https://archive.org/details/modeltheory0000hodg), [Cambridge University Press](/source/Cambridge_University_Press), ISBN 978-0-521-30442-9 - Huntington, E. V. (1904), "The continuum and other types of serial order, with an introduction to Cantor's transfinite numbers", [Harvard University Press](/source/Harvard_University_Press) - Marker, David (2002), *Model Theory: An Introduction*, [Graduate Texts in Mathematics](/source/Graduate_Texts_in_Mathematics), Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), ISBN 978-0-387-98760-6 --- Adapted from the Wikipedia article [Back-and-forth method](https://en.wikipedia.org/wiki/Back-and-forth_method) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Back-and-forth_method?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.