Omnific integers are the counterpart in the Surreal numbers of the integers. The surreal numbers are usually defined using set theory, and then the omnific integers are defined as a particular subset (or rather subclass) 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.

The reason I'm interested is because of this question I asked a while back, about finding a nonstandard model of (Robinson) arithmetic whose field of fractions forms a real closed field. The Omnific integers form such a nonstandard model, so I want to find out whether we can axiomatize them.

Any help would be greatly appreciated.

Thank You in Advance.

EDIT: To be clear, I don't want an axiomatization of the Omnific Integers that's based on something else, like the real numbers, the surreal numbers, or set theory. I want a theory along the lines of Peano Arithmetic.

EDIT 2: As Emil said, it seems that a recursive axiomatization of the Omnific integers is impossible. So might we define them in some other way, without reference to the surreal numbers (or the real numbers)?

first-orderaxioms for the omnific integers, or are you just looking for a directaxiomatic definitionof the omnific integers that does not refer to surreals, etc.? In case the distinction I'm making is not clear, first-order Peano Arithmetic does notdefinethe natural numbers (there exist nonstandard models). $\endgroup$ – Timothy Chow Aug 15 '11 at 17:57