An infinity which depends on the axiom of choice

摘要

In the early years of set theory, Du Bois Reymond introduced a vague notion of infinitary pantachie meant to symbolize an infinity bigger than the infinity of real numbers. Hausdorff reformulated this concept rigorously as a maximal chain (a linearly ordered subset) in a partially ordered set of certain type, for instance, the set NN under eventual domination. Hausdorff proved the existence of a pantachy in any partially ordered set, using the axiom of choice AC. We show in this note that the pantachy existence theorem fails in the absense of AC, and moreover, even if AC is assumed, hence pantachies do exist, one may not be able to come up with an individual, effectively defined example of a pantachy.