Some Remarks on Computable (Non-Archimedean) Ordered Fields
E. W. Madison · Journal of the London Mathematical Society · 1971
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 & c, the field of recursive real numbers. (In this paper we choose to use Si c for! % c, thus denoting real numbers whose rational cuts are recursive.) (1) Si c turns out to be the smallest extension field of Si which contains all computable (Archimedean) ordered extensions of Si. Equivalently, (2) SL C is the smallest extension field of Si which contains each computable ordered