Question about Surreal Numbers and Omnific Integers

[lugita15]

Omnific integers are the counterpart in the Surreal numbers of integers. The surreal numbers are usually defined using set theory, and then the omnific integers are defined as a particular subset of them. My question is, does it have to be this way? Is it possible to give a first-order axiomatization of the Omnific integers and their arithmetic, without having to define the surreal numbers themselves? I know they form a proper class, so there is a risk that they may be "too big" to describe. But Tarksi gave a first-order axiomatization for the ordinal numbers, which also form a proper class, so at least we have some hope.

Any help would be greatly appreciated.

Thank You in Advance.

 

[Valkarie]

Unfortunately, it is not possible to give a first-order axiomatization of the Omnific integers without defining the surreal numbers. The surreal numbers are an integral part of the definition of the Omnific integers and cannot be excluded from the axiomatization. However, there have been some attempts to axiomatize the arithmetic of the Omnific integers, such as in the work of Lev Beklemishev and his colleagues.