Some Remarks on Computable (Non‐Archimedean) Ordered Fields

Let 2F and Jf be ordered fields such that & is a sub-field of Jf. JT is said to be Archimedean over ^ if, for any aeK, there exists a (leF such that a < /?. When ^ is the field of rational numbers SI and Jf is Archimedean over Si then one simply says that Jf is an Archimedean ordered field. $F is said to be dense in X if, for any a, ft e K such that a < /?, there is a y e F such that a < y < /?. Clearly, if OF is dense in Jf then X is Archimedean over 3F. For ^ = .2 the converse is also true; however, in general the converse is false. Indeed, one can give an example of an ordered field #" such that #" is not dense in its real-closure, say # . On the other hand, for any ordered field 2F, its real-closure #" is Archimedean over &'. See [7]. In [4], we showed that if Jf is a computable (Archimedean) ordered extension of SI then Jf is isomorphic to a subfield of &, the field of recursive real numbers. (In this paper we choose to use Si for !%, thus denoting real numbers whose rational cuts are recursive.) (1) Si turns out to be the smallest extension field of Si which contains all computable (Archimedean) ordered extensions of Si. Equivalently, (2) SL is the smallest extension field of Si which contains each computable ordered extension & of SI such that Si is dense in J". In [1], Dubrovski discusses an analogue of (2) for computable p-adic valued extensions of SI. In this paper we seek an analogue for (2) where Si is replaced by a computable ordered field. This is to say we raise the following question: Given a computable ordered field J", is there a smallest extension & of 9> such that & contains each computable ordered extension $F y of 8F such that $F is dense in 8* y1 Of course, if !F is any Archimedean ordered field then we can clearly choose & = Si. Thus our concern is only for an analogue of (2) where & is computable (non-Archimedean) ordered. At this point we are unable to prove an analogue of (1) for non-Archimedean ordered fields. Let (^", i/f) be a computable ordered field with \j/ an admissible indexing of SF. This is to say ̂ is a bijection from F to the natural numbers Jf such that if A(x, y) is addition or multiplication or the characteristic function for order then 1: Jf x JT -*Jf defined by X(i,j) = ^U(^~(0>^~0'))) * recursive. If ^ and 4> are two admissible indices of 2F then we say that i]/ is equivalent to (J)(\{/ = 4>) if ij/ofi' is a recursive permutation of Jf. If all indices of a computable ordered field $F are equivalent then we say that 3F has precisely one indexing. Clearly, the field Si has precisely one indexing. Frolich and Shepherdson show that the field St{aiXi ...,an) (for any oc's algebraic or transcendental over Si) has precisely one indexing. Also they show that the field Si{a.l,..., an,...) has more than one indexing, where {al5..., an...} is an infinite set of independent transcendentals. Let {<*!, ...,an,...} be an r.e. set of recursive reals which is an infinite set of independent transcendentals. Clearly, the ordered field Si(au ...,an) is computable