Can we axiomatize Omnific Integers without the Surreal Number system? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T12:10:59Z http://mathoverflow.net/feeds/question/72691 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72691/can-we-axiomatize-omnific-integers-without-the-surreal-number-system Can we axiomatize Omnific Integers without the Surreal Number system? Keshav Srinivasan 2011-08-11T15:23:06Z 2011-10-10T00:10:29Z <p>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.</p> <p>The reason I'm interested is because of <a href="http://mathoverflow.net/questions/54526/fulfilling-pythagoras-dream-using-nonstandard-models-of-arithmetic-and-or-surrea" rel="nofollow">this question</a> 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. </p> <p>Any help would be greatly appreciated.</p> <p>Thank You in Advance.</p> <p>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.</p> <p>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)?</p> http://mathoverflow.net/questions/72691/can-we-axiomatize-omnific-integers-without-the-surreal-number-system/72942#72942 Answer by Emil Jeřábek for Can we axiomatize Omnific Integers without the Surreal Number system? Emil Jeřábek 2011-08-15T17:30:50Z 2011-08-15T17:30:50Z <p>Here’s a couple of observations from my comment above.</p> <p>First, the theory of omnific integers is a complete extension of Robinson’s arithmetic, hence it is not recursively axiomatizable. This makes it rather unlikely that we can describe its full axiomatization in any reasonable way.</p> <p>Surreal numbers <strong>No</strong> form a real-closed field, and omnific integers <strong>Oz</strong> are its subring, hence they make an ordered ring. In fact, it is known that <strong>Oz</strong> is an integer part of <strong>No</strong> (i.e., for any surreal number $r$ there exists a unique omnific integer $n$ such that $n\le r&lt; n+1$), which—by a well-known result of Shepherdson—means that <strong>Oz</strong> is a model of IOpen (the theory of discretely ordered rings + induction for open formulas in the language of ordered rings). Moreover, the fraction field of <strong>Oz</strong> (namely, <strong>No</strong>) is real-closed; this can be expressed by a first-order axiom schema (let’s call it A), with one axiom for each degree. (This set of axioms can be simplified: in the presence of A, IOpen is equivalent to the theory of discretely ordered rings + division with remainder.)</p> <p>On the other hand, <strong>Oz</strong> does not satisfy induction for larger classes such as $E_1$ (bounded existential formulas), nor does it satisfy algebraic axioms such as normality or gcd. The reason is that such axioms contradict A (or even its corollary that $\sqrt2$ is in the fraction field of <strong>Oz</strong>).</p>