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.

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  • 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.
  • 数学基礎論、特に集合論とモデル理論において、カントールの往復論法（カントールのおうふくろんぽう、英：Cantor's back-and-forth method）とは、特定の条件を満たす可算無限濃度を有する構造の間に同型写像が存在することを示す論法であり、ゲオルク・カントールから命名された。 特に、以下の証明に使用される。 カントールは、任意の 2 つの可算無限な稠密全順序集合（全順序集合であって、任意の異なる 2 つの元の間に異なる元が存在するもの）に両端が存在しない（最小元・最大元を持たない）場合、両者が順序同型であることを示すために、この論法を用いた。 全順序集合の同型は、狭義単調増加な全単射である。 従って例えば、有理数全体の集合と実代数的数全体の集合の間には、狭義単調増加な全単射が存在する。 原子元を有しない可算無限濃度のブール代数が互いに同型であることを証明するために、この論法を使用できる。
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  • 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.
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  • Back-and-forth method
  • カントールの往復論法
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