I hope to return soon (perhaps after I finish my taxes) to address some of the interesting questions raised by Vladimir. In some cases I will expand on the answers I provided Vladimir in response to his recent private letter to me, responses which turned out to be incorporated into the motivation and formulation of some of his questions. At that time, I will also explain why the omnific integers referred to by Joel is not a full model of PA and make a few points about them as well.

For the time being, I merely wish to correct misconceptions about Hausdorff’s great writings on $\eta_{\alpha}$-orderings that one might be apt to walk away with after reading Vladimir’s remarks. Since I treat these matters with some care in Section 8 of my paper, 

*The Absolute Arithmetic Continuum and the Unification of All Numbers Great and Small*, **The Bulletin of Symbolic Logic** 18 (1) 2012, pp. 1-45

I refer interested parties to that paper for the requisite definitions and details. Even greater detail will be found in a forthcoming work of mine entitled *From du Bois-Reymond’s Infinitary Patachie to the Surreal Numbers*.
 
 To begin with, in Hausdorff's first great paper on ordered sets of 1906, he introduces the idea of an $\eta_{\alpha}$-ordering in precisely the way we use it today and he continued to use it in the same fashion in all of his subsequent writings. The definition is given on page 132 of the original paper and on page 150 of the wonderful recent English translation by Plotkin. As I explain my aforementioned BSL paper, Hausdorff was motivated to introduce the idea of an $\eta_{1}$-ordering to characterize the order type of his very insightful reconfiguration of Paul du Bois-Reymond's flawed conception of an infinitary pantichie. In fact, in he proves:


HAUSDORFF 1 [1907]: Infinitary pantachies exist. If P is an infinitary pantachie, then P is an $\eta_{1}$-ordering of power $2^{\aleph_{0}}$; in fact, P is (up to isomorphism) the unique $\eta_{1}$-ordering of power $\aleph_{1}$, assuming (the Continuum Hypothesis) CH.

In his investigation of 1907, Hausdorff also raises the question of the existence of a pantachie that is algebraically a field, but he only makes partial headway in providing an answer. However, in 1909 he returned to the problem and provided a stunning positive answer. Indeed, beginning with the ordered set of numerical sequences of the form *r, r , r, …, r, …* where   is a rational number, and utilizing what appears to be the very first algebraic application of his maximal principle, Hausdorff proves the following little-known, remarkable result.

HAUSDORFF 2 [1909]. There is a pantachie H of numerical sequences of real numbers indexed over the natural numbers (with operations suitably defined) that is an ordered field. Any such pantachie is, in fact, a real-closed ordered field.

Writing before Artin and Schrier [1926], Hausdorff of course does not refer to H as real closed; but he essentially establishes H is real-closed by showing it is the union of a chain of ordered fields, each of which admits no algebraic extension to a more inclusive ordered field.

Thus, contrary to what Vladimir contends, Hausdorff does not formulate his theory of  $\eta_{\alpha}$-orderings in terms of pantachies, but rather in the manner we know and love; moreover, he uses the special case of an $\eta_{1}$--ordering to characterize the order type of his pantachies.