ABSTRACT
Two equivalent sets, each ordered in a certain way, are similar or have equal ordertypes if they can be paired off so that each set preserves its own order; the apostles in any
given order can be paired off with the months in any given order so that at each stage the next apostle goes with the next month. Finite equivalent ordered sets are always similar, but infinite ones (sometimes also called transfinite in mathematics) need not be; the set of positive integers in ascending order is not similar to the same set in the reverse or descending order (…, 2, 1, 0), because the latter has no first term to go with the first term (0) of the former. Ordered sets of which every non-empty subset has a first member are called well-ordered, and the order-type of a well-ordered set is an ordinal number or ordinal. The ordinal of one well-ordered set is greater than that of another if the second set as a whole is similar to an initial segment (only) of the first.
