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Abstract. We show that there are 0-definably complete ordered fields which are not real closed. Therefore, the theory of definably with parameters complete ordered fields does not follow from the theory of 0-definably complete ordered fields. The mentioned completeness notions for ordered fields are the definable versions of completeness in the sense of Dedekind cuts. In earlier joint work, we had shown that it would become successively weakened if we just required nonexistence of definable regular gaps and then disallowing parameters. The result in this note shows reducing in the opposite order, at least one side is sharp.
Keywords and phrases: ordered field; 0-definably complete; real closed field.
1. Introduction
Real closed fields are precisely the definably with parameters Dedekind complete ordered fields. We show that a non-real-closed field may still be 0-definably complete. This answers a natural question, as in [3, Problem 17(i)], we raised earlier with J. S. Eivazloo. In the main step, for a certain ordered field of generalized power series, it is shown that the set of infinitesimals is not 0-definable (the latter is due to Lou van den Dries).
In [5, 4] we had already shown the non-implication for the regular cut variant. Recall that for an ordered field K, a gap C C K (namely, a cut which has no least upper bound in K) is regular if for each e K>0 we have that C + e ^ C (i.e., it is of zero distance to its complement). The ordered field K is called Scott complete if it does not have any proper extensions to an ordered field in which it is dense, equivalently it does not have any regular gaps.
Among the notions of definably (with or without parameters) Scott complete ordered fields we considered in [5] were DpSrcCOF in the parameter case and D$SrcCOF in the parameter-free case. Their models are ordered fields with no definable regular gaps with or without parameters respectively. We showed there that if an ordered field K is a proper dense sub-field of its real closure, then KR DpSrcCOF (see [5, Lemma 3.1]).
As we mentioned already, definably complete (with parameters) ordered fields are exactly the real closed fields, in notation: RCF = DpDCOF. A weaker form of half of...