11. Thus the example Z above is an example of a GO-space that is not a linearly ordered topological space. 12. Every cyclically ordered group can also be expressed as a subgroup of a product, where is a linearly ordered group. 13. This result is analogous to Otto H�lder's 1901 theorem that every Archimedean linearly ordered group is a subgroup of. 14. The elements of any model of Peano arithmetic are linearly ordered and possess an initial segment isomorphic to the standard natural numbers. 15. In Easton's model, " V " cannot be linearly ordered , so it cannot be well-ordered. 16. There are in fact many ways to construct such a linearly ordered set of numbers, but fundamentally, there are two different approaches: 17. A convex set is linearly ordered by the cut for any not in the set; this ordering is independent of the choice of. 18. For example, given two linearly ordered sets and, one may form a circle by joining them together at positive and negative infinity. 19. Every cyclically ordered group can be expressed as a quotient, where is a linearly ordered group and is a cyclic cofinal subgroup of. 20. Furthermore, Chang showed that every linearly ordered MV-algebra is isomorphic to an MV-algebra constructed from a group in this way.
