Georg Cantor proved that every two densely totally ordered countable sets without lower or upper bounds are order-isomorphic. In particular, there exists an order-isomorphism between the rational numbers and other densely ordered countable sets including the dyadic rationals and the algebraic numbers. The proof of this result uses the back-and-forth method.
Minkowski's question mark function can be used to determine the order isomorphisms between the quadratic algebraic numbers and the rational numbers, and between the rationals and the dyadic rationals.
Any binary relation R is said to be dense if, for all R-related x and y, there is a z such that x and z and also z and y are R-related. Formally:
- Alternatively, in terms of composition of R with itself, the dense condition may be expressed as R ⊆ R R.
Sufficient conditions for a binary relation R in a set X to be dense are:
- R is reflexive;
- R is coreflexive;
- R is quasireflexive;
- R is left or right Euclidean; or
- R is symmetric and semi-connex and X has at least 3 elements.
- Roitman, Judith (1990), "Theorem 27, p. 123", Introduction to Modern Set Theory, Pure and Applied Mathematics, 8, John Wiley & Sons, ISBN 9780471635192.
- Dasgupta, Abhijit (2013), Set Theory: With an Introduction to Real Point Sets, Springer-Verlag, p. 161, ISBN 9781461488545.
- Gunter Schmidt (2011) Relational Mathematics, page 212, Cambridge University Press ISBN 978-0-521-76268-7