**Surreals and NSA: some foundational issues.**

A.
Leaving aside the whole internal machinery of surreals (with funny questions like is $\omega$ an entire number and if yes is it odd or even, simple, a factorial, etc.), it is a principal foundational achievement of surreals that they give **concrete, well-defined examples** of saturated fields, in particular,

1) a concrete, well-defined countably saturated rcof,

2) a concrete, well-defined set-size saturated rcof (No itself), by necessity of class size.

The saturation properties in this context were first explicitly observed probably by Ehrlich in 1980s, but they follow from the $\eta_\alpha$-properties (obvious by construction) and a result (CK, Ex 5.4.4 on p. 369, later reproved by Simpson) which likely was "of common knowledge" in model theory in 1970s if not earlier.

Note that Hausdorff studied his $\eta_\alpha$ fields most successfully as "pantachies", that is, linearly ordered subsets of a certain partial order of R^N, and with a heavy dose of the axiom of choice - which does not yeild any single, well-defined, concrete example. Thus the countably saturated rcof which emerges as a certain initial part of No is probably the closest thing to the notoriously inconsistent "infinitaire pantachie" of DuBoisReymond known so far.

(As a marginal remark, a countably saturated rcof cannot be Borel - even cannot have a Borel set as the domain and Borel relation as the order - so it cannot be *too concretely well-defined*.)

B.
In the context of foundations of infinitesimals, the surreals have a major defect: they are just a rcof, w/o an adequate subsystem of "surnatural" numbers, which is a *sine qua non* for any full-scale treatment of infinitesimals.

C. Nonstandard extensions of R commonly denoted by R* do not suffer from this, of course, and moreover, along with a related system of hyperintegers N* they are equipped with the asterisk of the whole $\omega$-high superstructure over R. Yet for a long time they used to have an own foundational issue: unique, concretely defined examples of R* were not known. Such examples of OD (ordinally definable) models R* of any amount of saturation were first defined by Kanovei-Shelah (JSL, 2004) - including

(i) the full set-size saturated R* of class-size,

and in fact

(ii) an ordinal-definable full set-size saturated elementary extension of the whole set universe of ZFC considered in detail in Kanovei-Reeken, Nonstandard analysis axiomatically, Springer 2004.

D. As any two full set-size saturated rcof (both of class size, of course) are isomorphic under any suitable class theory with GC (global choice) by means af a standard application of the b&f method, No and the ground rcof structure of R* as in (i) above are isomorphic (first observed by Ehrlich). This means that, postfactum, the surreals No do indeed contain (under GC) a suitable system of surintegers and do allow the whole system of real functions and much more - simply inherited from R* as in (i) by means of the mentioned isomorphism.

This leads to the following problems of foundational importance, yet to be solved.

Problem 1. Note that both surreals and R* of type (i) are OD, well-defined ZFC classes, but the isomorphism between them is **not** such - it is an application of GC. So we ask: does there exist a ZFC-well-defined isomorphism between them?

Problem 2. Sharper, does there exist a ZFC-well-defined adequate system of surintegers, satiafying PA at least and preferably making (No,surintegers) to be an elementary extension of (R,N) ?

Problem 2 can be answered in the affirmative both by affirmatively answering Problem 1, and by means of an own surrealistic construction.

9more comments